tag:blogger.com,1999:blog-5741857851181907392024-03-06T00:48:26.888-08:00Trigonometric formulasThanh Ahttp://www.blogger.com/profile/16564310847356090654noreply@blogger.comBlogger23125tag:blogger.com,1999:blog-574185785118190739.post-60400789025380713872023-04-25T21:24:00.002-07:002023-04-25T21:24:54.473-07:00Make money online with LoadTeam <h2 class="h4 pi-weight-700 pi-uppercase pi-letter-spacing pi-has-bg pi-margin-bottom-20" style="background-color: white; box-sizing: border-box; color: #21252b; font-family: "Open Sans", Arial, sans-serif; font-size: 20px; letter-spacing: 1px; line-height: 1.3em; margin-bottom: 20px !important; margin-left: 0px; margin-right: 0px; margin-top: 0px; margin: 0px 0px 20px; overflow: hidden; padding: 0px; text-transform: uppercase;">WHAT IS LOADTEAM? </h2><p style="background-color: white; box-sizing: border-box; color: #666e70; font-family: "Open Sans", Arial, sans-serif; font-size: 14px; margin-bottom: 20px; margin-top: 0px; padding-bottom: 0px;"><a href="Make money online with LoadTeam " target="_blank">LoadTeam</a> is a Windows app that helps you make money by harvesting your computer’s idle power.<br style="box-sizing: border-box;" />The app runs quietly on the background, processing jobs one by one. When a job is completed, your computer sends the result to LoadTeam and you get paid for the job. You can transfer your money from LoadTeam to your PayPal account as soon as your LoadTeam balance reaches $1.00.<br style="box-sizing: border-box;" />You can run LoadTeam on as many computers as you’d like and the longer and harder your computers work at completing jobs, the more you get paid.<br style="box-sizing: border-box;" />LoadTeam is trusted by tens of thousands of users from all over the globe and our community keeps growing.<br style="box-sizing: border-box;" /></p><div class="separator" style="clear: both; text-align: center;"><a href="Make money online with LoadTeam " style="margin-left: 1em; margin-right: 1em;" target="_blank"><img border="0" data-original-height="178" data-original-width="283" height="178" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEitgBJWjZiIeEXEUEBcRoyU3MUFhAEm0_oGUYEZcri0O100P_BOZUnpwTvFXxxzbYYWalzh6va0KrG3u4N9ndfaqCFIsLUwH7RukNDUoOeUW7Czs0U3avSvRFxEVsXlS6X52Kjup5qX5wIo_oWFOggZQeV42rQYBb6yIUh2LiINDAtD0U27I65FtOPCFw/s1600/download.jpg" width="283" /></a></div><br /><p style="background-color: white; box-sizing: border-box; color: #666e70; font-family: "Open Sans", Arial, sans-serif; font-size: 14px; margin-bottom: 20px; margin-top: 0px; padding-bottom: 0px;"><br /></p><h2 class="h4 pi-weight-700 pi-uppercase pi-letter-spacing pi-has-bg pi-margin-bottom-20" style="background-color: white; box-sizing: border-box; color: #21252b; font-family: "Open Sans", Arial, sans-serif; font-size: 20px; letter-spacing: 1px; line-height: 1.3em; margin-bottom: 20px !important; margin-left: 0px; margin-right: 0px; margin-top: 0px; margin: 0px 0px 20px; overflow: hidden; padding: 0px; text-transform: uppercase;">WHY LOADTEAM? </h2><p style="background-color: white; box-sizing: border-box; color: #666e70; font-family: "Open Sans", Arial, sans-serif; font-size: 14px; margin-bottom: 20px; margin-top: 0px; padding-bottom: 0px;">LoadTeam pays you for doing nothing. All you need to do is leave your computer running for as long as possible, check your balance and transfer money to your PayPal account every now and then.</p><h2 class="h4 pi-weight-700 pi-uppercase pi-letter-spacing pi-has-bg pi-margin-bottom-20" style="background-color: white; box-sizing: border-box; color: #21252b; font-family: "Open Sans", Arial, sans-serif; font-size: 20px; letter-spacing: 1px; line-height: 1.3em; margin-bottom: 20px !important; margin-left: 0px; margin-right: 0px; margin-top: 0px; margin: 0px 0px 20px; overflow: hidden; padding: 0px; text-transform: uppercase;">HOW DOES IT WORK? </h2><p style="background-color: white; box-sizing: border-box; color: #666e70; font-family: "Open Sans", Arial, sans-serif; font-size: 14px; margin-bottom: 20px; margin-top: 0px; padding-bottom: 0px;">Carla explains how LoadTeam works and how you can make money selling your computer's idle power.</p><p style="background-color: white; box-sizing: border-box; color: #666e70; font-family: "Open Sans", Arial, sans-serif; font-size: 14px; margin-bottom: 20px; margin-top: 0px; padding-bottom: 0px;"><b>Register here: h<a href="ttps://www.loadteam.com/signup?referral=KP0CEWNRVL">ttps://www.loadteam.com/signup?referral=KP0CEWNRVL</a></b></p><p style="background-color: white; box-sizing: border-box; color: #666e70; font-family: "Open Sans", Arial, sans-serif; font-size: 14px; margin-bottom: 20px; margin-top: 0px; padding-bottom: 0px;"><b><br /></b></p><p style="background-color: white; box-sizing: border-box; color: #666e70; font-family: "Open Sans", Arial, sans-serif; font-size: 14px; margin-bottom: 20px; margin-top: 0px; padding-bottom: 0px;"><b><br />
<iframe allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share" allowfullscreen="" frameborder="0" height="393" src="https://www.youtube.com/embed/p8n32CaQMjk" title="Make money using your computer" width="699"></iframe>
</b></p>Thanh Ahttp://www.blogger.com/profile/16564310847356090654noreply@blogger.com0tag:blogger.com,1999:blog-574185785118190739.post-78821667498385799942016-12-01T22:49:00.005-08:002023-03-05T23:17:22.748-08:001. Angles<div dir="ltr" style="text-align: left;" trbidi="on">
This article uses Greek letters<img alt="" border="0" height="0" id="amznPsBmPixel_8250025" src="https://ir-na.amazon-adsystem.com/e/ir?source=bk&t=learnjavaporgramming-20&bm-id=default&l=ktl&linkId=451051929954d26607643a1d3c0dfc61&_cb=1539528780176" style="border: none; height: 0px; margin: 0px; padding: 0px; width: 0px;" width="0" /> such as alpha (α), beta (β), gamma (γ), and theta (θ) to represent angles. Several different units of angle measure are widely used, including degrees, radians, and gradians (gons):<br />
<div>
<br /></div>
<div>
1 full circle (turn) = 360 degrees = 2π radians = 400 gons.<br />
<br />
The following table shows the conversions and values for some common angles:<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a amzn-ps-bm-asin="B001L46NWI" class="amzn_ps_bm_il" data-amzn-link-id="78c46544e6d7ad16c53545b8e90bc20e" data-amzn-ps-bm-keyword="angels" href="http://www.amazon.com/Gifts-Decor-39694-Desert-Figurine/dp/B001L46NWI/ref=as_li_bk_ia/?tag=learnjavaporgramming-20&linkId=78c46544e6d7ad16c53545b8e90bc20e&linkCode=kia" id="amznPsBmLink_3115984" rel="nofollow" target="_blank"><img border="0" height="640" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg7Z5ZiLQ_jnbc6HT0zkR_XSF336oky2hZNUmmw0LXjyHohE6zcZS0yezsCMPKrai_4Pt5FSw8OBh6bOQ_ZndnSlw3geG95FInVLo7EHZjuiTA2Otn_w9onXem3KTE9ctnmAxuVmFasVV_L/s640/11.png" width="486" /></a><img alt="" border="0" height="0" id="amznPsBmPixel_3115984" src="https://ir-na.amazon-adsystem.com/e/ir?source=bk&t=learnjavaporgramming-20&bm-id=default&l=kia&linkId=78c46544e6d7ad16c53545b8e90bc20e&_cb=1539528798062" style="border: none; height: 0px; margin: 0px; padding: 0px; width: 0px;" width="0" /></div>
<div style="text-align: center;">
<br /></div>
<br />
<iframe sandbox="allow-popups allow-scripts allow-modals allow-forms allow-same-origin" style="width:120px;height:240px;" marginwidth="0" marginheight="0" scrolling="no" frameborder="0" src="//ws-na.amazon-adsystem.com/widgets/q?ServiceVersion=20070822&OneJS=1&Operation=GetAdHtml&MarketPlace=US&source=ss&ref=as_ss_li_til&ad_type=product_link&tracking_id=appaha00-20&language=en_US&marketplace=amazon®ion=US&placement=1941691382&asins=1941691382&linkId=9f8cd6b17f5c204e5cac0fdd2091c797&show_border=true&link_opens_in_new_window=true"></iframe>
<br />
Results for other angles can be found at Trigonometric<img alt="" border="0" height="0" id="amznPsBmPixel_3025979" src="https://ir-na.amazon-adsystem.com/e/ir?source=bk&t=learnjavaporgramming-20&bm-id=default&l=ktl&linkId=a75b16392907fc557e7f78d81cb7c407&_cb=1539528808292" style="border: none; height: 0px; margin: 0px; padding: 0px; width: 0px;" width="0" /> constants expressed in real radicals.<br />
<br />
Unless otherwise specified, all angles in this article are assumed to be in radians, but angles ending in a degree symbol (°) are in degrees. Per Niven's theorem multiples of 30° are the only angles that are a rational multiple of one degree and also have a rational sine or cosine, which may account for their popularity in examples.</div>
</div>
Thanh Ahttp://www.blogger.com/profile/16564310847356090654noreply@blogger.comtag:blogger.com,1999:blog-574185785118190739.post-15205626356646308502016-11-30T22:50:00.001-08:002023-03-05T23:17:44.576-08:002. Trigonometric functions<div dir="ltr" style="text-align: left;" trbidi="on">
The primary trigonometric functions<img alt="" border="0" height="0" id="amznPsBmPixel_8458319" src="https://ir-na.amazon-adsystem.com/e/ir?source=bk&t=learnjavaporgramming-20&bm-id=default&l=ktl&linkId=cf8eee0792172b6a26a6a54244bb5df0&_cb=1539528833496" style="border: none !important; height: 0px !important; margin: 0px !important; padding: 0px !important; width: 0px !important;" width="0" /> are the sine and cosine of an angle. These are sometimes abbreviated sin(θ) and cos(θ), respectively, where θ is the angle, but the parentheses around the angle are often omitted, e.g., sin θ and cos θ.<br />
<br />
The sine of an angle is defined in the context of a right triangle, as the ratio of the length of the side that is opposite to the angle divided by the length of the longest side of the triangle (the hypotenuse).<br />
<br />
The cosine of an angle is also defined in the context of a right triangle, as the ratio of the length of the side that is adjacent to the angle divided by the length of the longest side of the triangle (the hypotenuse).<br />
<br />
The tangent (tan) of an angle is the ratio of the sine to the cosine:<br />
<div>
<br /></div>
<div>
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiM9LNhY8yb5YwvBd56MwC1o_MNcCcNrKlxcsqGvmirCOBoX2-XS2crxTLAZV8e8w69Cl74W4tJCXNApuWSvNML96hxeuFUIyMCSRh2wcrt888Qbvn_DAmQjOwYGByVpI_zeJvLtDY1hXQx/s1600/1.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiM9LNhY8yb5YwvBd56MwC1o_MNcCcNrKlxcsqGvmirCOBoX2-XS2crxTLAZV8e8w69Cl74W4tJCXNApuWSvNML96hxeuFUIyMCSRh2wcrt888Qbvn_DAmQjOwYGByVpI_zeJvLtDY1hXQx/s1600/1.png" /></a></div>
<br />
<br />
Finally, the reciprocal functions secant (sec), cosecant (csc), and cotangent (cot) are the reciprocals of the cosine, sine, and tangent:<br />
<div>
<br /></div>
<div>
</div>
<div>
<div class="separator" style="clear: both; text-align: center;">
<a amzn-ps-bm-asin="0134217438" class="amzn_ps_bm_il" data-amzn-link-id="15129ad5201a142c8a6de9272a6dce78" data-amzn-ps-bm-keyword="trigonometry" href="http://www.amazon.com/Trigonometry-11th-Margaret-L-Lial/dp/0134217438/ref=as_li_bk_ia/?tag=learnjavaporgramming-20&linkId=15129ad5201a142c8a6de9272a6dce78&linkCode=kia" id="amznPsBmLink_2919348" rel="nofollow" target="_blank"><img border="0" height="112" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh_jrri7U8VDP_m6H_1kvJ5J2tRkGby90kInjJOhbB8zMyzKv3Qr6mX-AZsfivDspizvV7R7IpxpOLTJEW87FsA_gN_9UfZDmZ-PhH7qs2RAhKskQz7ZTjZC40PkMpiBVxUKzK7WZ2HgP1W/s640/2.png" width="640" /></a><img alt="" border="0" height="0" id="amznPsBmPixel_2919348" src="https://ir-na.amazon-adsystem.com/e/ir?source=bk&t=learnjavaporgramming-20&bm-id=default&l=kia&linkId=15129ad5201a142c8a6de9272a6dce78&_cb=1539528854318" style="border: none !important; height: 0px !important; margin: 0px !important; padding: 0px !important; width: 0px !important;" width="0" /></div>
<br />
<iframe sandbox="allow-popups allow-scripts allow-modals allow-forms allow-same-origin" style="width:120px;height:240px;" marginwidth="0" marginheight="0" scrolling="no" frameborder="0" src="//ws-na.amazon-adsystem.com/widgets/q?ServiceVersion=20070822&OneJS=1&Operation=GetAdHtml&MarketPlace=US&source=ss&ref=as_ss_li_til&ad_type=product_link&tracking_id=appaha00-20&language=en_US&marketplace=amazon®ion=US&placement=1941691382&asins=1941691382&linkId=9f8cd6b17f5c204e5cac0fdd2091c797&show_border=true&link_opens_in_new_window=true"></iframe>
<br />
These definitions are sometimes referred to as ratio identities</div>
</div>
</div>
Thanh Ahttp://www.blogger.com/profile/16564310847356090654noreply@blogger.comtag:blogger.com,1999:blog-574185785118190739.post-89565811486078569732016-11-29T22:56:00.001-08:002023-03-05T23:18:34.093-08:003. Inverse functions<div dir="ltr" style="text-align: left;" trbidi="on">
<i>Main article: Inverse trigonometric functions</i><br />
<br />
The inverse trigonometric functions are partial inverse functions for the trigonometric functions. For example, the inverse function for the sine, known as the inverse sine (sin^−1) or arcsine (arcsin or asin), satisfies<br />
<div>
<iframe sandbox="allow-popups allow-scripts allow-modals allow-forms allow-same-origin" style="width:120px;height:240px;" marginwidth="0" marginheight="0" scrolling="no" frameborder="0" src="//ws-na.amazon-adsystem.com/widgets/q?ServiceVersion=20070822&OneJS=1&Operation=GetAdHtml&MarketPlace=US&source=ss&ref=as_ss_li_til&ad_type=product_link&tracking_id=appaha00-20&language=en_US&marketplace=amazon®ion=US&placement=1941691382&asins=1941691382&linkId=9f8cd6b17f5c204e5cac0fdd2091c797&show_border=true&link_opens_in_new_window=true"></iframe>
<br /></div>
<div class="separator" style="clear: both; text-align: left;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiXYniDrBSex2x8V4VoED1WT_qZanb2QRtl6Hv2Bz_9Cq2O0t_XIWRtq4W7rLpCwpbOzGwAoLtx-SwmOpqjcJmww-fZdfULUJeW-tEpwp7-efTHM9Bij-gQs8mJ5j4vrDO6Gs2gd4oXpIfe/s1600/3.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="288" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiXYniDrBSex2x8V4VoED1WT_qZanb2QRtl6Hv2Bz_9Cq2O0t_XIWRtq4W7rLpCwpbOzGwAoLtx-SwmOpqjcJmww-fZdfULUJeW-tEpwp7-efTHM9Bij-gQs8mJ5j4vrDO6Gs2gd4oXpIfe/s640/3.png" width="640" /></a></div>
<div>
<br /></div>
Thanh Ahttp://www.blogger.com/profile/16564310847356090654noreply@blogger.comtag:blogger.com,1999:blog-574185785118190739.post-31014758552948060272016-11-28T22:59:00.000-08:002018-10-19T19:23:46.030-07:004. Pythagorean identity<div dir="ltr" style="text-align: left;" trbidi="on">
<div style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; line-height: inherit; margin-bottom: 0.5em; margin-top: 0.5em;">
In trigonometry, the basic relationship between the sine and the cosine is known as the Pythagorean identity:</div>
<dl style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; margin-bottom: 0.5em; margin-top: 0.2em;"><div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgHT7W22siDcJ3sh5ybe_7UePAUXOss3DBNZqE5OLosls7puWKWhKVWDya1INoBbQ0BWhPsRIpGBIY5mBwX2s0AzFa6dGwjtbegFu5T5xhKzFYrTo2uU6NA_EIVTLWCtBecbjn12BlgK5Mp/s1600/4.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgHT7W22siDcJ3sh5ybe_7UePAUXOss3DBNZqE5OLosls7puWKWhKVWDya1INoBbQ0BWhPsRIpGBIY5mBwX2s0AzFa6dGwjtbegFu5T5xhKzFYrTo2uU6NA_EIVTLWCtBecbjn12BlgK5Mp/s1600/4.png" /></a></div>
<dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 20.65px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle \sin ^{2}\theta +\cos ^{2}\theta =1\!}</annotation></semantics></math></span></dd></dl>
<div style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; line-height: inherit; margin-bottom: 0.5em; margin-top: 0.5em;">
where <span class="texhtml" style="font-family: "nimbus roman no9 l" , "times new roman" , "times" , serif; font-size: 20.65px; line-height: 1; white-space: nowrap;">cos<sup style="font-size: 16.52px; line-height: 1;">2</sup> <i>θ</i></span> means <span class="texhtml" style="font-family: "nimbus roman no9 l" , "times new roman" , "times" , serif; font-size: 20.65px; line-height: 1; white-space: nowrap;">(cos(<i>θ</i>))<sup style="font-size: 16.52px; line-height: 1;">2</sup></span> and <span class="texhtml" style="font-family: "nimbus roman no9 l" , "times new roman" , "times" , serif; font-size: 20.65px; line-height: 1; white-space: nowrap;">sin<sup style="font-size: 16.52px; line-height: 1;">2</sup> <i>θ</i></span> means <span class="texhtml" style="font-family: "nimbus roman no9 l" , "times new roman" , "times" , serif; font-size: 20.65px; line-height: 1; white-space: nowrap;">(sin(<i>θ</i>))<sup style="font-size: 16.52px; line-height: 1;">2</sup></span>.</div>
<div style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; line-height: inherit; margin-bottom: 0.5em; margin-top: 0.5em;">
This can be viewed as a version of the Pythagorean theorem, and follows from the equation <span class="texhtml" style="font-family: "nimbus roman no9 l" , "times new roman" , "times" , serif; font-size: 20.65px; line-height: 1; white-space: nowrap;"><i>x</i><sup style="font-size: 16.52px; line-height: 1;">2</sup> + <i>y</i><sup style="font-size: 16.52px; line-height: 1;">2</sup> = 1</span> for the unit circle. This equation can be solved for either the sine or the cosine:</div>
<dl style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; margin-bottom: 0.5em; margin-top: 0.2em;"><div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgqT3Xg9IyH8adbC3jqvium6hzRoHZubSU3xomKotasAtB6FrsNqFTS05pmYU7cCTJdNy32wB6OCzenCKn1XXiPrSyoIvuNneDvcDuSuiAwixZaYy13q4CdyDXmL8Qk3T7wFZ-L90jqxmbQ/s1600/5.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgqT3Xg9IyH8adbC3jqvium6hzRoHZubSU3xomKotasAtB6FrsNqFTS05pmYU7cCTJdNy32wB6OCzenCKn1XXiPrSyoIvuNneDvcDuSuiAwixZaYy13q4CdyDXmL8Qk3T7wFZ-L90jqxmbQ/s1600/5.png" /></a></div>
<dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 20.65px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\sin \theta &=\pm {\sqrt {1-\cos ^{2}\theta }},\\\cos \theta &=\pm {\sqrt {1-\sin ^{2}\theta }}.\end{aligned}}}</annotation></semantics></math></span></dd></dl>
<div style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; line-height: inherit; margin-bottom: 0.5em; margin-top: 0.5em;">
where the sign depends on the quadrant of <span class="texhtml mvar" style="font-family: "nimbus roman no9 l" , "times new roman" , "times" , serif; font-size: 20.65px; font-style: italic; line-height: 1; white-space: nowrap;">θ</span>.</div>
<h3 style="background: none rgb(255, 255, 255); border-bottom: none; font-family: sans-serif; font-size: 1.2em; line-height: 1.6; margin: 0.3em 0px 0px; overflow: hidden; padding-bottom: 0px; padding-top: 0.5em;">
<span class="mw-headline" id="Related_identities"><span id="Trigonometric_conversions"></span>Related identities</span></h3>
<div style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; line-height: inherit; margin-bottom: 0.5em; margin-top: 0.5em;">
Dividing the Pythagorean identity by either <span class="texhtml" style="font-family: "nimbus roman no9 l" , "times new roman" , "times" , serif; font-size: 20.65px; line-height: 1; white-space: nowrap;">cos<sup style="font-size: 16.52px; line-height: 1;">2</sup> <i>θ</i></span> or <span class="texhtml" style="font-family: "nimbus roman no9 l" , "times new roman" , "times" , serif; font-size: 20.65px; line-height: 1; white-space: nowrap;">sin<sup style="font-size: 16.52px; line-height: 1;">2</sup> <i>θ</i></span> yields two other identities:</div>
<dl style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjmtixORIPdL21q9CoqGz-N4kfDDtweK5dg03fBh24RMJsMxMu85ccWdxKd0lrPq-V0FksEDyIN6w2nVT4n7e3R1DUcWkrWOw-TzPpXx7R1as9BNpqVK-b3K6d9lkV0Cyx-7SAkltgv9X42/s1600/6.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="51" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjmtixORIPdL21q9CoqGz-N4kfDDtweK5dg03fBh24RMJsMxMu85ccWdxKd0lrPq-V0FksEDyIN6w2nVT4n7e3R1DUcWkrWOw-TzPpXx7R1as9BNpqVK-b3K6d9lkV0Cyx-7SAkltgv9X42/s640/6.png" width="640" /></a></div>
<span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 20.65px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle 1+\tan ^{2}\theta =\sec ^{2}\theta \quad {\text{and}}\quad 1+\cot ^{2}\theta =\csc ^{2}\theta .\!}</annotation></semantics></math></span></dd></dl>
<div style="line-height: inherit; margin-bottom: 0.5em; margin-top: 0.5em;">
<div style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px;">
Using these identities together with the ratio identities, it is possible to express any trigonometric function in terms of any other (up to a plus or minus sign):</div>
<div style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px;">
<br /></div>
<div style="text-align: center;">
<b>Each trigonometric function in terms of the other five</b></div>
</div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjOmJRzj3jeRwQkc-lQvd40mSz0jOgChTdA6H8k_deR6FhFBqXdO7eV-kyU_4YUKbRqc6PFSLtYDMTpFgxc-mEnQ6_M0w15Ot1zZwSvir5RO9ic2g8RbSYvWcDD96_U1DWdMxzyC3VgFqX7/s1600/7.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="292" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjOmJRzj3jeRwQkc-lQvd40mSz0jOgChTdA6H8k_deR6FhFBqXdO7eV-kyU_4YUKbRqc6PFSLtYDMTpFgxc-mEnQ6_M0w15Ot1zZwSvir5RO9ic2g8RbSYvWcDD96_U1DWdMxzyC3VgFqX7/s640/7.png" width="640" /></a></div>
<center style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px;">
<br />
</center>
</div>
Thanh Ahttp://www.blogger.com/profile/16564310847356090654noreply@blogger.comtag:blogger.com,1999:blog-574185785118190739.post-45313122953537712662016-11-27T23:07:00.000-08:002018-10-19T19:23:55.079-07:005. Historical shorthands<div dir="ltr" style="text-align: left;" trbidi="on">
The versine, coversine, haversine, and exsecant were used in navigation. For example, the haversine formula was used to calculate the distance between two points on a sphere. They are rarely used today.<br />
<div>
<br /></div>
<div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjSybMF4JjD5_0bLNvwoMehqFioZVbsCJfSe-K3R7P8V81aSsyVbogT5sF20GT9eTQxuMAEqagHBaW7kHJuqI-w2yhdOivoCHew4ypAlA66qEWOr-n9_2hxirahoX5O_VJ5DUvmbVkOqgj_/s1600/9.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="393" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjSybMF4JjD5_0bLNvwoMehqFioZVbsCJfSe-K3R7P8V81aSsyVbogT5sF20GT9eTQxuMAEqagHBaW7kHJuqI-w2yhdOivoCHew4ypAlA66qEWOr-n9_2hxirahoX5O_VJ5DUvmbVkOqgj_/s400/9.png" width="400" /></a></div>
<div style="text-align: center;">
<br /></div>
<div>
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj6USqgd64jz9992UHLL9t8rcQ46eHLjFXgEmjMSzIq2YbirdueI2bzenzLJixfx1qE1xpBGlssY5Wz5E51BkG8e2cWPkqSkrbNguYnwCGzHykswbqJXLNb9IcZN6A1FnppOE6TLN0wY2Hz/s1600/8.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="640" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj6USqgd64jz9992UHLL9t8rcQ46eHLjFXgEmjMSzIq2YbirdueI2bzenzLJixfx1qE1xpBGlssY5Wz5E51BkG8e2cWPkqSkrbNguYnwCGzHykswbqJXLNb9IcZN6A1FnppOE6TLN0wY2Hz/s640/8.png" width="366" /></a></div>
<div>
<br /></div>
</div>
</div>
</div>
Thanh Ahttp://www.blogger.com/profile/16564310847356090654noreply@blogger.comtag:blogger.com,1999:blog-574185785118190739.post-69526484259014893202016-11-26T01:40:00.000-08:002018-10-19T19:24:08.797-07:006. Symmetry, shifts, and periodicity<div dir="ltr" style="text-align: left;" trbidi="on">
<div style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; line-height: inherit; margin-bottom: 0.5em; margin-top: 0.5em;">
By examining the unit circle, the following properties of the trigonometric functions can be established.</div>
<h3 style="background: none rgb(255, 255, 255); border-bottom: none; font-family: sans-serif; font-size: 1.2em; line-height: 1.6; margin: 0.3em 0px 0px; overflow: hidden; padding-bottom: 0px; padding-top: 0.5em;">
<span class="mw-headline" id="Symmetry">Symmetry<img alt="" border="0" height="0" id="amznPsBmPixel_3313246" src="https://ir-na.amazon-adsystem.com/e/ir?source=bk&t=learnjavaporgramming-20&bm-id=default&l=ktl&linkId=ec3df8d84644fbe41c8c3a2b927e9557&_cb=1539529039672" style="border: none !important; height: 0px !important; margin: 0px !important; padding: 0px !important; width: 0px !important;" width="0" /></span></h3>
<div class="separator" style="clear: both; text-align: center;">
<a amzn-ps-bm-asin="0823427625" class="amzn_ps_bm_il" data-amzn-link-id="6706386c3574f1984586cb33c969c992" data-amzn-ps-bm-keyword="symmetry" href="http://www.amazon.com/Seeing-Symmetry-Loreen-Leedy/dp/0823427625/ref=as_li_bk_ia/?tag=learnjavaporgramming-20&linkId=6706386c3574f1984586cb33c969c992&linkCode=kia" id="amznPsBmLink_4813148" rel="nofollow" target="_blank"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhyC7z0T8s4Tf33SzFhsw2wtbr9SAZqKPiMNxHqQzTAsbYW0kqgmaMnKVop9RddV34GfhLWtJ2UYjH7o0cybL0S6mMOho_to9ZCzsUCyziEqMPDT0yZvuqbTXm1LVdQ86dRDSyfQRRuqFKT/s1600/13.png" /></a><img alt="" border="0" height="0" id="amznPsBmPixel_4813148" src="https://ir-na.amazon-adsystem.com/e/ir?source=bk&t=learnjavaporgramming-20&bm-id=default&l=kia&linkId=6706386c3574f1984586cb33c969c992&_cb=1539529026099" style="border: none !important; height: 0px !important; margin: 0px !important; padding: 0px !important; width: 0px !important;" width="0" /></div>
<div class="floatright" style="background-color: white; border: 0px; clear: right; color: #252525; float: right; font-family: sans-serif; font-size: 17.5px; margin: 0px 0px 0.5em 0.5em; position: relative;">
</div>
<div style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; line-height: inherit; margin-bottom: 0.5em; margin-top: 0.5em;">
When the trigonometric functions are reflected from certain angles, the result is often one of the other trigonometric functions. This leads to the following identities:</div>
<div style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; line-height: inherit; margin-bottom: 0.5em; margin-top: 0.5em;">
<br /></div>
<div class="separator" style="clear: both; text-align: center;">
<a amzn-ps-bm-asin="1568812205" class="amzn_ps_bm_il" data-amzn-link-id="25e6736e43a2c892e83537d137356211" data-amzn-ps-bm-keyword="symmetry" href="http://www.amazon.com/Symmetries-Things-John-H-Conway/dp/1568812205/ref=as_li_bk_ia/?tag=learnjavaporgramming-20&linkId=25e6736e43a2c892e83537d137356211&linkCode=kia" id="amznPsBmLink_4983392" rel="nofollow" target="_blank"><img border="0" height="300" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhD0Tb8laf60FJgFb8ZIylDgnxw_y4qnOeqeLt2oh7a9A1PBqpjwLsTL_07HkxlZu_I8QSqjB-cl36Ot9LGoAWLQr9qQXMYBA43dlNaXSJL5yH4EjcXUkHMY_4-lg4wGR11_3is9SnVhFw8/s640/12.png" width="640" /></a><img alt="" border="0" height="0" id="amznPsBmPixel_4983392" src="https://ir-na.amazon-adsystem.com/e/ir?source=bk&t=learnjavaporgramming-20&bm-id=default&l=kia&linkId=25e6736e43a2c892e83537d137356211&_cb=1539529058834" style="border: none !important; height: 0px !important; margin: 0px !important; padding: 0px !important; width: 0px !important;" width="0" /></div>
<div style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; line-height: inherit; margin-bottom: 0.5em; margin-top: 0.5em; text-align: center;">
<br /></div>
<div style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; line-height: inherit; margin-bottom: 0.5em; margin-top: 0.5em;">
<br /></div>
<div style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; line-height: inherit; margin-bottom: 0.5em; margin-top: 0.5em;">
Note that the sign in front of the trig function does not necessarily indicate the sign of the value. For example, <span class="texhtml" style="font-family: "nimbus roman no9 l" , "times new roman" , "times" , serif; font-size: 20.65px; line-height: 1; white-space: nowrap;">+cos <i>θ</i></span> does not always mean that <span class="texhtml" style="font-family: "nimbus roman no9 l" , "times new roman" , "times" , serif; font-size: 20.65px; line-height: 1; white-space: nowrap;">cos <i>θ</i></span> is positive. In particular, if <span class="texhtml mvar" style="font-family: "nimbus roman no9 l" , "times new roman" , "times" , serif; font-size: 20.65px; font-style: italic; line-height: 1; white-space: nowrap;">θ</span> = <span class="texhtml" style="font-family: "nimbus roman no9 l" , "times new roman" , "times" , serif; font-size: 20.65px; line-height: 1; white-space: nowrap;">π</span>, then <span class="texhtml" style="font-family: "nimbus roman no9 l" , "times new roman" , "times" , serif; font-size: 20.65px; line-height: 1; white-space: nowrap;">+cos <i>θ</i></span> = −1.</div>
<h3 style="background: none rgb(255, 255, 255); border-bottom: none; font-family: sans-serif; font-size: 1.2em; line-height: 1.6; margin: 0.3em 0px 0px; overflow: hidden; padding-bottom: 0px; padding-top: 0.5em;">
<span class="mw-headline" id="Shifts_and_periodicity"><a amzn-ps-bm-asin="027308478X" class="amzn_ps_bm_tl" data-amzn-link-id="3e881fdc7e3eafdbe8fe69cad22c9183" data-amzn-ps-bm-keyword="Shifts and periodicity" href="http://www.amazon.com/Periodicity-Right-invertible-Operators-Research-Mathematics/dp/027308478X/ref=as_li_bk_tl/?tag=learnjavaporgramming-20&linkId=3e881fdc7e3eafdbe8fe69cad22c9183&linkCode=ktl" id="amznPsBmLink_1409228" rel="nofollow" target="_blank">Shifts and periodicity</a><img alt="" border="0" height="0" id="amznPsBmPixel_1409228" src="https://ir-na.amazon-adsystem.com/e/ir?source=bk&t=learnjavaporgramming-20&bm-id=default&l=ktl&linkId=3e881fdc7e3eafdbe8fe69cad22c9183&_cb=1539529065703" style="border: none !important; height: 0px !important; margin: 0px !important; padding: 0px !important; width: 0px !important;" width="0" /></span></h3>
<div style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; line-height: inherit; margin-bottom: 0.5em; margin-top: 0.5em;">
By shifting the function round by certain angles, it is often possible to find different trigonometric functions that express particular results more simply. Some examples of this are shown by shifting functions round by <span class="sfrac nowrap" style="display: inline-block; font-size: 14.875px; text-align: center; vertical-align: -0.5em; white-space: nowrap;"><span style="display: block; line-height: 1em; margin: 0px 0.1em;"><span class="texhtml" style="font-family: "nimbus roman no9 l" , "times new roman" , "times" , serif; font-size: 17.5525px; line-height: 1;">π</span></span><span class="visualhide" style="height: 1px; left: -10000px; overflow: hidden; position: absolute; top: auto; width: 1px;">/</span><span style="border-top: 1px solid; display: block; line-height: 1em; margin: 0px 0.1em;">2</span></span>, <span class="texhtml" style="font-family: "nimbus roman no9 l" , "times new roman" , "times" , serif; font-size: 20.65px; line-height: 1; white-space: nowrap;">π</span> and 2<span class="texhtml" style="font-family: "nimbus roman no9 l" , "times new roman" , "times" , serif; font-size: 20.65px; line-height: 1; white-space: nowrap;">π</span> radians. Because the periods of these functions are either <span class="texhtml" style="font-family: "nimbus roman no9 l" , "times new roman" , "times" , serif; font-size: 20.65px; line-height: 1; white-space: nowrap;">π</span> or 2<span class="texhtml" style="font-family: "nimbus roman no9 l" , "times new roman" , "times" , serif; font-size: 20.65px; line-height: 1; white-space: nowrap;">π</span>, there are cases where the new function is exactly the same as the old function without the shift.</div>
<div style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; line-height: inherit; margin-bottom: 0.5em; margin-top: 0.5em;">
<br /></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgLLIbxx1bED2VunnA1hsvcdzcs5Zpv6bmGClAzgbqZ4pUfdTxjabEW1R9ReZ43vVYQL1KzB5mgWyMb08xwCWUulKN_JTWNxyfsN16UZCDYkkyFpLcwTFWg1rtElhMCUx-kErV_lyRUcoj2/s1600/14.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="236" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgLLIbxx1bED2VunnA1hsvcdzcs5Zpv6bmGClAzgbqZ4pUfdTxjabEW1R9ReZ43vVYQL1KzB5mgWyMb08xwCWUulKN_JTWNxyfsN16UZCDYkkyFpLcwTFWg1rtElhMCUx-kErV_lyRUcoj2/s640/14.png" width="640" /></a></div>
<div style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; line-height: inherit; margin-bottom: 0.5em; margin-top: 0.5em;">
<br /></div>
</div>
Thanh Ahttp://www.blogger.com/profile/16564310847356090654noreply@blogger.comtag:blogger.com,1999:blog-574185785118190739.post-9398462135741086152016-11-25T01:49:00.000-08:002018-10-19T19:24:17.467-07:007. Angle sum and difference identities<div dir="ltr" style="text-align: left;" trbidi="on">
<div style="text-align: left;">
<span style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px;">These are also known as the </span><i style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px;">addition and subtraction theorems</i><span style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px;"> or </span><i style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px;">formulae</i><span style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px;">. The identities can be derived by combining right triangles such as in the adjacent diagram, or by considering the invariance of the length of a chord on a unit circle given a particular central angle. Furthermore, it is even possible to derive the identities using </span>Euler's identity<span style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px;"> although this would be a more obscure approach given that complex numbers are used.</span><br />
<div class="separator" style="clear: both; text-align: center;">
<span style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiDB0LO0-RF37AcOcDR9tEKQjHSBQp3dbnV3zI4SggXrqIIfad11_yoyhVjczdWDkSh46-XoSWvGiIDE-a1wouQTaAKT_FdZHWbvQEZLydL4sWrLNeFIuW3P3ZtxmtxcYk-3Psj7EiVN7_u/s1600/a1.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="400" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiDB0LO0-RF37AcOcDR9tEKQjHSBQp3dbnV3zI4SggXrqIIfad11_yoyhVjczdWDkSh46-XoSWvGiIDE-a1wouQTaAKT_FdZHWbvQEZLydL4sWrLNeFIuW3P3ZtxmtxcYk-3Psj7EiVN7_u/s400/a1.png" width="256" /></a></span></div>
<span style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px;">
<br />
</span><br />
<div class="separator" style="clear: both; text-align: center;">
<span style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjuj-tmesnGeS134NPWstFcVUH61WVSsKcTVTdz7g2P3USQBA77uSkbhiR4yq6VSi2ejSkO9mA72h5hnwx5K_m7Itlc-QjA8UyGdlB5K3-A9WOq26pVhQXGT5AmsxszYAz9U_71jJWTPy0p/s1600/a2.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="400" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjuj-tmesnGeS134NPWstFcVUH61WVSsKcTVTdz7g2P3USQBA77uSkbhiR4yq6VSi2ejSkO9mA72h5hnwx5K_m7Itlc-QjA8UyGdlB5K3-A9WOq26pVhQXGT5AmsxszYAz9U_71jJWTPy0p/s400/a2.png" width="257" /></a></span></div>
<span style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px;">
</span></div>
<div style="text-align: center;">
<div style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; line-height: inherit; margin-bottom: 0.5em; margin-top: 0.5em; text-align: start;">
For the angle addition diagram for the sine and cosine, the line in bold with the 1 on it is of length 1. It is the hypotenuse of a right angle triangle with angle <span class="texhtml mvar" style="font-family: "nimbus roman no9 l" , "times new roman" , "times" , serif; font-size: 20.65px; font-style: italic; line-height: 1; white-space: nowrap;">β</span> which gives the <span class="texhtml" style="font-family: "nimbus roman no9 l" , "times new roman" , "times" , serif; font-size: 20.65px; line-height: 1; white-space: nowrap;">sin <i>β</i></span> and <span class="texhtml" style="font-family: "nimbus roman no9 l" , "times new roman" , "times" , serif; font-size: 20.65px; line-height: 1; white-space: nowrap;">cos <i>β</i></span>. The <span class="texhtml" style="font-family: "nimbus roman no9 l" , "times new roman" , "times" , serif; font-size: 20.65px; line-height: 1; white-space: nowrap;">cos <i>β</i></span> line is the hypotenuse of a right angle triangle with angle <span class="texhtml mvar" style="font-family: "nimbus roman no9 l" , "times new roman" , "times" , serif; font-size: 20.65px; font-style: italic; line-height: 1; white-space: nowrap;">α</span> so it has sides <span class="texhtml" style="font-family: "nimbus roman no9 l" , "times new roman" , "times" , serif; font-size: 20.65px; line-height: 1; white-space: nowrap;">sin <i>α</i></span> and <span class="texhtml" style="font-family: "nimbus roman no9 l" , "times new roman" , "times" , serif; font-size: 20.65px; line-height: 1; white-space: nowrap;">cos <i>α</i></span> both multiplied by <span class="texhtml" style="font-family: "nimbus roman no9 l" , "times new roman" , "times" , serif; font-size: 20.65px; line-height: 1; white-space: nowrap;">cos <i>β</i></span>. This is the same for the <span class="texhtml" style="font-family: "nimbus roman no9 l" , "times new roman" , "times" , serif; font-size: 20.65px; line-height: 1; white-space: nowrap;">sin <i>β</i></span> line. The original line is also the hypotenuse of a right angle triangle with angle <span class="texhtml" style="font-family: "nimbus roman no9 l" , "times new roman" , "times" , serif; font-size: 20.65px; line-height: 1; white-space: nowrap;"><i>α</i> + <i>β</i></span>, the opposite side is the <span class="texhtml" style="font-family: "nimbus roman no9 l" , "times new roman" , "times" , serif; font-size: 20.65px; line-height: 1; white-space: nowrap;">sin(<i>α</i> + <i>β</i>)</span> line up from the origin and the adjacent side is the <span class="texhtml" style="font-family: "nimbus roman no9 l" , "times new roman" , "times" , serif; font-size: 20.65px; line-height: 1; white-space: nowrap;">cos(<i>α</i> + <i>β</i>)</span> segment going horizontally from the top left.</div>
<div style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; line-height: inherit; margin-bottom: 0.5em; margin-top: 0.5em; text-align: start;">
Overall the diagram can be used to show the sine and cosine of sum identities</div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjW-REoEV8J1DrBocwxPhwSJowwKVSPxvMiqF172xYZoOtx69Eh3-VZC_54H_12YIwZUwqrLX9I9ZmpTXKl0n_uWkYuaDXv1qm0Ms-ukEoJXABzKQDjg86AO6dBAlHuGeOOnzrXFZFFyfph/s1600/a3.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="66" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjW-REoEV8J1DrBocwxPhwSJowwKVSPxvMiqF172xYZoOtx69Eh3-VZC_54H_12YIwZUwqrLX9I9ZmpTXKl0n_uWkYuaDXv1qm0Ms-ukEoJXABzKQDjg86AO6dBAlHuGeOOnzrXFZFFyfph/s400/a3.png" width="400" /></a></div>
<div style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; line-height: inherit; margin-bottom: 0.5em; margin-top: 0.5em; text-align: start;">
because the opposite sides of the rectangle are equal.</div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhmV34IQ5SQ1Xw_hc__NcY0fkDkgArwe1idJJlXOtlxY7DapmrvmA8RzreaO0dLS0G4K8xHn_wAvH_s2wXhSYJwU2X5S190MAWNo9F5KTCxuN2ZsI5ADEGvuG0gl_MFr9-J6pfvIjc29CNd/s1600/a4.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="416" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhmV34IQ5SQ1Xw_hc__NcY0fkDkgArwe1idJJlXOtlxY7DapmrvmA8RzreaO0dLS0G4K8xHn_wAvH_s2wXhSYJwU2X5S190MAWNo9F5KTCxuN2ZsI5ADEGvuG0gl_MFr9-J6pfvIjc29CNd/s640/a4.png" width="640" /></a></div>
<div style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; line-height: inherit; margin-bottom: 0.5em; margin-top: 0.5em; text-align: start;">
<br /></div>
<h3 style="background: none rgb(255, 255, 255); border-bottom: none; font-family: sans-serif; font-size: 1.2em; line-height: 1.6; margin: 0.3em 0px 0px; overflow: hidden; padding-bottom: 0px; padding-top: 0.5em; text-align: start;">
<span class="mw-headline" id="Matrix_form">Matrix form</span></h3>
<div class="hatnote" role="note" style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; font-style: italic; margin-bottom: 0.5em; padding-left: 1.6em; text-align: start;">
<br /></div>
<div style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; line-height: inherit; margin-bottom: 0.5em; margin-top: 0.5em; text-align: start;">
The sum and difference formulae for sine and cosine can be written in matrix form as:</div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgLVv7Wo77YQqb3MZ7zu5iWRzCU36yeL07oP28VakEcxBBMRCrk5-Byru0__MZ7osJPolIrppje_DilqPEtePE_AcdIOccEQ5cbUK1XQh34GaFtLB6wwQ-icBRqc3ZeJOum1blRzJAL0Yxx/s1600/a5.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="274" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgLVv7Wo77YQqb3MZ7zu5iWRzCU36yeL07oP28VakEcxBBMRCrk5-Byru0__MZ7osJPolIrppje_DilqPEtePE_AcdIOccEQ5cbUK1XQh34GaFtLB6wwQ-icBRqc3ZeJOum1blRzJAL0Yxx/s640/a5.png" width="640" /></a></div>
<div style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; line-height: inherit; margin-bottom: 0.5em; margin-top: 0.5em; text-align: center;">
<br /></div>
<dl style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; margin-bottom: 0.5em; margin-top: 0.2em; text-align: start;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 20.65px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}&{}\quad \left({\begin{array}{rr}\cos \alpha &-\sin \alpha \\\sin \alpha &\cos \alpha \end{array}}\right)\left({\begin{array}{rr}\cos \beta &-\sin \beta \\\sin \beta &\cos \beta \end{array}}\right)\\[12pt]&=\left({\begin{array}{rr}\cos \alpha \cos \beta -\sin \alpha \sin \beta &-\cos \alpha \sin \beta -\sin \alpha \cos \beta \\\sin \alpha \cos \beta +\cos \alpha \sin \beta &-\sin \alpha \sin \beta +\cos \alpha \cos \beta \end{array}}\right)\\[12pt]&=\left({\begin{array}{rr}\cos(\alpha +\beta )&-\sin(\alpha +\beta )\\\sin(\alpha +\beta )&\cos(\alpha +\beta )\end{array}}\right).\end{aligned}}}</annotation></semantics></math></span></dd></dl>
<div style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; line-height: inherit; margin-bottom: 0.5em; margin-top: 0.5em; text-align: start;">
This shows that these matrices form a representation of the rotation group in the plane (technically, the special orthogonal group <i>SO</i>(2)), since the composition law is fulfilled: subsequent multiplications of a vector with these two matrices yields the same result as the rotation by the sum of the angles.</div>
<h3 style="background: none rgb(255, 255, 255); border-bottom: none; font-family: sans-serif; font-size: 1.2em; line-height: 1.6; margin: 0.3em 0px 0px; overflow: hidden; padding-bottom: 0px; padding-top: 0.5em; text-align: start;">
<span class="mw-headline" id="Sines_and_cosines_of_sums_of_infinitely_many_terms">Sines and cosines of sums of infinitely many terms</span></h3>
<div>
<span class="mw-headline"><br /></span></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj-W8HVKRAoqWT72dWdyzlIoUabLGQmw1DPNZm-0NLZuV-bh2VFvnfNLRbq6xWNaEy36Hrl8wG1eWNuagtnGKTJI_XlD0TIVjZsYqVSNmLvs-m-OAzS_UJO__BZW8JI5DPKFxrVHXAMaU4k/s1600/a6.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="200" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj-W8HVKRAoqWT72dWdyzlIoUabLGQmw1DPNZm-0NLZuV-bh2VFvnfNLRbq6xWNaEy36Hrl8wG1eWNuagtnGKTJI_XlD0TIVjZsYqVSNmLvs-m-OAzS_UJO__BZW8JI5DPKFxrVHXAMaU4k/s640/a6.png" width="640" /></a></div>
<div>
<span class="mw-headline"><br /></span></div>
<dl style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; margin-bottom: 0.5em; margin-top: 0.2em; text-align: start;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 20.65px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle \sin \left(\sum _{i=1}^{\infty }\theta _{i}\right)=\sum _{{\text{odd}}\ k\geq 1}(-1)^{\frac {k-1}{2}}\sum _{\begin{smallmatrix}A\subseteq \{\,1,2,3,\dots \,\}\\\left|A\right|=k\end{smallmatrix}}\left(\prod _{i\in A}\sin \theta _{i}\prod _{i\not \in A}\cos \theta _{i}\right)}</annotation></semantics></math></span></dd></dl>
<dl style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; margin-bottom: 0.5em; margin-top: 0.2em; text-align: start;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 20.65px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle \cos \left(\sum _{i=1}^{\infty }\theta _{i}\right)=\sum _{{\text{even}}\ k\geq 0}~(-1)^{\frac {k}{2}}~~\sum _{\begin{smallmatrix}A\subseteq \{\,1,2,3,\dots \,\}\\\left|A\right|=k\end{smallmatrix}}\left(\prod _{i\in A}\sin \theta _{i}\prod _{i\not \in A}\cos \theta _{i}\right)}</annotation></semantics></math></span></dd></dl>
<div style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; line-height: inherit; margin-bottom: 0.5em; margin-top: 0.5em; text-align: start;">
In these two identities an asymmetry appears that is not seen in the case of sums of finitely many terms: in each product, there are only finitely many sine factors and cofinitely many cosine factors.</div>
<div style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; line-height: inherit; margin-bottom: 0.5em; margin-top: 0.5em; text-align: start;">
If only finitely many of the terms <span class="texhtml mvar" style="font-family: "nimbus roman no9 l" , "times new roman" , "times" , serif; font-size: 20.65px; font-style: italic; line-height: 1; white-space: nowrap;">θi</span> are nonzero, then only finitely many of the terms on the right side will be nonzero because sine factors will vanish, and in each term, all but finitely many of the cosine factors will be unity.</div>
<h3 style="background: none rgb(255, 255, 255); border-bottom: none; font-family: sans-serif; font-size: 1.2em; line-height: 1.6; margin: 0.3em 0px 0px; overflow: hidden; padding-bottom: 0px; padding-top: 0.5em; text-align: start;">
<span class="mw-headline" id="Tangents_of_sums">Tangents of sums</span></h3>
<div style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; line-height: inherit; margin-bottom: 0.5em; margin-top: 0.5em; text-align: start;">
Let <span class="texhtml mvar" style="font-family: "nimbus roman no9 l" , "times new roman" , "times" , serif; font-size: 20.65px; font-style: italic; line-height: 1; white-space: nowrap;">e<sub style="font-size: 16.52px; line-height: 1;">k</sub></span> (for <span class="texhtml mvar" style="font-family: "nimbus roman no9 l" , "times new roman" , "times" , serif; font-size: 20.65px; font-style: italic; line-height: 1; white-space: nowrap;">k</span> = 0, 1, 2, 3, ...) be the <span class="texhtml mvar" style="font-family: "nimbus roman no9 l" , "times new roman" , "times" , serif; font-size: 20.65px; font-style: italic; line-height: 1; white-space: nowrap;">k</span>th-degree <a href="https://en.wikipedia.org/wiki/Elementary_symmetric_polynomial" style="background: none; color: #0b0080; text-decoration: none;" title="Elementary symmetric polynomial">elementary symmetric polynomial</a> in the variables</div>
<div style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; line-height: inherit; margin-bottom: 0.5em; margin-top: 0.5em; text-align: start;">
<br /></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgXGwHOIFfhRG-b8KJFjhRvvoh3mu-JeZjrsRk9gV0_g8A0xPVc9Qyt8zxD1uR1AGicdgJtgdb1RIEOsrPdEFKVi9M7ieJ3lhKAp1NlbZ6l6Itrfwava7sKClZQCJiH6BdKYU1WCOOqnONT/s1600/a8.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="546" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgXGwHOIFfhRG-b8KJFjhRvvoh3mu-JeZjrsRk9gV0_g8A0xPVc9Qyt8zxD1uR1AGicdgJtgdb1RIEOsrPdEFKVi9M7ieJ3lhKAp1NlbZ6l6Itrfwava7sKClZQCJiH6BdKYU1WCOOqnONT/s640/a8.png" width="640" /></a></div>
<div style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; line-height: inherit; margin-bottom: 0.5em; margin-top: 0.5em; text-align: start;">
<br /></div>
<div style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; line-height: inherit; margin-bottom: 0.5em; margin-top: 0.5em; text-align: start;">
The number of terms on the right side depends on the number of terms on the left side.</div>
<div style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; line-height: inherit; margin-bottom: 0.5em; margin-top: 0.5em; text-align: start;">
For example:</div>
<div style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; line-height: inherit; margin-bottom: 0.5em; margin-top: 0.5em; text-align: start;">
<br /></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiywzlj_GoafVB-DqKR0k79o6tRwJ8Qd8-UHL_6VrAstvuVtoLDZozE-7YIOCuhJe8i3uV6hCDvLeeeyKDu8Sl3lEFxpv-Epex5rz6HK3atR37-MwhWlyLVblORP2v5COjM0iN_-UwPBH43/s1600/a9.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="200" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiywzlj_GoafVB-DqKR0k79o6tRwJ8Qd8-UHL_6VrAstvuVtoLDZozE-7YIOCuhJe8i3uV6hCDvLeeeyKDu8Sl3lEFxpv-Epex5rz6HK3atR37-MwhWlyLVblORP2v5COjM0iN_-UwPBH43/s640/a9.png" width="640" /></a></div>
<div style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; line-height: inherit; margin-bottom: 0.5em; margin-top: 0.5em; text-align: start;">
<br /></div>
<div style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; line-height: inherit; margin-bottom: 0.5em; margin-top: 0.5em; text-align: start;">
<br /></div>
<dl style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; margin-bottom: 0.5em; margin-top: 0.2em; text-align: start;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 20.65px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\tan(\theta _{1}+\theta _{2})&={\frac {e_{1}}{e_{0}-e_{2}}}={\frac {x_{1}+x_{2}}{1\ -\ x_{1}x_{2}}}={\frac {\tan \theta _{1}+\tan \theta _{2}}{1\ -\ \tan \theta _{1}\tan \theta _{2}}},\\[8pt]\tan(\theta _{1}+\theta _{2}+\theta _{3})&={\frac {e_{1}-e_{3}}{e_{0}-e_{2}}}={\frac {(x_{1}+x_{2}+x_{3})\ -\ (x_{1}x_{2}x_{3})}{1\ -\ (x_{1}x_{2}+x_{1}x_{3}+x_{2}x_{3})}},\\[8pt]\tan(\theta _{1}+\theta _{2}+\theta _{3}+\theta _{4})&={\frac {e_{1}-e_{3}}{e_{0}-e_{2}+e_{4}}}\\[8pt]&={\frac {(x_{1}+x_{2}+x_{3}+x_{4})\ -\ (x_{1}x_{2}x_{3}+x_{1}x_{2}x_{4}+x_{1}x_{3}x_{4}+x_{2}x_{3}x_{4})}{1\ -\ (x_{1}x_{2}+x_{1}x_{3}+x_{1}x_{4}+x_{2}x_{3}+x_{2}x_{4}+x_{3}x_{4})\ +\ (x_{1}x_{2}x_{3}x_{4})}},\end{aligned}}}</annotation></semantics></math></span></dd></dl>
<div style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; line-height: inherit; margin-bottom: 0.5em; margin-top: 0.5em; text-align: start;">
and so on. The case of only finitely many terms can be proved by mathematical induction.</div>
<h3 style="background: none rgb(255, 255, 255); border-bottom: none; font-family: sans-serif; font-size: 1.2em; line-height: 1.6; margin: 0.3em 0px 0px; overflow: hidden; padding-bottom: 0px; padding-top: 0.5em; text-align: start;">
<span class="mw-headline" id="Secants_and_cosecants_of_sums">Secants and cosecants of sums</span></h3>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgBtQg3K6R0kwxBhmIoGXWKftXFH58k_ygk0BWSiEn3m3trtwAy_CeL5RyzdEEqYUabhS5ON4KcLku7HP6Tc7d-A7CM1n3iSn9PQddsXA34tfxOKzjxQh4Pz3uslCxyR92_46wCdsBaAoNh/s1600/a10.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="173" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgBtQg3K6R0kwxBhmIoGXWKftXFH58k_ygk0BWSiEn3m3trtwAy_CeL5RyzdEEqYUabhS5ON4KcLku7HP6Tc7d-A7CM1n3iSn9PQddsXA34tfxOKzjxQh4Pz3uslCxyR92_46wCdsBaAoNh/s400/a10.png" width="400" /></a></div>
<div>
<span class="mw-headline"><br /></span></div>
<dl style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; margin-bottom: 0.5em; margin-top: 0.2em; text-align: start;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 20.65px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\sec \left(\sum _{i}\theta _{i}\right)&={\frac {\prod _{i}\sec \theta _{i}}{e_{0}-e_{2}+e_{4}-\cdots }}\\[8pt]\csc \left(\sum _{i}\theta _{i}\right)&={\frac {\prod _{i}\sec \theta _{i}}{e_{1}-e_{3}+e_{5}-\cdots }}\end{aligned}}}</annotation></semantics></math></span></dd></dl>
<div style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; line-height: inherit; margin-bottom: 0.5em; margin-top: 0.5em; text-align: start;">
where <span class="texhtml mvar" style="font-family: "nimbus roman no9 l" , "times new roman" , "times" , serif; font-size: 20.65px; font-style: italic; line-height: 1; white-space: nowrap;">e<sub style="font-size: 16.52px; line-height: 1;">k</sub></span> is the <span class="texhtml mvar" style="font-family: "nimbus roman no9 l" , "times new roman" , "times" , serif; font-size: 20.65px; font-style: italic; line-height: 1; white-space: nowrap;">k</span>th-degree elementary symmetric polynomial in the <span class="texhtml mvar" style="font-family: "nimbus roman no9 l" , "times new roman" , "times" , serif; font-size: 20.65px; font-style: italic; line-height: 1; white-space: nowrap;">n</span> variables <span class="texhtml" style="font-family: "nimbus roman no9 l" , "times new roman" , "times" , serif; font-size: 20.65px; line-height: 1; white-space: nowrap;"><i>x</i><sub style="font-size: 16.52px; line-height: 1;"><i>i</i></sub> = tan <i>θ</i><sub style="font-size: 16.52px; line-height: 1;"><i>i</i></sub></span>, <span class="texhtml" style="font-family: "nimbus roman no9 l" , "times new roman" , "times" , serif; font-size: 20.65px; line-height: 1; white-space: nowrap;"><i>i</i> = 1, ..., <i>n</i></span>, and the number of terms in the denominator and the number of factors in the product in the numerator depend on the number of terms in the sum on the left. The case of only finitely many terms can be proved by mathematical induction on the number of such terms. The convergence of the series in the denominators can be shown by writing the secant identity in the form</div>
<dl style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; margin-bottom: 0.5em; margin-top: 0.2em; text-align: start;"><div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhKLvTYLsIRGrm3T1I6yr8_f8BmU8fmpiryipQNCAm-UgglNKOcL3MvSp-9qqM1oDy3dkwCI5wMfbX7yWTup-WHO2wTSGeFOoAOYHl_DZR8v2ACZ4ysIdpo0utc0p1LkzUB_h_1B39Yw_p1/s1600/a11.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="73" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhKLvTYLsIRGrm3T1I6yr8_f8BmU8fmpiryipQNCAm-UgglNKOcL3MvSp-9qqM1oDy3dkwCI5wMfbX7yWTup-WHO2wTSGeFOoAOYHl_DZR8v2ACZ4ysIdpo0utc0p1LkzUB_h_1B39Yw_p1/s320/a11.png" width="320" /></a></div>
<dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 20.65px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle e_{0}-e_{2}+e_{4}-\cdots ={\frac {\prod _{i}\sec \theta _{i}}{\sec \left(\sum _{i}\theta _{i}\right)}}}</annotation></semantics></math></span></dd></dl>
<div style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; line-height: inherit; margin-bottom: 0.5em; margin-top: 0.5em; text-align: start;">
and then observing that the left side converges if the right side converges, and similarly for the cosecant identity.</div>
<div style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; line-height: inherit; margin-bottom: 0.5em; margin-top: 0.5em; text-align: start;">
For example,</div>
<dl style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; margin-bottom: 0.5em; margin-top: 0.2em; text-align: start;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 20.65px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\sec(\alpha +\beta +\gamma )&={\frac {\sec \alpha \sec \beta \sec \gamma }{1-\tan \alpha \tan \beta -\tan \alpha \tan \gamma -\tan \beta \tan \gamma }}\\[8pt]\csc(\alpha +\beta +\gamma )&={\frac {\sec \alpha \sec \beta \sec \gamma }{\tan \alpha +\tan \beta +\tan \gamma -\tan \alpha \tan \beta \tan \gamma }}.\end{aligned}}}</annotation></semantics></math></span></dd></dl>
<div style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; line-height: inherit; margin-bottom: 0.5em; margin-top: 0.5em; text-align: start;">
<br /></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhbLlJ256HSDbmV0AinlEhOEs34MB4cgE6H66vccl3ZFJa42NnwYbJdWrllqCMTLztWJVDxGDX960DxLUoZqJtTHrqTokyUxePW_8qA9sVlxO6y2LjAKGlQ48M9kR5zdp2307gM8QgzJVfk/s1600/a12.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="168" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhbLlJ256HSDbmV0AinlEhOEs34MB4cgE6H66vccl3ZFJa42NnwYbJdWrllqCMTLztWJVDxGDX960DxLUoZqJtTHrqTokyUxePW_8qA9sVlxO6y2LjAKGlQ48M9kR5zdp2307gM8QgzJVfk/s640/a12.png" width="640" /></a></div>
<div style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; line-height: inherit; margin-bottom: 0.5em; margin-top: 0.5em; text-align: start;">
<br /></div>
</div>
</div>
Thanh Ahttp://www.blogger.com/profile/16564310847356090654noreply@blogger.comtag:blogger.com,1999:blog-574185785118190739.post-57436316502157664132016-11-24T02:12:00.000-08:002018-10-19T19:24:30.510-07:008. Multiple-angle formulae<div dir="ltr" style="text-align: left;" trbidi="on">
<h3 style="background: none rgb(255, 255, 255); border-bottom: none; line-height: 1.6; margin: 0.3em 0px 0px; overflow: hidden; padding-bottom: 0px; padding-top: 0.5em;">
<span class="mw-headline"><span style="font-family: sans-serif; font-size: large;">Multiple-angle formulae<img alt="" border="0" height="0" id="amznPsBmPixel_4145303" src="https://ir-na.amazon-adsystem.com/e/ir?source=bk&t=learnjavaporgramming-20&bm-id=default&l=ktl&linkId=e93a839e81f1f779289c244fde501aac&_cb=1539528890379" style="border: none !important; height: 0px !important; margin: 0px !important; padding: 0px !important; width: 0px !important;" width="0" /></span></span></h3>
<div class="separator" style="clear: both; text-align: center;">
<a amzn-ps-bm-asin="B01G686UNK" class="amzn_ps_bm_il" data-amzn-link-id="fd9dd17b3d92257ef0e86da85984fcae" data-amzn-ps-bm-keyword="multiple-angle formulae" href="http://www.amazon.com/8th-Grade-Math-MCQs-Questions-ebook/dp/B01G686UNK/ref=as_li_bk_ia/?tag=learnjavaporgramming-20&linkId=fd9dd17b3d92257ef0e86da85984fcae&linkCode=kia" id="amznPsBmLink_4171978" rel="nofollow" target="_blank"><img border="0" height="106" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEghyphenhyphenpKoX7QtW_A7Ziw5eOWXPqiu_CFo7alDvC1Pbq6cMXdYdHlCIdYQ-StcXrFeaw_b5Bbd69Y7sMkm2asarh2H6PKr8virHQEKMTtL00BGtMOts8YbkpIE44Mj0oAuKJnjETCu7AKf0evA/s640/b1.png" width="640" /></a><img alt="" border="0" height="0" id="amznPsBmPixel_4171978" src="https://ir-na.amazon-adsystem.com/e/ir?source=bk&t=learnjavaporgramming-20&bm-id=default&l=kia&linkId=fd9dd17b3d92257ef0e86da85984fcae&_cb=1539528924865" style="border: none !important; height: 0px !important; margin: 0px !important; padding: 0px !important; width: 0px !important;" width="0" /></div>
<h3 style="background: none rgb(255, 255, 255); border-bottom: none; font-family: sans-serif; font-size: 1.2em; line-height: 1.6; margin: 0.3em 0px 0px; overflow: hidden; padding-bottom: 0px; padding-top: 0.5em;">
<span class="mw-headline" id="Double-angle.2C_triple-angle.2C_and_half-angle_formulae">Double-angle, triple-angle, and half-angle formulae</span></h3>
<h4 style="background: none rgb(255, 255, 255); border-bottom: none; font-family: sans-serif; font-size: 17.5px; line-height: 1.6; margin: 0.3em 0px 0px; overflow: hidden; padding-bottom: 0px; padding-top: 0.5em;">
<span class="mw-headline" id="Double-angle_formulae">Double-angle formulae</span></h4>
<div>
<span class="mw-headline"><br /></span></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjOoZ2BSZa8Y0KwJHSNrbyr9E903PpECsFrtQ7b6dfTl0UAgc75UeSf3PWc2Lc9Y935W273GNhdPZuSxqgmZJOn7TsryEk3bKg9J1lHldsRNXdXmhrQWx2D-6j7TPK7G5_Dmuz3XVTieiS3/s1600/b2.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="226" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjOoZ2BSZa8Y0KwJHSNrbyr9E903PpECsFrtQ7b6dfTl0UAgc75UeSf3PWc2Lc9Y935W273GNhdPZuSxqgmZJOn7TsryEk3bKg9J1lHldsRNXdXmhrQWx2D-6j7TPK7G5_Dmuz3XVTieiS3/s640/b2.png" width="640" /></a></div>
<div>
<span class="mw-headline"><br /></span></div>
<h4 style="background: none rgb(255, 255, 255); border-bottom: none; font-family: sans-serif; font-size: 17.5px; line-height: 1.6; margin: 0.3em 0px 0px; overflow: hidden; padding-bottom: 0px; padding-top: 0.5em;">
<span class="mw-headline" id="Triple-angle_formulae">Triple-angle formulae</span></h4>
<div class="separator" style="clear: both; text-align: left;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEilZt4frTlfp_6m8ESbqwJ9fyLVNee1w_yXjOwE9cYpmfk3xYG_HAiVYeaVkHmP6KQ1GGTYTMHRBLm104yLKP0ITdaKHaweKW-BHWq2W3HvQB3l8fvnQVJt-O3ntSxzA_2gv2ekzA1ajfTB/s1600/b3.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="226" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEilZt4frTlfp_6m8ESbqwJ9fyLVNee1w_yXjOwE9cYpmfk3xYG_HAiVYeaVkHmP6KQ1GGTYTMHRBLm104yLKP0ITdaKHaweKW-BHWq2W3HvQB3l8fvnQVJt-O3ntSxzA_2gv2ekzA1ajfTB/s640/b3.png" width="640" /></a></div>
<div>
<br /></div>
<div>
<br /></div>
<div>
<h4 style="background: none rgb(255, 255, 255); border-bottom: none; font-family: sans-serif; font-size: 17.5px; line-height: 1.6; margin: 0.3em 0px 0px; overflow: hidden; padding-bottom: 0px; padding-top: 0.5em;">
<span class="mw-headline" id="Half-angle_formulae">Half-angle formulae</span></h4>
</div>
<div>
<span class="mw-headline"><br /></span></div>
<div class="separator" style="clear: both; text-align: left;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh8qQdhKdt7XtVeqrSYpT4T7Fs0pfn2cnQuCUr1JKpI6J730DW0TswbdbgBXZtuVNe9T1mooSPwxbwG5RDAydUvXHsmxFODBp0pB4kGS5dE2uuj9FGyK5WIBfY70sfvqGfAhesgl3VnYYE-/s1600/b4.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="450" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh8qQdhKdt7XtVeqrSYpT4T7Fs0pfn2cnQuCUr1JKpI6J730DW0TswbdbgBXZtuVNe9T1mooSPwxbwG5RDAydUvXHsmxFODBp0pB4kGS5dE2uuj9FGyK5WIBfY70sfvqGfAhesgl3VnYYE-/s640/b4.png" width="640" /></a></div>
<div>
<span class="mw-headline"><br /></span></div>
<div>
<span class="mw-headline"><br /></span></div>
<div>
<span class="mw-headline"></span><br />
<div style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; line-height: inherit; margin-bottom: 0.5em; margin-top: 0.5em;">
<span class="mw-headline">Also</span></div>
<span class="mw-headline">
</span>
<br />
<div style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; line-height: inherit; margin-bottom: 0.5em; margin-top: 0.5em;">
<span class="mw-headline"><br /></span></div>
<span class="mw-headline">
</span>
<div class="separator" style="clear: both; text-align: left;">
<span class="mw-headline"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgGfX5F98ihfAuZDpsA2T1ssAbznFs_CPUdC33fyypoDhk-2432GJKP4JrSUah8MT9rx146hGdMxzDd2-Ibl-8tdc61JRDiW40AUpC44xRVGqfeuMUiSqLdFpsaqmHG9uSTrm-NkUBtNxOl/s1600/b5.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="269" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgGfX5F98ihfAuZDpsA2T1ssAbznFs_CPUdC33fyypoDhk-2432GJKP4JrSUah8MT9rx146hGdMxzDd2-Ibl-8tdc61JRDiW40AUpC44xRVGqfeuMUiSqLdFpsaqmHG9uSTrm-NkUBtNxOl/s400/b5.png" width="400" /></a></span></div>
<span class="mw-headline">
<h4 style="background: none rgb(255, 255, 255); border-bottom: none; font-family: sans-serif; font-size: 17.5px; line-height: 1.6; margin: 0.3em 0px 0px; overflow: hidden; padding-bottom: 0px; padding-top: 0.5em;">
<span class="mw-headline"><br /></span></h4>
<h4 style="background: none rgb(255, 255, 255); border-bottom: none; font-family: sans-serif; font-size: 17.5px; line-height: 1.6; margin: 0.3em 0px 0px; overflow: hidden; padding-bottom: 0px; padding-top: 0.5em;">
<span class="mw-headline" id="Table">* Table</span></h4>
<div class="hatnote" role="note" style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; font-style: italic; margin-bottom: 0.5em; padding-left: 1.6em;">
<br /></div>
<div style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; line-height: inherit; margin-bottom: 0.5em; margin-top: 0.5em;">
These can be shown by using either the sum and difference identities or the multiple-angle formulae.</div>
<div style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; line-height: inherit; margin-bottom: 0.5em; margin-top: 0.5em;">
<br /></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEglta4ATNum57GKFUgi9o9uO4MtDvBfBBW3L3F4pETArCKiQVp494B5CUjxsY_Akm5XJuQ1m0aY21GlSsqkzyYNbIoFTWhxEh6Z4nb39rzBPmI12DVSzROzhXlQBxPAtmxUHeM00__CUlit/s1600/bang.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="336" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEglta4ATNum57GKFUgi9o9uO4MtDvBfBBW3L3F4pETArCKiQVp494B5CUjxsY_Akm5XJuQ1m0aY21GlSsqkzyYNbIoFTWhxEh6Z4nb39rzBPmI12DVSzROzhXlQBxPAtmxUHeM00__CUlit/s640/bang.png" width="640" /></a></div>
<div style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; line-height: inherit; margin-bottom: 0.5em; margin-top: 0.5em;">
<br /></div>
<div style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; line-height: inherit; margin-bottom: 0.5em; margin-top: 0.5em;">
The fact that the triple-angle formula for sine and cosine only involves powers of a single function allows one to relate the geometric problem of a compass and straightedge construction of angle trisection to the algebraic problem of solving a cubic equation, which allows one to prove that trisection is in general impossible using the given tools, by field theory.</div>
<div style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; line-height: inherit; margin-bottom: 0.5em; margin-top: 0.5em;">
A <a amzn-ps-bm-asin="B017RMGC3M" class="amzn_ps_bm_tl" data-amzn-link-id="68e7cbd372f6c16cde82fbbe174f8cc3" data-amzn-ps-bm-keyword="formula " href="http://www.amazon.com/Similac-Advance-Infant-Formula-Powder/dp/B017RMGC3M/ref=as_li_bk_tl/?tag=learnjavaporgramming-20&linkId=68e7cbd372f6c16cde82fbbe174f8cc3&linkCode=ktl" id="amznPsBmLink_2961694" rel="nofollow" target="_blank">formula </a><img alt="" border="0" height="0" id="amznPsBmPixel_2961694" src="https://ir-na.amazon-adsystem.com/e/ir?source=bk&t=learnjavaporgramming-20&bm-id=default&l=ktl&linkId=68e7cbd372f6c16cde82fbbe174f8cc3&_cb=1539528945937" style="border: none !important; height: 0px !important; margin: 0px !important; padding: 0px !important; width: 0px !important;" width="0" />for computing the trigonometric identities for the one-third angle exists, but it requires finding the zeroes of the cubic equation <span class="texhtml" style="font-family: "nimbus roman no9 l" , "times new roman" , "times" , serif; font-size: 20.65px; line-height: 1; white-space: nowrap;">4<i>x</i><sup style="font-size: 16.52px; line-height: 1;">3</sup> − 3<i>x</i> + d = 0</span>, where <span class="texhtml mvar" style="font-family: "nimbus roman no9 l" , "times new roman" , "times" , serif; font-size: 20.65px; font-style: italic; line-height: 1; white-space: nowrap;">x</span> is the value of the cosine function at the one-third angle and <span class="texhtml mvar" style="font-family: "nimbus roman no9 l" , "times new roman" , "times" , serif; font-size: 20.65px; font-style: italic; line-height: 1; white-space: nowrap;">d</span> is the known value of the cosine function at the full angle. However, the discriminant of this equation is positive, so this equation has three real roots (of which only one is the solution for the cosine of the one-third angle). None of these solutions is reducible to a real algebraic expression, as they use intermediate complex numbers under the cube roots.</div>
<h3 style="background: none rgb(255, 255, 255); border-bottom: none; font-family: sans-serif; font-size: 1.2em; line-height: 1.6; margin: 0.3em 0px 0px; overflow: hidden; padding-bottom: 0px; padding-top: 0.5em;">
<span class="mw-headline" id="Sine.2C_cosine.2C_and_tangent_of_multiple_angles">Sine, cosine, and tangent of multiple angles</span></h3>
<div style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; line-height: inherit; margin-bottom: 0.5em; margin-top: 0.5em;">
For specific multiples, these follow from the angle addition formulas, while the general formula was given by 16th-century French mathematician François Viète.</div>
<dl style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 20.65px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\sin(n\theta )&=\sum _{k{\text{ odd}}}(-1)^{\frac {k-1}{2}}{n \choose k}\cos ^{n-k}\theta \sin ^{k}\theta ,\\\cos(n\theta )&=\sum _{k{\text{ even}}}(-1)^{\frac {k}{2}}{n \choose k}\cos ^{n-k}\theta \sin ^{k}\theta \end{aligned}}}</annotation></semantics></math></span><img alt="{\displaystyle {\begin{aligned}\sin(n\theta )&=\sum _{k{\text{ odd}}}(-1)^{\frac {k-1}{2}}{n \choose k}\cos ^{n-k}\theta \sin ^{k}\theta ,\\\cos(n\theta )&=\sum _{k{\text{ even}}}(-1)^{\frac {k}{2}}{n \choose k}\cos ^{n-k}\theta \sin ^{k}\theta \end{aligned}}}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc2aca9a8182b5c47663f6c2f6c2a8e6ae754c47" style="border: none; display: inline-block; height: 13.843ex; vertical-align: -6.338ex; width: 43.52ex;" /></dd></dl>
<div style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; line-height: inherit; margin-bottom: 0.5em; margin-top: 0.5em;">
In each of these two equations, the first parenthesized term is a binomial coefficient, and the final trigonometric function equals one or minus one or zero so that half the entries in each of the sums are removed. <span class="texhtml" style="font-family: "nimbus roman no9 l" , "times new roman" , "times" , serif; font-size: 20.65px; line-height: 1; white-space: nowrap;">tan <i>nθ</i></span> can be written in terms of <span class="texhtml" style="font-family: "nimbus roman no9 l" , "times new roman" , "times" , serif; font-size: 20.65px; line-height: 1; white-space: nowrap;">tan <i>θ</i></span> using the recurrence relation:</div>
<dl style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 20.65px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle \tan \,{\big (}(n{+}1)\theta {\big )}={\frac {\tan(n\theta )+\tan \theta }{1-\tan(n\theta )\,\tan \theta }}.}</annotation></semantics></math></span><img alt="{\displaystyle \tan \,{\big (}(n{+}1)\theta {\big )}={\frac {\tan(n\theta )+\tan \theta }{1-\tan(n\theta )\,\tan \theta }}.}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a5337dac09e38ab178b881c2653d9b18121b1a54" style="border: none; display: inline-block; height: 6.509ex; vertical-align: -2.671ex; width: 35.65ex;" /></dd></dl>
<div style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; line-height: inherit; margin-bottom: 0.5em; margin-top: 0.5em;">
<span class="texhtml" style="font-family: "nimbus roman no9 l" , "times new roman" , "times" , serif; font-size: 20.65px; line-height: 1; white-space: nowrap;">cot <i>nθ</i></span> can be written in terms of <span class="texhtml" style="font-family: "nimbus roman no9 l" , "times new roman" , "times" , serif; font-size: 20.65px; line-height: 1; white-space: nowrap;">cot <i>θ</i></span> using the recurrence relation:</div>
<dl style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 20.65px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle \cot \,{\big (}(n{+}1)\theta {\big )}={\frac {\cot(n\theta )\,\cot \theta -1}{\cot(n\theta )+\cot \theta }}.}</annotation></semantics></math></span><img alt="{\displaystyle \cot \,{\big (}(n{+}1)\theta {\big )}={\frac {\cot(n\theta )\,\cot \theta -1}{\cot(n\theta )+\cot \theta }}.}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/61ed8f550789f7e9d285fc3c5516f48c662b241a" style="border: none; display: inline-block; height: 6.509ex; vertical-align: -2.671ex; width: 34.87ex;" /></dd></dl>
<h3 style="background: none rgb(255, 255, 255); border-bottom: none; font-family: sans-serif; font-size: 1.2em; line-height: 1.6; margin: 0.3em 0px 0px; overflow: hidden; padding-bottom: 0px; padding-top: 0.5em;">
<span class="mw-headline" id="Chebyshev_method">Chebyshev method</span></h3>
<div style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; line-height: inherit; margin-bottom: 0.5em; margin-top: 0.5em;">
The Chebyshev method is a recursive algorithm for finding the <span class="texhtml mvar" style="font-family: "nimbus roman no9 l" , "times new roman" , "times" , serif; font-size: 20.65px; font-style: italic; line-height: 1; white-space: nowrap;">n</span>th multiple angle formula knowing the <span class="texhtml" style="font-family: "nimbus roman no9 l" , "times new roman" , "times" , serif; font-size: 20.65px; line-height: 1; white-space: nowrap;">(<i>n</i> − 1)</span>th and <span class="texhtml" style="font-family: "nimbus roman no9 l" , "times new roman" , "times" , serif; font-size: 20.65px; line-height: 1; white-space: nowrap;">(<i>n</i> − 2)</span>th formulae.</div>
<div style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; line-height: inherit; margin-bottom: 0.5em; margin-top: 0.5em;">
<span class="texhtml" style="font-family: "nimbus roman no9 l" , "times new roman" , "times" , serif; font-size: 20.65px; line-height: 1; white-space: nowrap;">cos(<i>nx</i>)</span> can be computed from the cosine of <span class="texhtml" style="font-family: "nimbus roman no9 l" , "times new roman" , "times" , serif; font-size: 20.65px; line-height: 1; white-space: nowrap;">(<i>n</i> − 1)<i>x</i></span> and <span class="texhtml" style="font-family: "nimbus roman no9 l" , "times new roman" , "times" , serif; font-size: 20.65px; line-height: 1; white-space: nowrap;">(<i>n</i> − 2)<i>x</i></span> as follows:</div>
<dl style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 20.65px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle \cos(nx)=2\cdot \cos x\cdot \cos {\big (}(n-1)x{\big )}-\cos {\big (}(n-2)x{\big )}\,}</annotation></semantics></math></span><img alt="{\displaystyle \cos(nx)=2\cdot \cos x\cdot \cos {\big (}(n-1)x{\big )}-\cos {\big (}(n-2)x{\big )}\,}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3b1fcf7757b7742b3f5747b948fd9aaba3e4a8d" style="border: none; display: inline-block; height: 3.343ex; vertical-align: -1.171ex; width: 52.046ex;" /></dd></dl>
<div style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; line-height: inherit; margin-bottom: 0.5em; margin-top: 0.5em;">
Similarly <span class="texhtml" style="font-family: "nimbus roman no9 l" , "times new roman" , "times" , serif; font-size: 20.65px; line-height: 1; white-space: nowrap;">sin(<i>nx</i>)</span> can be computed from the sines of <span class="texhtml" style="font-family: "nimbus roman no9 l" , "times new roman" , "times" , serif; font-size: 20.65px; line-height: 1; white-space: nowrap;">(<i>n</i> − 1)<i>x</i></span> and <span class="texhtml" style="font-family: "nimbus roman no9 l" , "times new roman" , "times" , serif; font-size: 20.65px; line-height: 1; white-space: nowrap;">(<i>n</i> − 2)<i>x</i></span></div>
<dl style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 20.65px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle \sin(nx)=2\cdot \cos x\cdot \sin {\big (}(n-1)x{\big )}-\sin {\big (}(n-2)x{\big )}\,}</annotation></semantics></math></span><img alt="{\displaystyle \sin(nx)=2\cdot \cos x\cdot \sin {\big (}(n-1)x{\big )}-\sin {\big (}(n-2)x{\big )}\,}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/335ce4c8384a6315565b8ea43b8aca64075f6655" style="border: none; display: inline-block; height: 3.343ex; vertical-align: -1.171ex; width: 51.279ex;" /></dd></dl>
<div style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; line-height: inherit; margin-bottom: 0.5em; margin-top: 0.5em;">
For the tangent, we have:</div>
<dl style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 20.65px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle \tan(nx)={\frac {H+K\tan x}{K-H\tan x}}\,}</annotation></semantics></math></span><img alt="\tan(nx)={\frac {H+K\tan x}{K-H\tan x}}\," aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b9b4595c1030478d3a91970507197804c86270d6" style="border: none; display: inline-block; height: 5.509ex; vertical-align: -2.005ex; width: 24.805ex;" /></dd></dl>
<div style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; line-height: inherit; margin-bottom: 0.5em; margin-top: 0.5em;">
where <span class="texhtml" style="font-family: "nimbus roman no9 l" , "times new roman" , "times" , serif; font-size: 20.65px; line-height: 1; white-space: nowrap;"><i><span class="sfrac nowrap" style="display: inline-block; font-size: 17.5525px; text-align: center; vertical-align: -0.5em;"><span style="display: block; line-height: 1em; margin: 0px 0.1em;">H</span><span class="visualhide" style="height: 1px; left: -10000px; overflow: hidden; position: absolute; top: auto; width: 1px;">/</span><span style="border-top: 1px solid; display: block; line-height: 1em; margin: 0px 0.1em;">K</span></span></i> = tan(<i>n</i> − 1)<i>x</i></span>.</div>
<h3 style="background: none rgb(255, 255, 255); border-bottom: none; font-family: sans-serif; font-size: 1.2em; line-height: 1.6; margin: 0.3em 0px 0px; overflow: hidden; padding-bottom: 0px; padding-top: 0.5em;">
<span class="mw-headline" id="Tangent_of_an_average"><a amzn-ps-bm-asin="B00SN852CM" class="amzn_ps_bm_tl" data-amzn-link-id="de2316dd0f76e46defc743f408d22c12" data-amzn-ps-bm-keyword="Tangent of an average" href="http://www.amazon.com/UlmDesign-Your-Calculator/dp/B00SN852CM/ref=as_li_bk_tl/?tag=learnjavaporgramming-20&linkId=de2316dd0f76e46defc743f408d22c12&linkCode=ktl" id="amznPsBmLink_6691248" rel="nofollow" target="_blank">Tangent of an average</a><img alt="" border="0" height="0" id="amznPsBmPixel_6691248" src="https://ir-na.amazon-adsystem.com/e/ir?source=bk&t=learnjavaporgramming-20&bm-id=default&l=ktl&linkId=de2316dd0f76e46defc743f408d22c12&_cb=1539528961124" style="border: none !important; height: 0px !important; margin: 0px !important; padding: 0px !important; width: 0px !important;" width="0" /></span></h3>
<dl style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 20.65px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle \tan \left({\frac {\alpha +\beta }{2}}\right)={\frac {\sin \alpha +\sin \beta }{\cos \alpha +\cos \beta }}=-\,{\frac {\cos \alpha -\cos \beta }{\sin \alpha -\sin \beta }}}</annotation></semantics></math></span><img alt="\tan \left({\frac {\alpha +\beta }{2}}\right)={\frac {\sin \alpha +\sin \beta }{\cos \alpha +\cos \beta }}=-\,{\frac {\cos \alpha -\cos \beta }{\sin \alpha -\sin \beta }}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e0059ec572ecb7f76446d6d49406212189d349cf" style="border: none; display: inline-block; height: 6.176ex; vertical-align: -2.505ex; width: 48.957ex;" /></dd></dl>
<div style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; line-height: inherit; margin-bottom: 0.5em; margin-top: 0.5em;">
Setting either <span class="texhtml mvar" style="font-family: "nimbus roman no9 l" , "times new roman" , "times" , serif; font-size: 20.65px; font-style: italic; line-height: 1; white-space: nowrap;">α</span> or <span class="texhtml mvar" style="font-family: "nimbus roman no9 l" , "times new roman" , "times" , serif; font-size: 20.65px; font-style: italic; line-height: 1; white-space: nowrap;">β</span> to 0 gives the usual tangent half-angle formulae.</div>
<h3 style="background: none rgb(255, 255, 255); border-bottom: none; font-family: sans-serif; font-size: 1.2em; line-height: 1.6; margin: 0.3em 0px 0px; overflow: hidden; padding-bottom: 0px; padding-top: 0.5em;">
<span class="mw-headline" id="Vi.C3.A8te.27s_infinite_product">Viète's infinite product</span></h3>
<dl style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 20.65px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle \cos {\theta \over 2}\cdot \cos {\theta \over 4}\cdot \cos {\theta \over 8}\cdots =\prod _{n=1}^{\infty }\cos {\theta \over 2^{n}}={\sin \theta \over \theta }=\operatorname {sinc} \,\theta .}</annotation></semantics></math></span><img alt="\cos {\theta \over 2}\cdot \cos {\theta \over 4}\cdot \cos {\theta \over 8}\cdots =\prod _{n=1}^{\infty }\cos {\theta \over 2^{n}}={\sin \theta \over \theta }=\operatorname {sinc} \,\theta ." aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/022878829d7a84ac00fe178d59f760ff668002ce" style="border: none; display: inline-block; height: 7.009ex; vertical-align: -3.005ex; width: 54.376ex;" /></dd></dl>
<div style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; line-height: inherit; margin-bottom: 0.5em; margin-top: 0.5em;">
(Refer to sinc function.)</div>
<div style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; line-height: inherit; margin-bottom: 0.5em; margin-top: 0.5em;">
<br /></div>
<div style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; line-height: inherit; margin-bottom: 0.5em; margin-top: 0.5em;">
<br /></div>
</span></div>
</div>
Thanh Ahttp://www.blogger.com/profile/16564310847356090654noreply@blogger.comtag:blogger.com,1999:blog-574185785118190739.post-72648452736426310502016-11-23T02:23:00.000-08:002016-12-02T02:41:25.943-08:009. Power-reduction formula<span style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px;">Obtained by solving the second and third versions of the cosine double-angle formula.</span><br />
<span style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px;"><br /></span>
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhvjBEdu479Ku0NgZZ8A_tx9tCIUs2B_7Bm8l3kKMAFCUAC_MIIRccwDYKuPmknaQa5GHNVf9z5O4cUnO_F5_1Fc2m_zeHTz7ETP-CsxgA8vhpQoXPno2gfKvQC4tJC7I8anP_6CE9lKsF9/s1600/c1.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="160" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhvjBEdu479Ku0NgZZ8A_tx9tCIUs2B_7Bm8l3kKMAFCUAC_MIIRccwDYKuPmknaQa5GHNVf9z5O4cUnO_F5_1Fc2m_zeHTz7ETP-CsxgA8vhpQoXPno2gfKvQC4tJC7I8anP_6CE9lKsF9/s640/c1.png" width="640" /></a></div>
<span style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px;"><br /></span>
<span style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px;"><br /></span>
<span style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px;">and in general terms of powers of </span><span class="texhtml" style="background-color: white; color: #252525; font-family: "nimbus roman no9 l" , "times new roman" , "times" , serif; font-size: 20.65px; line-height: 1; white-space: nowrap;">sin <i>θ</i></span><span style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px;"> or </span><span class="texhtml" style="background-color: white; color: #252525; font-family: "nimbus roman no9 l" , "times new roman" , "times" , serif; font-size: 20.65px; line-height: 1; white-space: nowrap;">cos <i>θ</i></span><span style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px;"> the following is true, and can be deduced using </span>De Moivre's formula<span style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px;">, </span>Euler's formula<span style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px;"> and the </span>binomial theorem<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi2dBA8LRyx3LHE-hWqDDj32HgdI1etZkdGafHDiCRvJCHhbWunk-oM9iBtUJgibGwzdtflTmKGRQUNdGmNyyJ4Z7NOGyle20U0Rh0hYPyKoEegkgH2rF9zQgzuoINZbExBaNd6J-3rkOY0/s1600/c2.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="128" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi2dBA8LRyx3LHE-hWqDDj32HgdI1etZkdGafHDiCRvJCHhbWunk-oM9iBtUJgibGwzdtflTmKGRQUNdGmNyyJ4Z7NOGyle20U0Rh0hYPyKoEegkgH2rF9zQgzuoINZbExBaNd6J-3rkOY0/s640/c2.png" width="640" /></a></div>
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />Thanh Ahttp://www.blogger.com/profile/16564310847356090654noreply@blogger.comtag:blogger.com,1999:blog-574185785118190739.post-90757342225711055652016-11-22T02:25:00.000-08:002016-12-02T02:41:18.036-08:0010. Product-to-sum and sum-to-product identities<div style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; line-height: inherit; margin-bottom: 0.5em; margin-top: 0.5em;">
The product-to-sum identities or prosthaphaeresis formulas can be proven by expanding their right-hand sides using the angle addition theorems. See amplitude modulation for an application of the product-to-sum formulae, and beat (acoustics) and phase detector for applications of the sum-to-product formulae.</div>
<table class="wikitable" style="background-color: white; border-collapse: collapse; border: 1px solid rgb(170, 170, 170); color: black; font-family: sans-serif; font-size: 17.5px; margin: 1em 0px;"><tbody>
<tr><th style="background-color: #f2f2f2; border: 1px solid rgb(170, 170, 170); padding: 0.2em 0.4em; text-align: center;">Product-to-sum</th></tr>
<tr><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em 0.4em;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 20.65px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle 2\cos \theta \cos \varphi ={\cos(\theta -\varphi )+\cos(\theta +\varphi )}}</annotation></semantics></math></span><img alt="2\cos \theta \cos \varphi ={\cos(\theta -\varphi )+\cos(\theta +\varphi )}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c4f58abb83a587513760f60577944e6a251d510" style="border: none; display: inline-block; height: 2.843ex; vertical-align: -0.838ex; width: 38.507ex;" /></td></tr>
<tr><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em 0.4em;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 20.65px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle 2\sin \theta \sin \varphi ={\cos(\theta -\varphi )-\cos(\theta +\varphi )}}</annotation></semantics></math></span><img alt="2\sin \theta \sin \varphi ={\cos(\theta -\varphi )-\cos(\theta +\varphi )}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b3da7a71f9c2934f6afdde2be1739585d1d0188f" style="border: none; display: inline-block; height: 2.843ex; vertical-align: -0.838ex; width: 37.997ex;" /></td></tr>
<tr><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em 0.4em;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 20.65px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle 2\sin \theta \cos \varphi ={\sin(\theta +\varphi )+\sin(\theta -\varphi )}}</annotation></semantics></math></span><img alt="2\sin \theta \cos \varphi ={\sin(\theta +\varphi )+\sin(\theta -\varphi )}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eda03d0bff71e76338d847953e6fd2d3d6b0f790" style="border: none; display: inline-block; height: 2.843ex; vertical-align: -0.838ex; width: 37.741ex;" /></td></tr>
<tr><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em 0.4em;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 20.65px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle 2\cos \theta \sin \varphi ={\sin(\theta +\varphi )-\sin(\theta -\varphi )}}</annotation></semantics></math></span><img alt="2\cos \theta \sin \varphi ={\sin(\theta +\varphi )-\sin(\theta -\varphi )}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c9b6ce7925bc17fc1adb25a612ff873613e9aef" style="border: none; display: inline-block; height: 2.843ex; vertical-align: -0.838ex; width: 37.741ex;" /></td></tr>
<tr><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em 0.4em;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 20.65px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle \tan \theta \tan \varphi ={\frac {\cos(\theta -\varphi )-\cos(\theta +\varphi )}{\cos(\theta -\varphi )+\cos(\theta +\varphi )}}}</annotation></semantics></math></span><img alt="\tan \theta \tan \varphi ={\frac {\cos(\theta -\varphi )-\cos(\theta +\varphi )}{\cos(\theta -\varphi )+\cos(\theta +\varphi )}}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b48833e08100bb76de7be938519d1987386ab9f2" style="border: none; display: inline-block; height: 6.509ex; vertical-align: -2.671ex; width: 38.281ex;" /></td></tr>
<tr><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em 0.4em;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 20.65px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\prod _{k=1}^{n}\cos \theta _{k}&={\frac {1}{2^{n}}}\sum _{e\in S}\cos(e_{1}\theta _{1}+\cdots +e_{n}\theta _{n})\\[6pt]&{\text{where }}S=\{1,-1\}^{n}\end{aligned}}}</annotation></semantics></math></span><img alt="{\begin{aligned}\prod _{k=1}^{n}\cos \theta _{k}&={\frac {1}{2^{n}}}\sum _{e\in S}\cos(e_{1}\theta _{1}+\cdots +e_{n}\theta _{n})\\[6pt]&{\text{where }}S=\{1,-1\}^{n}\end{aligned}}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a86e836e8111fd80b7c225d1c4394bf4ac8459e5" style="border: none; display: inline-block; height: 11.676ex; margin-bottom: -0.334ex; vertical-align: -5.004ex; width: 42.701ex;" /></td></tr>
</tbody></table>
<table class="wikitable" style="background-color: white; border-collapse: collapse; border: 1px solid rgb(170, 170, 170); color: black; font-family: sans-serif; font-size: 17.5px; margin: 1em 0px;"><tbody>
<tr><th style="background-color: #f2f2f2; border: 1px solid rgb(170, 170, 170); padding: 0.2em 0.4em; text-align: center;">Sum-to-product</th></tr>
<tr><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em 0.4em;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 20.65px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle \sin \theta \pm \sin \varphi =2\sin \left({\frac {\theta \pm \varphi }{2}}\right)\cos \left({\frac {\theta \mp \varphi }{2}}\right)}</annotation></semantics></math></span><img alt="\sin \theta \pm \sin \varphi =2\sin \left({\frac {\theta \pm \varphi }{2}}\right)\cos \left({\frac {\theta \mp \varphi }{2}}\right)" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/838468c4f91a3bd2a833279c980893762e20931a" style="border: none; display: inline-block; height: 6.176ex; vertical-align: -2.505ex; width: 42.637ex;" /></td></tr>
<tr><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em 0.4em;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 20.65px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle \cos \theta +\cos \varphi =2\cos \left({\frac {\theta +\varphi }{2}}\right)\cos \left({\frac {\theta -\varphi }{2}}\right)}</annotation></semantics></math></span><img alt="\cos \theta +\cos \varphi =2\cos \left({\frac {\theta +\varphi }{2}}\right)\cos \left({\frac {\theta -\varphi }{2}}\right)" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/677cfdcfc861384c210778089fcc67c5a6b67676" style="border: none; display: inline-block; height: 6.176ex; vertical-align: -2.505ex; width: 43.404ex;" /></td></tr>
<tr><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em 0.4em;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 20.65px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle \cos \theta -\cos \varphi =-2\sin \left({\theta +\varphi \over 2}\right)\sin \left({\theta -\varphi \over 2}\right)}</annotation></semantics></math></span><img alt="\cos \theta -\cos \varphi =-2\sin \left({\theta +\varphi \over 2}\right)\sin \left({\theta -\varphi \over 2}\right)" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d0250efa8568bf43684451449d56d82487bdadf0" style="border: none; display: inline-block; height: 6.176ex; vertical-align: -2.505ex; width: 44.711ex;" /></td></tr>
</tbody></table>
<h3 style="background: none rgb(255, 255, 255); border-bottom: none; font-family: sans-serif; font-size: 1.2em; line-height: 1.6; margin: 0.3em 0px 0px; overflow: hidden; padding-bottom: 0px; padding-top: 0.5em;">
<span class="mw-headline" id="Other_related_identities">Other related identities</span></h3>
<ul style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; list-style-image: url("data:image/svg+xml,%3C%3Fxml%20version%3D%221.0%22%20encoding%3D%22UTF-8%22%3F%3E%0A%3Csvg%20xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F2000%2Fsvg%22%20version%3D%221.1%22%20width%3D%225%22%20height%3D%2213%22%3E%0A%3Ccircle%20cx%3D%222.5%22%20cy%3D%229.5%22%20r%3D%222.5%22%20fill%3D%22%2300528c%22%2F%3E%0A%3C%2Fsvg%3E%0A"); margin: 0.3em 0px 0px 1.6em; padding: 0px;">
<li style="margin-bottom: 0.1em;">If <span class="texhtml" style="font-family: "nimbus roman no9 l" , "times new roman" , "times" , serif; font-size: 20.65px; line-height: 1; white-space: nowrap;"><i>x</i> + <i>y</i> + <i>z</i> = <span class="texhtml" style="font-feature-settings: 'lnum' 1, 'tnum' 1, 'kern' 0; font-kerning: none; font-size: 20.65px; font-variant-numeric: lining-nums tabular-nums; line-height: 1;">π</span></span> (half circle), then
<dl style="margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 20.65px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle \sin(2x)+\sin(2y)+\sin(2z)=4\sin x\sin y\sin z.\,}</annotation></semantics></math></span><img alt="\sin(2x)+\sin(2y)+\sin(2z)=4\sin x\sin y\sin z.\," aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cfd6c54c18ce35f99fd06e63f3e9a6a4e0775f8f" style="border: none; display: inline-block; height: 2.843ex; vertical-align: -0.838ex; width: 46.891ex;" /></dd></dl>
</li>
<li style="margin-bottom: 0.1em;"><b>Triple tangent identity:</b> If <span class="texhtml" style="font-family: "nimbus roman no9 l" , "times new roman" , "times" , serif; font-size: 20.65px; line-height: 1; white-space: nowrap;"><i>x</i> + <i>y</i> + <i>z</i> = <span class="texhtml" style="font-feature-settings: 'lnum' 1, 'tnum' 1, 'kern' 0; font-kerning: none; font-size: 20.65px; font-variant-numeric: lining-nums tabular-nums; line-height: 1;">π</span></span> (half circle), then
<dl style="margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 20.65px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle \tan x+\tan y+\tan z=\tan x\tan y\tan z.\,}</annotation></semantics></math></span><img alt="{\displaystyle \tan x+\tan y+\tan z=\tan x\tan y\tan z.\,}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d4b40fbd090d0b49a34c40def60f1ee1788e4b39" style="border: none; display: inline-block; height: 2.343ex; vertical-align: -0.671ex; width: 40.507ex;" /></dd><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;">In particular, the formula holds when <span class="texhtml mvar" style="font-family: "nimbus roman no9 l" , "times new roman" , "times" , serif; font-size: 20.65px; font-style: italic; line-height: 1; white-space: nowrap;">x</span>, <span class="texhtml mvar" style="font-family: "nimbus roman no9 l" , "times new roman" , "times" , serif; font-size: 20.65px; font-style: italic; line-height: 1; white-space: nowrap;">y</span>, and <span class="texhtml mvar" style="font-family: "nimbus roman no9 l" , "times new roman" , "times" , serif; font-size: 20.65px; font-style: italic; line-height: 1; white-space: nowrap;">z</span> are the three angles of any triangle.</dd><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;">(If any of <span class="texhtml" style="font-family: "nimbus roman no9 l" , "times new roman" , "times" , serif; font-size: 20.65px; line-height: 1; white-space: nowrap;"><i>x</i>, <i>y</i>, <i>z</i></span> is a right angle, one should take both sides to be <span class="texhtml" style="font-family: "nimbus roman no9 l" , "times new roman" , "times" , serif; font-size: 20.65px; line-height: 1; white-space: nowrap;">∞</span>. This is neither <span class="texhtml" style="font-family: "nimbus roman no9 l" , "times new roman" , "times" , serif; font-size: 20.65px; line-height: 1; white-space: nowrap;">+∞</span> nor <span class="texhtml" style="font-family: "nimbus roman no9 l" , "times new roman" , "times" , serif; font-size: 20.65px; line-height: 1; white-space: nowrap;">−∞</span>; for present purposes it makes sense to add just one point at infinity to the real line, that is approached by <span class="texhtml" style="font-family: "nimbus roman no9 l" , "times new roman" , "times" , serif; font-size: 20.65px; line-height: 1; white-space: nowrap;">tan <i>θ</i></span> as <span class="texhtml" style="font-family: "nimbus roman no9 l" , "times new roman" , "times" , serif; font-size: 20.65px; line-height: 1; white-space: nowrap;">tan <i>θ</i></span> either increases through positive values or decreases through negative values. This is a one-point compactification of the real line.)</dd></dl>
</li>
<li style="margin-bottom: 0.1em;"><b>Triple cotangent identity:</b> If <span class="texhtml" style="font-family: "nimbus roman no9 l" , "times new roman" , "times" , serif; font-size: 20.65px; line-height: 1; white-space: nowrap;"><i>x</i> + <i>y</i> + <i>z</i> = <span class="sfrac nowrap" style="display: inline-block; font-size: 17.5525px; text-align: center; vertical-align: -0.5em;"><span style="display: block; line-height: 1em; margin: 0px 0.1em;"><span class="texhtml" style="font-feature-settings: 'lnum' 1, 'tnum' 1, 'kern' 0; font-kerning: none; font-size: 17.5525px; font-variant-numeric: lining-nums tabular-nums; line-height: 1;">π</span></span><span class="visualhide" style="height: 1px; left: -10000px; overflow: hidden; position: absolute; top: auto; width: 1px;">/</span><span style="border-top: 1px solid; display: block; line-height: 1em; margin: 0px 0.1em;">2</span></span></span> (right angle or quarter circle), then
<dl style="margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 20.65px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle \cot x+\cot y+\cot z=\cot x\cot y\cot z.\,}</annotation></semantics></math></span><img alt="{\displaystyle \cot x+\cot y+\cot z=\cot x\cot y\cot z.\,}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a509b32e9c598b2535599029009d50cbc54e8985" style="border: none; display: inline-block; height: 2.343ex; vertical-align: -0.671ex; width: 38.946ex;" /></dd></dl>
</li>
</ul>
<h3 style="background: none rgb(255, 255, 255); border-bottom: none; font-family: sans-serif; font-size: 1.2em; line-height: 1.6; margin: 0.3em 0px 0px; overflow: hidden; padding-bottom: 0px; padding-top: 0.5em;">
<span class="mw-headline" id="Hermite.27s_cotangent_identity">Hermite's cotangent identity</span></h3>
<div class="hatnote" role="note" style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; font-style: italic; margin-bottom: 0.5em; padding-left: 1.6em;">
<br /></div>
<div style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; line-height: inherit; margin-bottom: 0.5em; margin-top: 0.5em;">
Charles Hermite demonstrated the following identity. Suppose <span class="texhtml" style="font-family: "nimbus roman no9 l" , "times new roman" , "times" , serif; font-size: 20.65px; line-height: 1; white-space: nowrap;"><i>a</i><sub style="font-size: 16.52px; line-height: 1;">1</sub>, ..., <i>a</i><sub style="font-size: 16.52px; line-height: 1;"><i>n</i></sub></span> are complex numbers, no two of which differ by an integer multiple of <span class="texhtml" style="font-family: "nimbus roman no9 l" , "times new roman" , "times" , serif; font-size: 20.65px; line-height: 1; white-space: nowrap;">π</span>. Let</div>
<dl style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 20.65px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle A_{n,k}=\prod _{\begin{smallmatrix}1\leq j\leq n\\j\neq k\end{smallmatrix}}\cot(a_{k}-a_{j})}</annotation></semantics></math></span><img alt="A_{n,k}=\prod _{\begin{smallmatrix}1\leq j\leq n\\j\neq k\end{smallmatrix}}\cot(a_{k}-a_{j})" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add4d3e70d16463f3a6941af196afc1e6c337ad1" style="border: none; display: inline-block; height: 8.009ex; vertical-align: -5.505ex; width: 25.941ex;" /></dd></dl>
<div style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; line-height: inherit; margin-bottom: 0.5em; margin-top: 0.5em;">
(in particular, <span class="texhtml" style="font-family: "nimbus roman no9 l" , "times new roman" , "times" , serif; font-size: 20.65px; line-height: 1; white-space: nowrap;"><i>A</i><sub style="font-size: 16.52px; line-height: 1;">1,1</sub></span>, being an empty product, is 1). Then</div>
<dl style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 20.65px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle \cot(z-a_{1})\cdots \cot(z-a_{n})=\cos {\frac {n\pi }{2}}+\sum _{k=1}^{n}A_{n,k}\cot(z-a_{k}).}</annotation></semantics></math></span><img alt="\cot(z-a_{1})\cdots \cot(z-a_{n})=\cos {\frac {n\pi }{2}}+\sum _{k=1}^{n}A_{n,k}\cot(z-a_{k})." aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9db60873e93178b3b00e9cdea04ca45727748be1" style="border: none; display: inline-block; height: 7.176ex; vertical-align: -3.171ex; width: 59.522ex;" /></dd></dl>
<div style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; line-height: inherit; margin-bottom: 0.5em; margin-top: 0.5em;">
The simplest non-trivial example is the case <span class="texhtml" style="font-family: "nimbus roman no9 l" , "times new roman" , "times" , serif; font-size: 20.65px; line-height: 1; white-space: nowrap;"><i>n</i> = 2</span>:</div>
<dl style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 20.65px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle \cot(z-a_{1})\cot(z-a_{2})=-1+\cot(a_{1}-a_{2})\cot(z-a_{1})+\cot(a_{2}-a_{1})\cot(z-a_{2}).}</annotation></semantics></math></span><img alt="\cot(z-a_{1})\cot(z-a_{2})=-1+\cot(a_{1}-a_{2})\cot(z-a_{1})+\cot(a_{2}-a_{1})\cot(z-a_{2})." aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b24df9b1cd3d225a90be137a1e195b9314bd873c" style="border: none; display: inline-block; height: 2.843ex; vertical-align: -0.838ex; width: 83.302ex;" /></dd></dl>
<h3 style="background: none rgb(255, 255, 255); border-bottom: none; font-family: sans-serif; font-size: 1.2em; line-height: 1.6; margin: 0.3em 0px 0px; overflow: hidden; padding-bottom: 0px; padding-top: 0.5em;">
<span class="mw-headline" id="Ptolemy.27s_theorem">Ptolemy's theorem</span></h3>
<div class="hatnote" role="note" style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; font-style: italic; margin-bottom: 0.5em; padding-left: 1.6em;">
<br /></div>
<div style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; line-height: inherit; margin-bottom: 0.5em; margin-top: 0.5em;">
Ptolemy's theorem can be expressed in the language of modern trigonometry as:</div>
<dl style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;">If <span class="texhtml" style="font-family: "nimbus roman no9 l" , "times new roman" , "times" , serif; font-size: 20.65px; line-height: 1; white-space: nowrap;"><i>w</i> + <i>x</i> + <i>y</i> + <i>z</i> = <span class="texhtml" style="font-feature-settings: 'lnum' 1, 'tnum' 1, 'kern' 0; font-kerning: none; font-size: 20.65px; font-variant-numeric: lining-nums tabular-nums; line-height: 1;">π</span></span>, then:
<br />
<dl style="margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 20.65px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\sin(w+x)\sin(x+y)&=\sin(x+y)\sin(y+z)&{\text{(trivial)}}\\&=\sin(y+z)\sin(z+w)&{\text{(trivial)}}\\&=\sin(z+w)\sin(w+x)&{\text{(trivial)}}\\&=\sin w\sin y+\sin x\sin z.&{\text{(significant)}}\end{aligned}}}</annotation></semantics></math></span><img alt="{\displaystyle {\begin{aligned}\sin(w+x)\sin(x+y)&=\sin(x+y)\sin(y+z)&{\text{(trivial)}}\\&=\sin(y+z)\sin(z+w)&{\text{(trivial)}}\\&=\sin(z+w)\sin(w+x)&{\text{(trivial)}}\\&=\sin w\sin y+\sin x\sin z.&{\text{(significant)}}\end{aligned}}}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d9b7e3a838b0df774622005150cabacff7c15d56" style="border: none; display: inline-block; height: 12.509ex; vertical-align: -5.671ex; width: 64.568ex;" /></dd></dl>
</dd></dl>
<div style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; line-height: inherit; margin-bottom: 0.5em; margin-top: 0.5em;">
(The first three equalities are trivial rearrangements; the fourth is the substance of this identity.)</div>
Thanh Ahttp://www.blogger.com/profile/16564310847356090654noreply@blogger.comtag:blogger.com,1999:blog-574185785118190739.post-64617038665169316882016-11-21T02:27:00.000-08:002016-12-02T02:41:35.486-08:0011. Linear combinations<div style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; line-height: inherit; margin-bottom: 0.5em; margin-top: 0.5em;">
For some purposes it is important to know that any linear combination of sine waves of the same period or frequency but different phase shifts is also a sine wave with the same period or frequency, but a different phase shift. This is useful in sinusoid data fitting, because the measured or observed data are linearly related to the <span class="texhtml mvar" style="font-family: "nimbus roman no9 l" , "times new roman" , "times" , serif; font-size: 20.65px; font-style: italic; line-height: 1; white-space: nowrap;">a</span> and <span class="texhtml mvar" style="font-family: "nimbus roman no9 l" , "times new roman" , "times" , serif; font-size: 20.65px; font-style: italic; line-height: 1; white-space: nowrap;">b</span> unknowns of the in-phase and quadrature components basis below, resulting in a simpler Jacobian, compared to that of <span class="texhtml mvar" style="font-family: "nimbus roman no9 l" , "times new roman" , "times" , serif; font-size: 20.65px; font-style: italic; line-height: 1; white-space: nowrap;">c</span> and <span class="texhtml mvar" style="font-family: "nimbus roman no9 l" , "times new roman" , "times" , serif; font-size: 20.65px; font-style: italic; line-height: 1; white-space: nowrap;">φ</span>.</div>
<h3 style="background: none rgb(255, 255, 255); border-bottom: none; font-family: sans-serif; font-size: 1.2em; line-height: 1.6; margin: 0.3em 0px 0px; overflow: hidden; padding-bottom: 0px; padding-top: 0.5em;">
<span class="mw-headline" id="Sine_and_cosine">Sine and cosine</span></h3>
<div style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; line-height: inherit; margin-bottom: 0.5em; margin-top: 0.5em;">
In the case of a non-zero linear combination of a sine and cosine wave (which is just a sine wave with a phase shift of <span class="sfrac nowrap" style="display: inline-block; font-size: 14.875px; text-align: center; vertical-align: -0.5em; white-space: nowrap;"><span style="display: block; line-height: 1em; margin: 0px 0.1em;"><span class="texhtml" style="font-family: "nimbus roman no9 l" , "times new roman" , "times" , serif; font-size: 17.5525px; line-height: 1;">π</span></span><span class="visualhide" style="height: 1px; left: -10000px; overflow: hidden; position: absolute; top: auto; width: 1px;">/</span><span style="border-top: 1px solid; display: block; line-height: 1em; margin: 0px 0.1em;">2</span></span>), we have</div>
<dl style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 20.65px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle a\sin x+b\cos x=c\cdot \sin(x+\varphi )\,}</annotation></semantics></math></span><img alt="a\sin x+b\cos x=c\cdot \sin(x+\varphi )\," aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aba98b9f2b5a622d98065385f0856d516bd829fc" style="border: none; display: inline-block; height: 2.843ex; vertical-align: -0.838ex; width: 31.999ex;" /></dd></dl>
<div style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; line-height: inherit; margin-bottom: 0.5em; margin-top: 0.5em;">
where</div>
<dl style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 20.65px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle c={\sqrt {a^{2}+b^{2}}},\,}</annotation></semantics></math></span><img alt="c={\sqrt {a^{2}+b^{2}}},\," aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/30ffa7f6ad69897f8d8cb481d9c309224979aa6a" style="border: none; display: inline-block; height: 3.509ex; vertical-align: -0.838ex; width: 14.727ex;" /></dd></dl>
<div style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; line-height: inherit; margin-bottom: 0.5em; margin-top: 0.5em;">
and (using the atan2 function)</div>
<dl style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 20.65px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle \varphi =\operatorname {atan2} \left(b,a\right).}</annotation></semantics></math></span><img alt="\varphi =\operatorname {atan2} \left(b,a\right)." aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82f4cbfd53cdb9e82120b9b8bbaa25dcb3af941" style="border: none; display: inline-block; height: 2.843ex; vertical-align: -0.838ex; width: 16.156ex;" /></dd></dl>
<h3 style="background: none rgb(255, 255, 255); border-bottom: none; font-family: sans-serif; font-size: 1.2em; line-height: 1.6; margin: 0.3em 0px 0px; overflow: hidden; padding-bottom: 0px; padding-top: 0.5em;">
<span class="mw-headline" id="Arbitrary_phase_shift">Arbitrary phase shift</span></h3>
<div style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; line-height: inherit; margin-bottom: 0.5em; margin-top: 0.5em;">
More generally, for an arbitrary phase shift, we have</div>
<dl style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 20.65px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle a\sin x+b\sin(x+\theta )=c\sin(x+\varphi )\,}</annotation></semantics></math></span><img alt="a\sin x+b\sin(x+\theta )=c\sin(x+\varphi )\," aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/79789e2176370a37c1f85dddffefb4736cb02d55" style="border: none; display: inline-block; height: 2.843ex; vertical-align: -0.838ex; width: 35.836ex;" /></dd></dl>
<div style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; line-height: inherit; margin-bottom: 0.5em; margin-top: 0.5em;">
where</div>
<dl style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 20.65px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle c={\sqrt {a^{2}+b^{2}+2ab\cos \theta }},\,}</annotation></semantics></math></span><img alt="c={\sqrt {a^{2}+b^{2}+2ab\cos \theta }},\," aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bb0003596acd93bf31d08dc6987ad4d79d2e0a12" style="border: none; display: inline-block; height: 3.509ex; vertical-align: -0.838ex; width: 26.017ex;" /></dd></dl>
<div style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; line-height: inherit; margin-bottom: 0.5em; margin-top: 0.5em;">
and</div>
<dl style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 20.65px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle \varphi =\operatorname {atan} {\frac {b\,\sin \theta }{a+b\cos \theta }}.}</annotation></semantics></math></span><img alt="{\displaystyle \varphi =\operatorname {atan} {\frac {b\,\sin \theta }{a+b\cos \theta }}.}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a1247a1c7e768e6dcf3d5d004645789f64ee289" style="border: none; display: inline-block; height: 5.676ex; vertical-align: -2.171ex; width: 21.201ex;" /></dd></dl>
<h3 style="background: none rgb(255, 255, 255); border-bottom: none; font-family: sans-serif; font-size: 1.2em; line-height: 1.6; margin: 0.3em 0px 0px; overflow: hidden; padding-bottom: 0px; padding-top: 0.5em;">
<span class="mw-headline" id="More_than_two_sinusoids">More than two sinusoids</span></h3>
<div style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; line-height: inherit; margin-bottom: 0.5em; margin-top: 0.5em;">
The general case reads<sup class="noprint Inline-Template Template-Fact" style="font-size: 14px; line-height: 1; white-space: nowrap;">[<i><span title="This claim needs references to reliable sources. (October 2012)">citation needed</span></i>]</sup></div>
<dl style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 20.65px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle \sum _{i}a_{i}\sin(x+\theta _{i})=a\sin(x+\theta ),}</annotation></semantics></math></span><img alt="\sum _{i}a_{i}\sin(x+\theta _{i})=a\sin(x+\theta )," aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e0f58cc33fbd8f27ae1221a7cab1d60a49f6986" style="border: none; display: inline-block; height: 5.509ex; vertical-align: -3.005ex; width: 32.406ex;" /></dd></dl>
<div style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; line-height: inherit; margin-bottom: 0.5em; margin-top: 0.5em;">
where</div>
<dl style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 20.65px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle a^{2}=\sum _{i,j}a_{i}a_{j}\cos(\theta _{i}-\theta _{j})}</annotation></semantics></math></span><img alt="a^{2}=\sum _{i,j}a_{i}a_{j}\cos(\theta _{i}-\theta _{j})" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54e2a23b18ed4268ef834d3dad3dbf03a19b176b" style="border: none; display: inline-block; height: 5.843ex; vertical-align: -3.338ex; width: 25.505ex;" /></dd></dl>
<div style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; line-height: inherit; margin-bottom: 0.5em; margin-top: 0.5em;">
and</div>
<dl style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 20.65px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle \tan \theta ={\frac {\sum _{i}a_{i}\sin \theta _{i}}{\sum _{i}a_{i}\cos \theta _{i}}}.}</annotation></semantics></math></span><img alt="\tan \theta ={\frac {\sum _{i}a_{i}\sin \theta _{i}}{\sum _{i}a_{i}\cos \theta _{i}}}." aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/108579447fd00d2a5d3f4b97b76f484ccd13a9a0" style="border: none; display: inline-block; height: 6.843ex; vertical-align: -2.838ex; width: 21.012ex;" /></dd></dl>
<div style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; line-height: inherit; margin-bottom: 0.5em; margin-top: 0.5em;">
See also Phasor addition.</div>
Thanh Ahttp://www.blogger.com/profile/16564310847356090654noreply@blogger.comtag:blogger.com,1999:blog-574185785118190739.post-85729811671168335032016-11-20T02:28:00.000-08:002018-10-19T19:25:34.884-07:0012. Lagrange's trigonometric identities<div dir="ltr" style="text-align: left;" trbidi="on">
<div style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; line-height: inherit; margin-bottom: 0.5em; margin-top: 0.5em;">
These identities, named after Joseph Louis Lagrange<img alt="" border="0" height="0" id="amznPsBmPixel_5915627" src="https://ir-na.amazon-adsystem.com/e/ir?source=bk&t=learnjavaporgramming-20&bm-id=default&l=ktl&linkId=f199a4ebca009393e35b5fc1e7db171a&_cb=1539529002105" style="border: none !important; height: 0px !important; margin: 0px !important; padding: 0px !important; width: 0px !important;" width="0" />, are:<sup class="reference" id="cite_ref-32" style="font-size: 14px; line-height: 1; unicode-bidi: isolate; white-space: nowrap;"></sup></div>
<dl style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><div style="text-align: center;">
<span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 20.65px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\sum _{n=1}^{N}\sin(n\theta )&={\frac {1}{2}}\cot {\frac {\theta }{2}}-{\frac {\cos \left(\left(N+{\frac {1}{2}}\right)\theta \right)}{2\sin \left({\frac {\theta }{2}}\right)}}\\\sum _{n=1}^{N}\cos(n\theta )&=-{\frac {1}{2}}+{\frac {\sin \left(\left(N+{\frac {1}{2}}\right)\theta \right)}{2\sin \left({\frac {\theta }{2}}\right)}}\end{aligned}}}</annotation></semantics></math></span><a amzn-ps-bm-asin="GATEWAY" class="amzn_ps_bm_il" data-amzn-link-id="6661eed4b9fd0d1a2882c333afac1103" data-amzn-ps-bm-keyword="{\displaystyle {\begin{aligned}\sum _{n=1}^{N}\sin(n\theta )&={\frac {1}{2}}\cot {\frac {\theta }{2}}-{\frac {\cos \left(\left(N+{\frac {1}{2}}\right)\theta \right)}{2\sin \left({\frac {\theta }{2}}\right)}}\\\sum _{n=1}^{N}\cos(n\theta )&=-{\frac {1}{2}}+{\frac {\sin \left(\left(N+{\frac {1}{2}}\right)\theta \right)}{2\sin \left({\frac {\theta }{2}}\right)}}\end{aligned}}}" href="http://amazon.com/ref=as_li_bk_ia/?tag=learnjavaporgramming-20&linkId=6661eed4b9fd0d1a2882c333afac1103&linkCode=kia" id="amznPsBmLink_1061366" rel="nofollow" target="_blank"><img alt="{\displaystyle {\begin{aligned}\sum _{n=1}^{N}\sin(n\theta )&={\frac {1}{2}}\cot {\frac {\theta }{2}}-{\frac {\cos \left(\left(N+{\frac {1}{2}}\right)\theta \right)}{2\sin \left({\frac {\theta }{2}}\right)}}\\\sum _{n=1}^{N}\cos(n\theta )&=-{\frac {1}{2}}+{\frac {\sin \left(\left(N+{\frac {1}{2}}\right)\theta \right)}{2\sin \left({\frac {\theta }{2}}\right)}}\end{aligned}}}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/11fa89e9cf41dd5e296f66e9286f3283e9c7c43c" style="border: none; display: inline-block; height: 20.509ex; vertical-align: -9.671ex; width: 43.515ex;" /></a><img alt="" border="0" height="0" id="amznPsBmPixel_1061366" src="https://ir-na.amazon-adsystem.com/e/ir?source=bk&t=learnjavaporgramming-20&bm-id=default&l=kia&linkId=6661eed4b9fd0d1a2882c333afac1103&_cb=1539528991040" style="border: none !important; height: 0px !important; margin: 0px !important; padding: 0px !important; width: 0px !important;" width="0" /></div>
</dd></dl>
<div style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; line-height: inherit; margin-bottom: 0.5em; margin-top: 0.5em;">
A related function is the following function of <span class="texhtml mvar" style="font-family: "nimbus roman no9 l" , "times new roman" , "times" , serif; font-size: 20.65px; font-style: italic; line-height: 1; white-space: nowrap;">x</span>, called the Dirichlet<img alt="" border="0" height="0" id="amznPsBmPixel_2825905" src="https://ir-na.amazon-adsystem.com/e/ir?source=bk&t=learnjavaporgramming-20&bm-id=default&l=ktl&linkId=8ae78eb43e6d404e2aba9dba6535ce70&_cb=1539528982743" style="border: none !important; height: 0px !important; margin: 0px !important; padding: 0px !important; width: 0px !important;" width="0" /> kernel.</div>
<dl style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 20.65px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle 1+2\cos x+2\cos(2x)+2\cos(3x)+\cdots +2\cos(nx)={\frac {\sin \left(\left(n+{\frac {1}{2}}\right)x\right)}{\sin(x/2)}}.}</annotation></semantics></math></span><img alt="1+2\cos x+2\cos(2x)+2\cos(3x)+\cdots +2\cos(nx)={\frac {\sin \left(\left(n+{\frac {1}{2}}\right)x\right)}{\sin(x/2)}}." aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/acfaf0395fd5a4483962461489b4b2aa3afb9402" style="border: none; display: inline-block; height: 8.343ex; vertical-align: -2.671ex; width: 72.691ex;" /></dd></dl>
</div>
Thanh Ahttp://www.blogger.com/profile/16564310847356090654noreply@blogger.comtag:blogger.com,1999:blog-574185785118190739.post-87803326530487067512016-11-19T02:29:00.000-08:002018-10-19T19:25:42.307-07:0013. Other sums of trigonometric functions<div dir="ltr" style="text-align: left;" trbidi="on">
<div style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; line-height: inherit; margin-bottom: 0.5em; margin-top: 0.5em;">
Sum of sines and cosines with arguments in arithmetic progression:<sup class="reference" id="cite_ref-33" style="font-size: 14px; line-height: 1; unicode-bidi: isolate; white-space: nowrap;"><a href="https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-33" style="background: none; color: #0b0080; text-decoration: none;">]</a></sup> if <span class="texhtml" style="font-family: "nimbus roman no9 l" , "times new roman" , "times" , serif; font-size: 20.65px; line-height: 1; white-space: nowrap;"><i>α</i> ≠ 0</span>, then</div>
<dl style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 20.65px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}&\sin \varphi +\sin(\varphi +\alpha )+\sin(\varphi +2\alpha )+\cdots \\[8pt]&{}\qquad \qquad \cdots +\sin(\varphi +n\alpha )={\frac {\sin {\frac {(n+1)\alpha }{2}}\cdot \sin(\varphi +{\frac {n\alpha }{2}})}{\sin {\frac {\alpha }{2}}}}\quad {\text{and}}\\[10pt]&\cos \varphi +\cos(\varphi +\alpha )+\cos(\varphi +2\alpha )+\cdots \\[8pt]&{}\qquad \qquad \cdots +\cos(\varphi +n\alpha )={\frac {\sin {\frac {(n+1)\alpha }{2}}\cdot \cos(\varphi +{\frac {n\alpha }{2}})}{\sin {\frac {\alpha }{2}}}}.\end{aligned}}}</annotation></semantics></math></span><img alt="{\begin{aligned}&\sin \varphi +\sin(\varphi +\alpha )+\sin(\varphi +2\alpha )+\cdots \\[8pt]&{}\qquad \qquad \cdots +\sin(\varphi +n\alpha )={\frac {\sin {\frac {(n+1)\alpha }{2}}\cdot \sin(\varphi +{\frac {n\alpha }{2}})}{\sin {\frac {\alpha }{2}}}}\quad {\text{and}}\\[10pt]&\cos \varphi +\cos(\varphi +\alpha )+\cos(\varphi +2\alpha )+\cdots \\[8pt]&{}\qquad \qquad \cdots +\cos(\varphi +n\alpha )={\frac {\sin {\frac {(n+1)\alpha }{2}}\cdot \cos(\varphi +{\frac {n\alpha }{2}})}{\sin {\frac {\alpha }{2}}}}.\end{aligned}}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b43456ae300839ced3594491868e6125c7b2e3d" style="border: none; display: inline-block; height: 29.176ex; vertical-align: -14.005ex; width: 61.312ex;" /></dd></dl>
<div style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; line-height: inherit; margin-bottom: 0.5em; margin-top: 0.5em;">
For any <span class="texhtml mvar" style="font-family: "nimbus roman no9 l" , "times new roman" , "times" , serif; font-size: 20.65px; font-style: italic; line-height: 1; white-space: nowrap;">a</span> and <span class="texhtml mvar" style="font-family: "nimbus roman no9 l" , "times new roman" , "times" , serif; font-size: 20.65px; font-style: italic; line-height: 1; white-space: nowrap;">b</span>:</div>
<dl style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 20.65px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle a\cos x+b\sin x={\sqrt {a^{2}+b^{2}}}\cos {\big (}x-\operatorname {atan2} \,(b,a){\big )}\;}</annotation></semantics></math></span><img alt="{\displaystyle a\cos x+b\sin x={\sqrt {a^{2}+b^{2}}}\cos {\big (}x-\operatorname {atan2} \,(b,a){\big )}\;}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a7f7166f0f1c2330000ca1296eca34dfd4dac3b9" style="border: none; display: inline-block; height: 3.843ex; vertical-align: -1.171ex; width: 50.173ex;" /></dd></dl>
<div style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; line-height: inherit; margin-bottom: 0.5em; margin-top: 0.5em;">
where <span class="texhtml" style="font-family: "nimbus roman no9 l" , "times new roman" , "times" , serif; font-size: 20.65px; line-height: 1; white-space: nowrap;">atan2(<i>y</i>, <i>x</i>)</span> is the generalization of <span class="texhtml" style="font-family: "nimbus roman no9 l" , "times new roman" , "times" , serif; font-size: 20.65px; line-height: 1; white-space: nowrap;">arctan(<i><span class="sfrac nowrap" style="display: inline-block; font-size: 17.5525px; text-align: center; vertical-align: -0.5em;"><span style="display: block; line-height: 1em; margin: 0px 0.1em;">y</span><span class="visualhide" style="height: 1px; left: -10000px; overflow: hidden; position: absolute; top: auto; width: 1px;">/</span><span style="border-top: 1px solid; display: block; line-height: 1em; margin: 0px 0.1em;">x</span></span></i>)</span> that covers the entire circular range.</div>
<dl style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 20.65px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle \sec x\pm \tan x=\tan \left({\frac {\pi }{4}}\pm {\frac {x}{2}}\right).}</annotation></semantics></math></span><img alt="\sec x\pm \tan x=\tan \left({\frac {\pi }{4}}\pm {\frac {x}{2}}\right)." aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd5930c8e9ffe4e059640a99b1e1b68d23905be4" style="border: none; display: inline-block; height: 4.843ex; vertical-align: -1.838ex; width: 29.868ex;" /></dd></dl>
<div style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; line-height: inherit; margin-bottom: 0.5em; margin-top: 0.5em;">
The above identity is sometimes convenient to know when thinking about the Gudermannian function, which relates the circular and hyperbolic trigonometric functions without resorting to complex numbers.</div>
<div style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; line-height: inherit; margin-bottom: 0.5em; margin-top: 0.5em;">
If <span class="texhtml mvar" style="font-family: "nimbus roman no9 l" , "times new roman" , "times" , serif; font-size: 20.65px; font-style: italic; line-height: 1; white-space: nowrap;">x</span>, <span class="texhtml mvar" style="font-family: "nimbus roman no9 l" , "times new roman" , "times" , serif; font-size: 20.65px; font-style: italic; line-height: 1; white-space: nowrap;">y</span>, and <span class="texhtml mvar" style="font-family: "nimbus roman no9 l" , "times new roman" , "times" , serif; font-size: 20.65px; font-style: italic; line-height: 1; white-space: nowrap;">z</span> are the three angles of any triangle, i.e. if <span class="texhtml" style="font-family: "nimbus roman no9 l" , "times new roman" , "times" , serif; font-size: 20.65px; line-height: 1; white-space: nowrap;"><i>x</i> + <i>y</i> + <i>z</i> = π</span>, then</div>
<dl style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 20.65px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle \cot x\cot y+\cot y\cot z+\cot z\cot x=1.\,}</annotation></semantics></math></span><img alt="\cot x\cot y+\cot y\cot z+\cot z\cot x=1.\," aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8222a98468c1d74c163f1cbdf39007ecb310e80" style="border: none; display: inline-block; height: 2.509ex; vertical-align: -0.671ex; width: 40.506ex;" /></dd></dl>
</div>
Thanh Ahttp://www.blogger.com/profile/16564310847356090654noreply@blogger.comtag:blogger.com,1999:blog-574185785118190739.post-64719214475939309452016-11-18T02:30:00.000-08:002016-12-02T02:41:56.576-08:0014. Certain linear fractional transformations<div style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; line-height: inherit; margin-bottom: 0.5em; margin-top: 0.5em;">
If <span class="texhtml" style="font-family: "nimbus roman no9 l" , "times new roman" , "times" , serif; font-size: 20.65px; line-height: 1; white-space: nowrap;"><i>f</i>(<i>x</i>)</span> is given by the linear fractional transformation</div>
<dl style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 20.65px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle f(x)={\frac {(\cos \alpha )x-\sin \alpha }{(\sin \alpha )x+\cos \alpha }},}</annotation></semantics></math></span><img alt="f(x)={\frac {(\cos \alpha )x-\sin \alpha }{(\sin \alpha )x+\cos \alpha }}," aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cbc62585cae3c00794e045d021d6bb010afee79b" style="border: none; display: inline-block; height: 6.509ex; vertical-align: -2.671ex; width: 24.883ex;" /></dd></dl>
<div style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; line-height: inherit; margin-bottom: 0.5em; margin-top: 0.5em;">
and similarly</div>
<dl style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 20.65px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle g(x)={\frac {(\cos \beta )x-\sin \beta }{(\sin \beta )x+\cos \beta }},}</annotation></semantics></math></span><img alt="g(x)={\frac {(\cos \beta )x-\sin \beta }{(\sin \beta )x+\cos \beta }}," aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dd4ca80eca844d6808928b5c9c6bbfeec9d09f53" style="border: none; display: inline-block; height: 6.509ex; vertical-align: -2.671ex; width: 24.409ex;" /></dd></dl>
<div style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; line-height: inherit; margin-bottom: 0.5em; margin-top: 0.5em;">
then</div>
<dl style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 20.65px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle f{\big (}g(x){\big )}=g{\big (}f(x){\big )}={\frac {{\big (}\cos(\alpha +\beta ){\big )}x-\sin(\alpha +\beta )}{{\big (}\sin(\alpha +\beta ){\big )}x+\cos(\alpha +\beta )}}.}</annotation></semantics></math></span><img alt="{\displaystyle f{\big (}g(x){\big )}=g{\big (}f(x){\big )}={\frac {{\big (}\cos(\alpha +\beta ){\big )}x-\sin(\alpha +\beta )}{{\big (}\sin(\alpha +\beta ){\big )}x+\cos(\alpha +\beta )}}.}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/733b8bbdcbd48e17c233d2afc1a1aebe527d3e84" style="border: none; display: inline-block; height: 7.509ex; vertical-align: -3.171ex; width: 50.986ex;" /></dd></dl>
<div style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; line-height: inherit; margin-bottom: 0.5em; margin-top: 0.5em;">
More tersely stated, if for all <span class="texhtml mvar" style="font-family: "nimbus roman no9 l" , "times new roman" , "times" , serif; font-size: 20.65px; font-style: italic; line-height: 1; white-space: nowrap;">α</span> we let <span class="texhtml mvar" style="font-family: "nimbus roman no9 l" , "times new roman" , "times" , serif; font-size: 20.65px; font-style: italic; line-height: 1; white-space: nowrap;">f<sub style="font-size: 16.52px; line-height: 1;">α</sub></span> be what we called <span class="texhtml mvar" style="font-family: "nimbus roman no9 l" , "times new roman" , "times" , serif; font-size: 20.65px; font-style: italic; line-height: 1; white-space: nowrap;">f</span> above, then</div>
<dl style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 20.65px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle f_{\alpha }\circ f_{\beta }=f_{\alpha +\beta }.\,}</annotation></semantics></math></span><img alt="f_{\alpha }\circ f_{\beta }=f_{\alpha +\beta }.\," aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c0e9bfaa32426a4ca6be9241b995716867550771" style="border: none; display: inline-block; height: 2.843ex; vertical-align: -1.005ex; width: 15.807ex;" /></dd></dl>
<div style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; line-height: inherit; margin-bottom: 0.5em; margin-top: 0.5em;">
If <span class="texhtml mvar" style="font-family: "nimbus roman no9 l" , "times new roman" , "times" , serif; font-size: 20.65px; font-style: italic; line-height: 1; white-space: nowrap;">x</span> is the slope of a line, then <span class="texhtml" style="font-family: "nimbus roman no9 l" , "times new roman" , "times" , serif; font-size: 20.65px; line-height: 1; white-space: nowrap;"><i>f</i>(<i>x</i>)</span> is the slope of its rotation through an angle of <span class="texhtml" style="font-family: "nimbus roman no9 l" , "times new roman" , "times" , serif; font-size: 20.65px; line-height: 1; white-space: nowrap;">−<i>α</i></span>.</div>
Thanh Ahttp://www.blogger.com/profile/16564310847356090654noreply@blogger.comtag:blogger.com,1999:blog-574185785118190739.post-48324475820447854432016-11-17T02:30:00.000-08:002018-10-19T19:26:02.179-07:0015. Inverse trigonometric functions<div dir="ltr" style="text-align: left;" trbidi="on">
<dl style="margin-bottom: 0.5em; margin-top: 0.2em;"><span style="font-size: large;"><b> Inverse trigonometric functions<img alt="" border="0" height="0" id="amznPsBmPixel_9094576" src="https://ir-na.amazon-adsystem.com/e/ir?source=bk&t=learnjavaporgramming-20&bm-id=default&l=ktl&linkId=6713044ae337ae2e7e8511048e159208&_cb=1539529115583" style="border: none !important; height: 0px !important; margin: 0px !important; padding: 0px !important; width: 0px !important;" width="0" /></b></span><dd style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><div style="text-align: center;">
<a amzn-ps-bm-asin="GATEWAY" class="amzn_ps_bm_il" data-amzn-link-id="b5b1b7f92efe859bed6aa15848812634" data-amzn-ps-bm-keyword="{\displaystyle {\begin{aligned}\arcsin x+\arccos x&={\tfrac {\pi }{2}}\\\arctan x+\operatorname {arccot} x&={\tfrac {\pi }{2}}\\\arctan x+\arctan {\frac {1}{x}}&={\begin{cases}{\tfrac {\pi }{2}},&{\text{if }}x>0\\-{\tfrac {\pi }{2}},&{\text{if }}x<0\end{cases}}\end{aligned}}}" href="http://amazon.com/ref=as_li_bk_ia/?tag=learnjavaporgramming-20&linkId=b5b1b7f92efe859bed6aa15848812634&linkCode=kia" id="amznPsBmLink_142516" rel="nofollow" target="_blank"><img alt="{\displaystyle {\begin{aligned}\arcsin x+\arccos x&={\tfrac {\pi }{2}}\\\arctan x+\operatorname {arccot} x&={\tfrac {\pi }{2}}\\\arctan x+\arctan {\frac {1}{x}}&={\begin{cases}{\tfrac {\pi }{2}},&{\text{if }}x>0\\-{\tfrac {\pi }{2}},&{\text{if }}x<0\end{cases}}\end{aligned}}}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b1c2c3294aba4a1004ba252bb09353508e103591" style="border: none; display: inline-block; height: 13.843ex; vertical-align: -6.338ex; width: 40.731ex;" /></a><img alt="" border="0" height="0" id="amznPsBmPixel_142516" src="https://ir-na.amazon-adsystem.com/e/ir?source=bk&t=learnjavaporgramming-20&bm-id=default&l=kia&linkId=b5b1b7f92efe859bed6aa15848812634&_cb=1539529102891" style="border: none !important; height: 0px !important; margin: 0px !important; padding: 0px !important; width: 0px !important;" width="0" /></div>
</dd></dl>
<h3 style="background: none rgb(255, 255, 255); border-bottom: none; font-family: sans-serif; font-size: 1.2em; line-height: 1.6; margin: 0.3em 0px 0px; overflow: hidden; padding-bottom: 0px; padding-top: 0.5em;">
<span class="mw-headline" id="Compositions_of_trig_and_inverse_trig_functions">Compositions of trig and inverse trig functions<img alt="" border="0" height="0" id="amznPsBmPixel_2848901" src="https://ir-na.amazon-adsystem.com/e/ir?source=bk&t=learnjavaporgramming-20&bm-id=default&l=ktl&linkId=07796dceed8fa8a897f4f4c530883528&_cb=1539529124262" style="border: none !important; height: 0px !important; margin: 0px !important; padding: 0px !important; width: 0px !important;" width="0" /></span></h3>
<dl style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 20.65px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\sin(\arccos x)&={\sqrt {1-x^{2}}}&\tan(\arcsin x)&={\frac {x}{\sqrt {1-x^{2}}}}\\\sin(\arctan x)&={\frac {x}{\sqrt {1+x^{2}}}}&\tan(\arccos x)&={\frac {\sqrt {1-x^{2}}}{x}}\\\cos(\arctan x)&={\frac {1}{\sqrt {1+x^{2}}}}&\cot(\arcsin x)&={\frac {\sqrt {1-x^{2}}}{x}}\\\cos(\arcsin x)&={\sqrt {1-x^{2}}}&\cot(\arccos x)&={\frac {x}{\sqrt {1-x^{2}}}}\end{aligned}}}</annotation></semantics></math></span><img alt="{\displaystyle {\begin{aligned}\sin(\arccos x)&={\sqrt {1-x^{2}}}&\tan(\arcsin x)&={\frac {x}{\sqrt {1-x^{2}}}}\\\sin(\arctan x)&={\frac {x}{\sqrt {1+x^{2}}}}&\tan(\arccos x)&={\frac {\sqrt {1-x^{2}}}{x}}\\\cos(\arctan x)&={\frac {1}{\sqrt {1+x^{2}}}}&\cot(\arcsin x)&={\frac {\sqrt {1-x^{2}}}{x}}\\\cos(\arcsin x)&={\sqrt {1-x^{2}}}&\cot(\arccos x)&={\frac {x}{\sqrt {1-x^{2}}}}\end{aligned}}}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b9410a14724601e3df93299115fd13c150f2274" style="border: none; display: inline-block; height: 26.509ex; vertical-align: -12.671ex; width: 56.876ex;" /></dd></dl>
</div>
Thanh Ahttp://www.blogger.com/profile/16564310847356090654noreply@blogger.comtag:blogger.com,1999:blog-574185785118190739.post-13356434879017855642016-11-16T02:31:00.000-08:002018-10-19T19:26:13.047-07:0016. Relation to the complex exponential function<div dir="ltr" style="text-align: left;" trbidi="on">
<dl style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt="e^{ix}=\cos x+i\sin x\," aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/62307b67d43ddfe413832df35fcde76bf9f2bad1" style="border: none; display: inline-block; height: 2.843ex; vertical-align: -0.505ex; width: 19.879ex;" /> (Euler's formula),</dd></dl>
<dl style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 20.65px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle e^{-ix}=\cos(-x)+i\sin(-x)=\cos x-i\sin x}</annotation></semantics></math></span><img alt="e^{-ix}=\cos(-x)+i\sin(-x)=\cos x-i\sin x" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3dc3a9f3bc659d8843d66528478d4bed85c2403" style="border: none; display: inline-block; height: 3.176ex; vertical-align: -0.838ex; width: 43.945ex;" /></dd></dl>
<dl style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 20.65px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle e^{i\pi }=-1}</annotation></semantics></math></span><img alt="e^{i\pi }=-1" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00ba628c5431f37a0be483c26111b1078c036f6b" style="border: none; display: inline-block; height: 2.676ex; vertical-align: -0.338ex; width: 8.951ex;" /> (Euler's identity),</dd></dl>
<dl style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 20.65px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle e^{2\pi i}=1}</annotation></semantics></math></span><img alt="e^{2\pi i}=1" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6da4586a614d29da6632bc76b2b4417996d62e88" style="border: none; display: inline-block; height: 2.676ex; vertical-align: -0.338ex; width: 7.961ex;" /></dd></dl>
<dl style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 20.65px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle \cos x={\frac {e^{ix}+e^{-ix}}{2}}}</annotation></semantics></math></span><img alt="\cos x={\frac {e^{ix}+e^{-ix}}{2}}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e16aed66a59735cc85653fb255bd6ba5b3e0db4" style="border: none; display: inline-block; height: 5.676ex; vertical-align: -1.838ex; width: 18.649ex;" /></dd></dl>
<dl style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 20.65px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle \sin x={\frac {e^{ix}-e^{-ix}}{2i}}}</annotation></semantics></math></span><img alt="\sin x={\frac {e^{ix}-e^{-ix}}{2i}}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a19a293592d5d5dda850bf2de5b92aba3c9764f" style="border: none; display: inline-block; height: 5.676ex; vertical-align: -1.838ex; width: 18.393ex;" /></dd></dl>
<div style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; line-height: inherit; margin-bottom: 0.5em; margin-top: 0.5em;">
and hence the corollary:</div>
<dl style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 20.65px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle \tan x={\frac {\sin x}{\cos x}}={\frac {e^{ix}-e^{-ix}}{i({e^{ix}+e^{-ix}})}}}</annotation></semantics></math></span><img alt="\tan x={\frac {\sin x}{\cos x}}={\frac {e^{ix}-e^{-ix}}{i({e^{ix}+e^{-ix}})}}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06950b618013903005efaa961c814b1f2caae35a" style="border: none; display: inline-block; height: 6.509ex; vertical-align: -2.671ex; width: 30.355ex;" /></dd></dl>
<div style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; line-height: inherit; margin-bottom: 0.5em; margin-top: 0.5em;">
where <span class="texhtml" style="font-family: "nimbus roman no9 l" , "times new roman" , "times" , serif; font-size: 20.65px; line-height: 1; white-space: nowrap;"><i>i</i><sup style="font-size: 16.52px; line-height: 1;">2</sup> = −1</span>.</div>
</div>
Thanh Ahttp://www.blogger.com/profile/16564310847356090654noreply@blogger.comtag:blogger.com,1999:blog-574185785118190739.post-63670110120721584962016-11-15T02:32:00.000-08:002016-12-02T02:42:24.691-08:0017. Infinite product formulae<div style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; line-height: inherit; margin-bottom: 0.5em; margin-top: 0.5em;">
For applications to special functions, the following <a href="https://en.wikipedia.org/wiki/Infinite_product" style="background: none; color: #0b0080; text-decoration: none;" title="Infinite product">infinite product</a> formulae for trigonometric functions are useful:<sup class="noprint Inline-Template Template-Fact" style="font-size: 14px; line-height: 1; white-space: nowrap;">[<i><span title="A&S doesn't have the last two formulae (June 2016)">citation needed</span></i>]</sup></div>
<dl style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 20.65px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\sin x&=x\prod _{n=1}^{\infty }\left(1-{\frac {x^{2}}{\pi ^{2}n^{2}}}\right)\\\sinh x&=x\prod _{n=1}^{\infty }\left(1+{\frac {x^{2}}{\pi ^{2}n^{2}}}\right)\\{\frac {\sin x}{x}}&=\prod _{n=1}^{\infty }\cos {\frac {x}{2^{n}}}\end{aligned}}\ \,{\begin{aligned}\cos x&=\prod _{n=1}^{\infty }\left(1-{\frac {x^{2}}{\pi ^{2}(n-{\frac {1}{2}})^{2}}}\right)\\\cosh x&=\prod _{n=1}^{\infty }\left(1+{\frac {x^{2}}{\pi ^{2}(n-{\frac {1}{2}})^{2}}}\right)\\|\sin x|&={\frac {1}{2}}\prod _{n=0}^{\infty }{\sqrt[{2^{n+1}}]{\left|\tan \left(2^{n}x\right)\right|}}\end{aligned}}}</annotation></semantics></math></span><img alt="{\displaystyle {\begin{aligned}\sin x&=x\prod _{n=1}^{\infty }\left(1-{\frac {x^{2}}{\pi ^{2}n^{2}}}\right)\\\sinh x&=x\prod _{n=1}^{\infty }\left(1+{\frac {x^{2}}{\pi ^{2}n^{2}}}\right)\\{\frac {\sin x}{x}}&=\prod _{n=1}^{\infty }\cos {\frac {x}{2^{n}}}\end{aligned}}\ \,{\begin{aligned}\cos x&=\prod _{n=1}^{\infty }\left(1-{\frac {x^{2}}{\pi ^{2}(n-{\frac {1}{2}})^{2}}}\right)\\\cosh x&=\prod _{n=1}^{\infty }\left(1+{\frac {x^{2}}{\pi ^{2}(n-{\frac {1}{2}})^{2}}}\right)\\|\sin x|&={\frac {1}{2}}\prod _{n=0}^{\infty }{\sqrt[{2^{n+1}}]{\left|\tan \left(2^{n}x\right)\right|}}\end{aligned}}}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c8c14aaad3e338d44553e8762837af4d44498529" style="border: none; display: inline-block; height: 22.843ex; vertical-align: -10.838ex; width: 62.598ex;" /></dd></dl>
Thanh Ahttp://www.blogger.com/profile/16564310847356090654noreply@blogger.comtag:blogger.com,1999:blog-574185785118190739.post-10184531107946016872016-11-14T02:32:00.000-08:002016-12-02T02:42:33.620-08:0018. Identities without variables<div style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; line-height: inherit; margin-bottom: 0.5em; margin-top: 0.5em;">
The curious identity known as Morrie's law</div>
<dl style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 20.65px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle \cos 20^{\circ }\cdot \cos 40^{\circ }\cdot \cos 80^{\circ }={\frac {1}{8}}}</annotation></semantics></math></span><img alt="\cos 20^{\circ }\cdot \cos 40^{\circ }\cdot \cos 80^{\circ }={\frac {1}{8}}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/263aea012268972c0a814dda0998b4d6a6973532" style="border: none; display: inline-block; height: 5.343ex; vertical-align: -2.005ex; width: 29.308ex;" /></dd></dl>
<div style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; line-height: inherit; margin-bottom: 0.5em; margin-top: 0.5em;">
is a special case of an identity that contains one variable:</div>
<dl style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 20.65px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle \prod _{j=0}^{k-1}\cos(2^{j}x)={\frac {\sin(2^{k}x)}{2^{k}\sin x}}.}</annotation></semantics></math></span><img alt="\prod _{j=0}^{k-1}\cos(2^{j}x)={\frac {\sin(2^{k}x)}{2^{k}\sin x}}." aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16d8686da401d2e26d09575d7d297fad784e3bf1" style="border: none; display: inline-block; height: 7.676ex; vertical-align: -3.338ex; width: 24.698ex;" /></dd></dl>
<div style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; line-height: inherit; margin-bottom: 0.5em; margin-top: 0.5em;">
The same cosine identity in radians is</div>
<dl style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 20.65px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle \cos {\frac {\pi }{9}}\cos {\frac {2\pi }{9}}\cos {\frac {4\pi }{9}}={\frac {1}{8}}.}</annotation></semantics></math></span><img alt="{\displaystyle \cos {\frac {\pi }{9}}\cos {\frac {2\pi }{9}}\cos {\frac {4\pi }{9}}={\frac {1}{8}}.}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1ec979b6590911bfc34ceed7f8b644e87b2b777e" style="border: none; display: inline-block; height: 5.509ex; vertical-align: -2.005ex; width: 26.02ex;" /></dd></dl>
<div style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; line-height: inherit; margin-bottom: 0.5em; margin-top: 0.5em;">
Similarly:</div>
<dl style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 20.65px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle \sin 20^{\circ }\cdot \sin 40^{\circ }\cdot \sin 80^{\circ }={\frac {\sqrt {3}}{8}}}</annotation></semantics></math></span><img alt="{\displaystyle \sin 20^{\circ }\cdot \sin 40^{\circ }\cdot \sin 80^{\circ }={\frac {\sqrt {3}}{8}}}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d48d9efac6609e756a9b6224248b123be594a1a6" style="border: none; display: inline-block; height: 6.009ex; vertical-align: -2.005ex; width: 30.488ex;" /></dd></dl>
<div style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; line-height: inherit; margin-bottom: 0.5em; margin-top: 0.5em;">
is a special case of an identity with the case x = 20:</div>
<dl style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 20.65px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle \sin x\cdot \sin(60^{\circ }-x)\cdot \sin(60^{\circ }+x)={\frac {\sin 3x}{4}}.}</annotation></semantics></math></span><img alt="{\displaystyle \sin x\cdot \sin(60^{\circ }-x)\cdot \sin(60^{\circ }+x)={\frac {\sin 3x}{4}}.}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/21c806f67579d4369819ca561f2f68524a55fa9e" style="border: none; display: inline-block; height: 5.176ex; vertical-align: -1.838ex; width: 43.014ex;" /></dd></dl>
<div style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; line-height: inherit; margin-bottom: 0.5em; margin-top: 0.5em;">
For the case <span class="texhtml mvar" style="font-family: "nimbus roman no9 l" , "times new roman" , "times" , serif; font-size: 20.65px; font-style: italic; line-height: 1; white-space: nowrap;">x</span> = 15:</div>
<dl style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 20.65px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle \sin 15^{\circ }\cdot \sin 45^{\circ }\cdot \sin 75^{\circ }={\frac {\sqrt {2}}{8}},}</annotation></semantics></math></span><img alt="{\displaystyle \sin 15^{\circ }\cdot \sin 45^{\circ }\cdot \sin 75^{\circ }={\frac {\sqrt {2}}{8}},}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/71b5e31b95b77dc1a23d69f1f13fb7e3d4de68c6" style="border: none; display: inline-block; height: 6.009ex; vertical-align: -2.005ex; width: 31.145ex;" /></dd></dl>
<dl style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 20.65px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle \sin 15^{\circ }\cdot \sin 75^{\circ }={\frac {1}{4}}.}</annotation></semantics></math></span><img alt="{\displaystyle \sin 15^{\circ }\cdot \sin 75^{\circ }={\frac {1}{4}}.}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2981af285400f288c3ed5ced7de0365d47a036a5" style="border: none; display: inline-block; height: 5.176ex; vertical-align: -1.838ex; width: 20.828ex;" /></dd></dl>
<div style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; line-height: inherit; margin-bottom: 0.5em; margin-top: 0.5em;">
For the case <span class="texhtml mvar" style="font-family: "nimbus roman no9 l" , "times new roman" , "times" , serif; font-size: 20.65px; font-style: italic; line-height: 1; white-space: nowrap;">x</span> = 10:</div>
<dl style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 20.65px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle \sin 10^{\circ }\cdot \sin 50^{\circ }\cdot \sin 70^{\circ }={\frac {1}{8}}.}</annotation></semantics></math></span><img alt="{\displaystyle \sin 10^{\circ }\cdot \sin 50^{\circ }\cdot \sin 70^{\circ }={\frac {1}{8}}.}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/481c028cc1989226779b0fd8083f4c7ecfaaa289" style="border: none; display: inline-block; height: 5.343ex; vertical-align: -2.005ex; width: 29.199ex;" /></dd></dl>
<div style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; line-height: inherit; margin-bottom: 0.5em; margin-top: 0.5em;">
The same cosine identity is</div>
<dl style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 20.65px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle \cos x\cdot \cos(60^{\circ }-x)\cdot \cos(60^{\circ }+x)={\frac {\cos 3x}{4}}.}</annotation></semantics></math></span><img alt="{\displaystyle \cos x\cdot \cos(60^{\circ }-x)\cdot \cos(60^{\circ }+x)={\frac {\cos 3x}{4}}.}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3a4cf9ebe4d25aae7a24bab9084ec63b174908b" style="border: none; display: inline-block; height: 5.176ex; vertical-align: -1.838ex; width: 44.036ex;" /></dd></dl>
<div style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; line-height: inherit; margin-bottom: 0.5em; margin-top: 0.5em;">
Similary:</div>
<dl style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 20.65px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle \cos 10^{\circ }\cdot \cos 50^{\circ }\cdot \cos 70^{\circ }={\frac {\sqrt {3}}{8}}.}</annotation></semantics></math></span><img alt="{\displaystyle \cos 10^{\circ }\cdot \cos 50^{\circ }\cdot \cos 70^{\circ }={\frac {\sqrt {3}}{8}}.}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3a40fcca60acd8a6ab20fea5ea41f58ddb9118a9" style="border: none; display: inline-block; height: 6.009ex; vertical-align: -2.005ex; width: 31.912ex;" /></dd></dl>
<dl style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 20.65px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle \cos 15^{\circ }\cdot \cos 45^{\circ }\cdot \cos 75^{\circ }={\frac {\sqrt {2}}{8}},}</annotation></semantics></math></span><img alt="{\displaystyle \cos 15^{\circ }\cdot \cos 45^{\circ }\cdot \cos 75^{\circ }={\frac {\sqrt {2}}{8}},}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd87e5350e7e1942e2e1f5462bb6570223d68025" style="border: none; display: inline-block; height: 6.009ex; vertical-align: -2.005ex; width: 31.912ex;" /></dd><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 20.65px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle \cos 15^{\circ }\cdot \cos 75^{\circ }={\frac {1}{4}}.}</annotation></semantics></math></span><img alt="{\displaystyle \cos 15^{\circ }\cdot \cos 75^{\circ }={\frac {1}{4}}.}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d7c489fa283280c5f06740085945de7763d04504" style="border: none; display: inline-block; height: 5.176ex; vertical-align: -1.838ex; width: 21.339ex;" /></dd></dl>
<div style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; line-height: inherit; margin-bottom: 0.5em; margin-top: 0.5em;">
Similarly:</div>
<dl style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 20.65px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle \tan 50^{\circ }\cdot \tan 60^{\circ }\cdot \tan 70^{\circ }=\tan 80^{\circ }.}</annotation></semantics></math></span><img alt="\tan 50^{\circ }\cdot \tan 60^{\circ }\cdot \tan 70^{\circ }=\tan 80^{\circ }." aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/13da7f8a1c19adb244916d9e8890321ffce7fb2a" style="border: none; display: inline-block; height: 2.343ex; vertical-align: -0.338ex; width: 35.887ex;" /></dd></dl>
<dl style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 20.65px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle \tan 40^{\circ }\cdot \tan 30^{\circ }\cdot \tan 20^{\circ }=\tan 10^{\circ }.}</annotation></semantics></math></span><img alt="\tan 40^{\circ }\cdot \tan 30^{\circ }\cdot \tan 20^{\circ }=\tan 10^{\circ }." aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a2f01965f7ec03a22a72a5676170d6a7de269aa5" style="border: none; display: inline-block; height: 2.343ex; vertical-align: -0.338ex; width: 35.887ex;" /></dd></dl>
<div style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; line-height: inherit; margin-bottom: 0.5em; margin-top: 0.5em;">
The following is perhaps not as readily generalized to an identity containing variables (but see explanation below):</div>
<dl style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 20.65px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle \cos 24^{\circ }+\cos 48^{\circ }+\cos 96^{\circ }+\cos 168^{\circ }={\frac {1}{2}}.}</annotation></semantics></math></span><img alt="\cos 24^{\circ }+\cos 48^{\circ }+\cos 96^{\circ }+\cos 168^{\circ }={\frac {1}{2}}." aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4006eb68677ca45758bbbb6eb2a348dfe881d4c8" style="border: none; display: inline-block; height: 5.176ex; vertical-align: -1.838ex; width: 43.249ex;" /></dd></dl>
<div style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; line-height: inherit; margin-bottom: 0.5em; margin-top: 0.5em;">
Degree measure ceases to be more felicitous than radian measure when we consider this identity with 21 in the denominators:</div>
<dl style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 20.65px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}&\cos {\frac {2\pi }{21}}+\cos \left(2\cdot {\frac {2\pi }{21}}\right)+\cos \left(4\cdot {\frac {2\pi }{21}}\right)\\[10pt]&{}\qquad {}+\cos \left(5\cdot {\frac {2\pi }{21}}\right)+\cos \left(8\cdot {\frac {2\pi }{21}}\right)+\cos \left(10\cdot {\frac {2\pi }{21}}\right)={\frac {1}{2}}.\end{aligned}}}</annotation></semantics></math></span><img alt="{\displaystyle {\begin{aligned}&\cos {\frac {2\pi }{21}}+\cos \left(2\cdot {\frac {2\pi }{21}}\right)+\cos \left(4\cdot {\frac {2\pi }{21}}\right)\\[10pt]&{}\qquad {}+\cos \left(5\cdot {\frac {2\pi }{21}}\right)+\cos \left(8\cdot {\frac {2\pi }{21}}\right)+\cos \left(10\cdot {\frac {2\pi }{21}}\right)={\frac {1}{2}}.\end{aligned}}}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8451fe2f10582cc02bfcee16c9f19b5d051c9c42" style="border: none; display: inline-block; height: 14.843ex; vertical-align: -6.838ex; width: 59.293ex;" /></dd></dl>
<div style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; line-height: inherit; margin-bottom: 0.5em; margin-top: 0.5em;">
The factors 1, 2, 4, 5, 8, 10 may start to make the pattern clear: they are those integers less than <span class="sfrac nowrap" style="display: inline-block; font-size: 14.875px; text-align: center; vertical-align: -0.5em; white-space: nowrap;"><span style="display: block; line-height: 1em; margin: 0px 0.1em;">21</span><span class="visualhide" style="height: 1px; left: -10000px; overflow: hidden; position: absolute; top: auto; width: 1px;">/</span><span style="border-top: 1px solid; display: block; line-height: 1em; margin: 0px 0.1em;">2</span></span> that are relatively prime to (or have no prime factors in common with) 21. The last several examples are corollaries of a basic fact about the irreducible cyclotomic polynomials: the cosines are the real parts of the zeroes of those polynomials; the sum of the zeroes is the Möbius function evaluated at (in the very last case above) 21; only half of the zeroes are present above. The two identities preceding this last one arise in the same fashion with 21 replaced by 10 and 15, respectively.</div>
<div style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; line-height: inherit; margin-bottom: 0.5em; margin-top: 0.5em;">
Other cosine identities include:</div>
<dl style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 20.65px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle 2\cos {\frac {\pi }{3}}=1,}</annotation></semantics></math></span><img alt="2\cos {\frac {\pi }{3}}=1," aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a016005fdbc5aa2c768dfc3f2a74df7e8c0ab7d7" style="border: none; display: inline-block; height: 4.843ex; vertical-align: -2.005ex; width: 12.207ex;" /></dd><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 20.65px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle 2\cos {\frac {\pi }{5}}\times 2\cos {\frac {2\pi }{5}}=1,}</annotation></semantics></math></span><img alt="2\cos {\frac {\pi }{5}}\times 2\cos {\frac {2\pi }{5}}=1," aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c291b6fa787e35fa3787246ba9ed0df6a809c567" style="border: none; display: inline-block; height: 5.343ex; vertical-align: -2.005ex; width: 23.499ex;" /></dd><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 20.65px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle 2\cos {\frac {\pi }{7}}\times 2\cos {\frac {2\pi }{7}}\times 2\cos {\frac {3\pi }{7}}=1,}</annotation></semantics></math></span><img alt="2\cos {\frac {\pi }{7}}\times 2\cos {\frac {2\pi }{7}}\times 2\cos {\frac {3\pi }{7}}=1," aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a094fc795f5d32d6f6847cd2febb1bdebc24f81c" style="border: none; display: inline-block; height: 5.343ex; vertical-align: -2.005ex; width: 34.791ex;" /></dd></dl>
<div style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; line-height: inherit; margin-bottom: 0.5em; margin-top: 0.5em;">
and so forth for all odd numbers, and hence</div>
<dl style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 20.65px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle \cos {\frac {\pi }{3}}+\cos {\frac {\pi }{5}}\times \cos {\frac {2\pi }{5}}+\cos {\frac {\pi }{7}}\times \cos {\frac {2\pi }{7}}\times \cos {\frac {3\pi }{7}}+\dots =1.}</annotation></semantics></math></span><img alt="\cos {\frac {\pi }{3}}+\cos {\frac {\pi }{5}}\times \cos {\frac {2\pi }{5}}+\cos {\frac {\pi }{7}}\times \cos {\frac {2\pi }{7}}\times \cos {\frac {3\pi }{7}}+\dots =1." aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d363b4652ea359de079e5e62c91809c3f5d142b" style="border: none; display: inline-block; height: 5.343ex; vertical-align: -2.005ex; width: 62.546ex;" /></dd></dl>
<div style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; line-height: inherit; margin-bottom: 0.5em; margin-top: 0.5em;">
Many of those curious identities stem from more general facts like the following:</div>
<dl style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 20.65px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle \prod _{k=1}^{n-1}\sin {\frac {k\pi }{n}}={\frac {n}{2^{n-1}}}}</annotation></semantics></math></span><img alt="\prod _{k=1}^{n-1}\sin {\frac {k\pi }{n}}={\frac {n}{2^{n-1}}}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3353129e74dab4d1c88156b3feaed39a87edecb2" style="border: none; display: inline-block; height: 7.509ex; vertical-align: -3.171ex; width: 18.629ex;" /></dd></dl>
<div style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; line-height: inherit; margin-bottom: 0.5em; margin-top: 0.5em;">
and</div>
<dl style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 20.65px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle \prod _{k=1}^{n-1}\cos {\frac {k\pi }{n}}={\frac {\sin {\frac {\pi n}{2}}}{2^{n-1}}}}</annotation></semantics></math></span><img alt="{\displaystyle \prod _{k=1}^{n-1}\cos {\frac {k\pi }{n}}={\frac {\sin {\frac {\pi n}{2}}}{2^{n-1}}}}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6097e77c87a535c4ddbf309777e4f561e8c032f0" style="border: none; display: inline-block; height: 7.509ex; vertical-align: -3.171ex; width: 20.424ex;" /></dd></dl>
<div style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; line-height: inherit; margin-bottom: 0.5em; margin-top: 0.5em;">
Combining these gives us</div>
<dl style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 20.65px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle \prod _{k=1}^{n-1}\tan {\frac {k\pi }{n}}={\frac {n}{\sin {\frac {\pi n}{2}}}}}</annotation></semantics></math></span><img alt="{\displaystyle \prod _{k=1}^{n-1}\tan {\frac {k\pi }{n}}={\frac {n}{\sin {\frac {\pi n}{2}}}}}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c859389844bfc0b7e89a545f1707d94f348a073e" style="border: none; display: inline-block; height: 7.509ex; vertical-align: -3.171ex; width: 20.672ex;" /></dd></dl>
<div style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; line-height: inherit; margin-bottom: 0.5em; margin-top: 0.5em;">
If <span class="texhtml mvar" style="font-family: "nimbus roman no9 l" , "times new roman" , "times" , serif; font-size: 20.65px; font-style: italic; line-height: 1; white-space: nowrap;">n</span> is an odd number (<span class="texhtml" style="font-family: "nimbus roman no9 l" , "times new roman" , "times" , serif; font-size: 20.65px; line-height: 1; white-space: nowrap;"><i>n</i> = 2<i>m</i> + 1</span>) we can make use of the symmetries to get</div>
<dl style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 20.65px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle \prod _{k=1}^{m}\tan {\frac {k\pi }{2m+1}}={\sqrt {2m+1}}}</annotation></semantics></math></span><img alt="\prod _{k=1}^{m}\tan {\frac {k\pi }{2m+1}}={\sqrt {2m+1}}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdd27a671ce4c2ace2f03869e2f93b6687ea362c" style="border: none; display: inline-block; height: 7.176ex; vertical-align: -3.171ex; width: 27.531ex;" /></dd></dl>
<div style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; line-height: inherit; margin-bottom: 0.5em; margin-top: 0.5em;">
The transfer function of the Butterworth low pass filter can be expressed in terms of polynomial and poles. By setting the frequency as the cutoff frequency, the following identity can be proved:</div>
<dl style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 20.65px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle \prod _{k=1}^{n}\sin {\frac {\left(2k-1\right)\pi }{4n}}=\prod _{k=1}^{n}\cos {\frac {\left(2k-1\right)\pi }{4n}}={\frac {\sqrt {2}}{2^{n}}}}</annotation></semantics></math></span><img alt="\prod _{k=1}^{n}\sin {\frac {\left(2k-1\right)\pi }{4n}}=\prod _{k=1}^{n}\cos {\frac {\left(2k-1\right)\pi }{4n}}={\frac {\sqrt {2}}{2^{n}}}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b1b66a3fb7f313ab994957a0a4356f061ac6758e" style="border: none; display: inline-block; height: 7.176ex; vertical-align: -3.171ex; width: 45.339ex;" /></dd></dl>
<h3 style="background: none rgb(255, 255, 255); border-bottom: none; font-family: sans-serif; font-size: 1.2em; line-height: 1.6; margin: 0.3em 0px 0px; overflow: hidden; padding-bottom: 0px; padding-top: 0.5em;">
<span class="mw-headline" id="Computing_.CF.80">Computing <span class="texhtml" style="font-family: "nimbus roman no9 l" , "times new roman" , "times" , serif; font-size: 24.78px; line-height: 1; white-space: nowrap;">π</span></span></h3>
<div style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; line-height: inherit; margin-bottom: 0.5em; margin-top: 0.5em;">
An efficient way to compute <span class="texhtml" style="font-family: "nimbus roman no9 l" , "times new roman" , "times" , serif; font-size: 20.65px; line-height: 1; white-space: nowrap;">π</span> is based on the following identity without variables, due to Machin:</div>
<dl style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 20.65px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle {\frac {\pi }{4}}=4\arctan {\frac {1}{5}}-\arctan {\frac {1}{239}}}</annotation></semantics></math></span><img alt="{\frac {\pi }{4}}=4\arctan {\frac {1}{5}}-\arctan {\frac {1}{239}}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba362ff207097dc35ca873f9a16bcda21a96b278" style="border: none; display: inline-block; height: 5.343ex; vertical-align: -2.005ex; width: 29.894ex;" /></dd></dl>
<div style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; line-height: inherit; margin-bottom: 0.5em; margin-top: 0.5em;">
or, alternatively, by using an identity of Leonhard Euler:</div>
<dl style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 20.65px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle {\frac {\pi }{4}}=5\arctan {\frac {1}{7}}+2\arctan {\frac {3}{79}}}</annotation></semantics></math></span><img alt="{\frac {\pi }{4}}=5\arctan {\frac {1}{7}}+2\arctan {\frac {3}{79}}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a92944742871304556f99c70f1d0dcddbfdd6d19" style="border: none; display: inline-block; height: 5.343ex; vertical-align: -2.005ex; width: 30.281ex;" /></dd></dl>
<div style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; line-height: inherit; margin-bottom: 0.5em; margin-top: 0.5em;">
or by using Pythagorean triples:</div>
<dl style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 20.65px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle \pi =\arccos {\frac {4}{5}}+\arccos {\frac {5}{13}}+\arccos {\frac {16}{65}}=\arcsin {\frac {3}{5}}+\arcsin {\frac {12}{13}}+\arcsin {\frac {63}{65}}.}</annotation></semantics></math></span><img alt="\pi =\arccos {\frac {4}{5}}+\arccos {\frac {5}{13}}+\arccos {\frac {16}{65}}=\arcsin {\frac {3}{5}}+\arcsin {\frac {12}{13}}+\arcsin {\frac {63}{65}}." aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/868e08992d38cdf1db57d2fe871465994374dece" style="border: none; display: inline-block; height: 5.509ex; vertical-align: -2.005ex; width: 75.605ex;" /></dd></dl>
<h3 style="background: none rgb(255, 255, 255); border-bottom: none; font-family: sans-serif; font-size: 1.2em; line-height: 1.6; margin: 0.3em 0px 0px; overflow: hidden; padding-bottom: 0px; padding-top: 0.5em;">
<span class="mw-headline" id="A_useful_mnemonic_for_certain_values_of_sines_and_cosines">A useful mnemonic for certain values of sines and cosines</span></h3>
<div style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; line-height: inherit; margin-bottom: 0.5em; margin-top: 0.5em;">
For certain simple angles, the sines and cosines take the form <span class="texhtml" style="font-family: "nimbus roman no9 l" , "times new roman" , "times" , serif; font-size: 20.65px; line-height: 1; white-space: nowrap;"><span class="sfrac nowrap" style="display: inline-block; font-size: 17.5525px; text-align: center; vertical-align: -0.5em;"><span style="display: block; line-height: 1em; margin: 0px 0.1em;"><span class="nowrap">√<span style="border-top: 1px solid; padding: 0px 0.1em;"><i>n</i></span></span></span><span class="visualhide" style="height: 1px; left: -10000px; overflow: hidden; position: absolute; top: auto; width: 1px;">/</span><span style="border-top: 1px solid; display: block; line-height: 1em; margin: 0px 0.1em;">2</span></span></span> for <span class="texhtml" style="font-family: "nimbus roman no9 l" , "times new roman" , "times" , serif; font-size: 20.65px; line-height: 1; white-space: nowrap;">0 ≤ <i>n</i> ≤ 4</span>, which makes them easy to remember.</div>
<dl style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 20.65px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle {\begin{matrix}\sin 0&=&\sin 0^{\circ }&=&{\frac {\sqrt {0}}{2}}&=&\cos 90^{\circ }&=&\cos {\frac {\pi }{2}}\\\sin {\frac {\pi }{6}}&=&\sin 30^{\circ }&=&{\frac {\sqrt {1}}{2}}&=&\cos 60^{\circ }&=&\cos {\frac {\pi }{3}}\\\sin {\frac {\pi }{4}}&=&\sin 45^{\circ }&=&{\frac {\sqrt {2}}{2}}&=&\cos 45^{\circ }&=&\cos {\frac {\pi }{4}}\\\sin {\frac {\pi }{3}}&=&\sin 60^{\circ }&=&{\frac {\sqrt {3}}{2}}&=&\cos 30^{\circ }&=&\cos {\frac {\pi }{6}}\\\sin {\frac {\pi }{2}}&=&\sin 90^{\circ }&=&{\frac {\sqrt {4}}{2}}&=&\cos 0^{\circ }&=&\cos 0\\[6pt]&&&&\uparrow \\&&&&{\text{These}}\\&&&&{\text{radicands}}\\&&&&{\text{are}}\\&&&&0,\,1,\,2,\,3,\,4.\end{matrix}}}</annotation></semantics></math></span><img alt="{\displaystyle {\begin{matrix}\sin 0&=&\sin 0^{\circ }&=&{\frac {\sqrt {0}}{2}}&=&\cos 90^{\circ }&=&\cos {\frac {\pi }{2}}\\\sin {\frac {\pi }{6}}&=&\sin 30^{\circ }&=&{\frac {\sqrt {1}}{2}}&=&\cos 60^{\circ }&=&\cos {\frac {\pi }{3}}\\\sin {\frac {\pi }{4}}&=&\sin 45^{\circ }&=&{\frac {\sqrt {2}}{2}}&=&\cos 45^{\circ }&=&\cos {\frac {\pi }{4}}\\\sin {\frac {\pi }{3}}&=&\sin 60^{\circ }&=&{\frac {\sqrt {3}}{2}}&=&\cos 30^{\circ }&=&\cos {\frac {\pi }{6}}\\\sin {\frac {\pi }{2}}&=&\sin 90^{\circ }&=&{\frac {\sqrt {4}}{2}}&=&\cos 0^{\circ }&=&\cos 0\\[6pt]&&&&\uparrow \\&&&&{\text{These}}\\&&&&{\text{radicands}}\\&&&&{\text{are}}\\&&&&0,\,1,\,2,\,3,\,4.\end{matrix}}}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e70d3193e0dd431c3b51f81064418d24a495a3a" style="border: none; display: inline-block; height: 41.009ex; vertical-align: -20.005ex; width: 62.847ex;" /></dd></dl>
<h3 style="background: none rgb(255, 255, 255); border-bottom: none; font-family: sans-serif; font-size: 1.2em; line-height: 1.6; margin: 0.3em 0px 0px; overflow: hidden; padding-bottom: 0px; padding-top: 0.5em;">
<span class="mw-headline" id="Miscellany">Miscellany</span></h3>
<div style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; line-height: inherit; margin-bottom: 0.5em; margin-top: 0.5em;">
With the golden ratio <span class="texhtml mvar" style="font-family: "nimbus roman no9 l" , "times new roman" , "times" , serif; font-size: 20.65px; font-style: italic; line-height: 1; white-space: nowrap;">φ</span>:</div>
<dl style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 20.65px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle \cos {\frac {\pi }{5}}=\cos 36^{\circ }={\tfrac {1}{4}}({\sqrt {5}}+1)={\tfrac {1}{2}}\varphi }</annotation></semantics></math></span><img alt="\cos {\frac {\pi }{5}}=\cos 36^{\circ }={\tfrac {1}{4}}({\sqrt {5}}+1)={\tfrac {1}{2}}\varphi " aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f634e97cb7eea62e26dd5f27c1e6623d869107af" style="border: none; display: inline-block; height: 4.843ex; vertical-align: -2.005ex; width: 35.807ex;" /></dd></dl>
<dl style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 20.65px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle \sin {\frac {\pi }{10}}=\sin 18^{\circ }={\tfrac {1}{4}}({\sqrt {5}}-1)={\tfrac {1}{2}}\varphi ^{-1}}</annotation></semantics></math></span><img alt="\sin {\frac {\pi }{10}}=\sin 18^{\circ }={\tfrac {1}{4}}({\sqrt {5}}-1)={\tfrac {1}{2}}\varphi ^{-1}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92efb93a33d62e8d54273a97948c3cbdb0cd3f76" style="border: none; display: inline-block; height: 4.843ex; vertical-align: -2.005ex; width: 38.647ex;" /></dd></dl>
<div style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; line-height: inherit; margin-bottom: 0.5em; margin-top: 0.5em;">
Also see trigonometric constants expressed in real radicals.</div>
<h3 style="background: none rgb(255, 255, 255); border-bottom: none; font-family: sans-serif; font-size: 1.2em; line-height: 1.6; margin: 0.3em 0px 0px; overflow: hidden; padding-bottom: 0px; padding-top: 0.5em;">
<span class="mw-headline" id="An_identity_of_Euclid">An identity of Euclid</span></h3>
<div style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; line-height: inherit; margin-bottom: 0.5em; margin-top: 0.5em;">
Euclid showed in Book XIII, Proposition 10 of his <i>Elements</i> that the area of the square on the side of a regular pentagon inscribed in a circle is equal to the sum of the areas of the squares on the sides of the regular hexagon and the regular decagon inscribed in the same circle. In the language of modern trigonometry, this says:</div>
<dl style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 20.65px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle \sin ^{2}18^{\circ }+\sin ^{2}30^{\circ }=\sin ^{2}36^{\circ }.\,}</annotation></semantics></math></span><img alt="\sin ^{2}18^{\circ }+\sin ^{2}30^{\circ }=\sin ^{2}36^{\circ }.\," aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00f875f5490d3e47c8b0291d16ae8ea2fe96c8c3" style="border: none; display: inline-block; height: 2.843ex; vertical-align: -0.505ex; width: 30.234ex;" /></dd></dl>
<div style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; line-height: inherit; margin-bottom: 0.5em; margin-top: 0.5em;">
Ptolemy used this proposition to compute some angles in his table of chords.</div>
Thanh Ahttp://www.blogger.com/profile/16564310847356090654noreply@blogger.comtag:blogger.com,1999:blog-574185785118190739.post-55018090916934689742016-11-13T02:34:00.000-08:002018-10-19T19:26:36.194-07:0019. Composition of trigonometric functions<div dir="ltr" style="text-align: left;" trbidi="on">
<div style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; line-height: inherit; margin-bottom: 0.5em; margin-top: 0.5em;">
This identity involves a trigonometric function of a trigonometric function:</div>
<dl style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 20.65px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle \cos(t\sin x)=J_{0}(t)+2\sum _{k=1}^{\infty }J_{2k}(t)\cos(2kx)}</annotation></semantics></math></span><img alt="\cos(t\sin x)=J_{0}(t)+2\sum _{k=1}^{\infty }J_{2k}(t)\cos(2kx)" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4405aef6354af781dd00f4e5d8ccd01e4077fa2d" style="border: none; display: inline-block; height: 7.176ex; vertical-align: -3.171ex; width: 42.14ex;" /></dd></dl>
<dl style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 20.65px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle \sin(t\sin x)=2\sum _{k=0}^{\infty }J_{2k+1}(t)\sin {\big (}(2k+1)x{\big )}}</annotation></semantics></math></span><img alt="{\displaystyle \sin(t\sin x)=2\sum _{k=0}^{\infty }J_{2k+1}(t)\sin {\big (}(2k+1)x{\big )}}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e7962c294ff1ea6846bbd8688106876c0fa37446" style="border: none; display: inline-block; height: 7.176ex; vertical-align: -3.171ex; width: 42.412ex;" /></dd></dl>
<dl style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 20.65px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle \cos(t\cos x)=J_{0}(t)+2\sum _{k=1}^{\infty }(-1)^{k}J_{2k}(t)\cos(2kx)}</annotation></semantics></math></span><img alt="\cos(t\cos x)=J_{0}(t)+2\sum _{k=1}^{\infty }(-1)^{k}J_{2k}(t)\cos(2kx)" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/94587da38f7f3c09db62e82c46485f69bb6fd4d5" style="border: none; display: inline-block; height: 7.176ex; vertical-align: -3.171ex; width: 47.926ex;" /></dd></dl>
<dl style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 20.65px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle \sin(t\cos x)=2\sum _{k=0}^{\infty }(-1)^{k}J_{2k+1}(t)\cos {\big (}(2k+1)x{\big )}}</annotation></semantics></math></span><img alt="{\displaystyle \sin(t\cos x)=2\sum _{k=0}^{\infty }(-1)^{k}J_{2k+1}(t)\cos {\big (}(2k+1)x{\big )}}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65a85f8ef4f0e10a54c2395cc598941841501a2a" style="border: none; display: inline-block; height: 7.176ex; vertical-align: -3.171ex; width: 48.454ex;" /></dd></dl>
<div style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; line-height: inherit; margin-bottom: 0.5em; margin-top: 0.5em;">
where <span class="texhtml mvar" style="font-family: "nimbus roman no9 l" , "times new roman" , "times" , serif; font-size: 20.65px; font-style: italic; line-height: 1; white-space: nowrap;">J<sub style="font-size: 16.52px; line-height: 1;">i</sub></span> are Bessel functions.</div>
</div>
Thanh Ahttp://www.blogger.com/profile/16564310847356090654noreply@blogger.comtag:blogger.com,1999:blog-574185785118190739.post-24100230691295819372016-11-12T02:35:00.000-08:002016-12-02T02:42:49.518-08:0020. Calculus<div style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; line-height: inherit; margin-bottom: 0.5em; margin-top: 0.5em;">
In calculus the relations stated below require angles to be measured in radians; the relations would become more complicated if angles were measured in another unit such as degrees. If the trigonometric functions are defined in terms of geometry, along with the definitions of arc length and area, their derivatives can be found by verifying two limits. The first is:</div>
<dl style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 20.65px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle \lim _{x\rightarrow 0}{\frac {\sin x}{x}}=1,}</annotation></semantics></math></span><img alt="\lim _{x\rightarrow 0}{\frac {\sin x}{x}}=1," aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cec799a02babc80e531aa1c085a0f9d1fafe2886" style="border: none; display: inline-block; height: 5.343ex; vertical-align: -2.005ex; width: 14.204ex;" /></dd></dl>
<div style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; line-height: inherit; margin-bottom: 0.5em; margin-top: 0.5em;">
verified using the unit circle and squeeze theorem. The second limit is:</div>
<dl style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 20.65px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle \lim _{x\rightarrow 0}{\frac {1-\cos x}{x}}=0,}</annotation></semantics></math></span><img alt="\lim _{x\rightarrow 0}{\frac {1-\cos x}{x}}=0," aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e1b6dad3145dbc7562aeb910faf6fd069860ecb1" style="border: none; display: inline-block; height: 5.343ex; vertical-align: -2.005ex; width: 18.483ex;" /></dd></dl>
<div style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; line-height: inherit; margin-bottom: 0.5em; margin-top: 0.5em;">
verified using the identity <span class="texhtml" style="font-family: "nimbus roman no9 l" , "times new roman" , "times" , serif; font-size: 20.65px; line-height: 1; white-space: nowrap;">tan <span class="sfrac nowrap" style="display: inline-block; font-size: 17.5525px; text-align: center; vertical-align: -0.5em;"><span style="display: block; line-height: 1em; margin: 0px 0.1em;"><i>x</i></span><span class="visualhide" style="height: 1px; left: -10000px; overflow: hidden; position: absolute; top: auto; width: 1px;">/</span><span style="border-top: 1px solid; display: block; line-height: 1em; margin: 0px 0.1em;">2</span></span> = <span class="sfrac nowrap" style="display: inline-block; font-size: 17.5525px; text-align: center; vertical-align: -0.5em;"><span style="display: block; line-height: 1em; margin: 0px 0.1em;">1 − cos <i>x</i></span><span class="visualhide" style="height: 1px; left: -10000px; overflow: hidden; position: absolute; top: auto; width: 1px;">/</span><span style="border-top: 1px solid; display: block; line-height: 1em; margin: 0px 0.1em;">sin <i>x</i></span></span></span>. Having established these two limits, one can use the limit definition of the derivative and the addition theorems to show that <span class="texhtml" style="font-family: "nimbus roman no9 l" , "times new roman" , "times" , serif; font-size: 20.65px; line-height: 1; white-space: nowrap;">(sin <i>x</i>)′ = cos <i>x</i></span> and <span class="texhtml" style="font-family: "nimbus roman no9 l" , "times new roman" , "times" , serif; font-size: 20.65px; line-height: 1; white-space: nowrap;">(cos <i>x</i>)′ = −sin <i>x</i></span>. If the sine and cosine functions are defined by their Taylor series, then the derivatives can be found by differentiating the power series term-by-term.</div>
<dl style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 20.65px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle {\mathrm {d} \over \mathrm {d} x}\sin x=\cos x}</annotation></semantics></math></span><img alt="{\mathrm {d} \over \mathrm {d} x}\sin x=\cos x" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/382c1e75cdcf37f11864f39cde1e2d5490714e4d" style="border: none; display: inline-block; height: 5.509ex; vertical-align: -2.005ex; width: 16.459ex;" /></dd></dl>
<div style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; line-height: inherit; margin-bottom: 0.5em; margin-top: 0.5em;">
The rest of the trigonometric functions can be differentiated using the above identities and the rules of differentiation:<sup class="reference" id="cite_ref-44" style="font-size: 14px; line-height: 1; unicode-bidi: isolate; white-space: nowrap;"></sup></div>
<dl style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 20.65px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\mathrm {d} \over \mathrm {d} x}\sin x&=\cos x,&{\mathrm {d} \over \mathrm {d} x}\arcsin x&={1 \over {\sqrt {1-x^{2}}}}\\\\{\mathrm {d} \over \mathrm {d} x}\cos x&=-\sin x,&{\mathrm {d} \over \mathrm {d} x}\arccos x&={-1 \over {\sqrt {1-x^{2}}}}\\\\{\mathrm {d} \over \mathrm {d} x}\tan x&=\sec ^{2}x,&{\mathrm {d} \over \mathrm {d} x}\arctan x&={1 \over 1+x^{2}}\\\\{\mathrm {d} \over \mathrm {d} x}\cot x&=-\csc ^{2}x,&{\mathrm {d} \over \mathrm {d} x}\operatorname {arccot} x&={-1 \over 1+x^{2}}\\\\{\mathrm {d} \over \mathrm {d} x}\sec x&=\tan x\sec x,&{\mathrm {d} \over \mathrm {d} x}\operatorname {arcsec} x&={1 \over |x|{\sqrt {x^{2}-1}}}\\\\{\mathrm {d} \over \mathrm {d} x}\csc x&=-\csc x\cot x,&{\mathrm {d} \over \mathrm {d} x}\operatorname {arccsc} x&={-1 \over |x|{\sqrt {x^{2}-1}}}\end{aligned}}}</annotation></semantics></math></span><img alt="{\displaystyle {\begin{aligned}{\mathrm {d} \over \mathrm {d} x}\sin x&=\cos x,&{\mathrm {d} \over \mathrm {d} x}\arcsin x&={1 \over {\sqrt {1-x^{2}}}}\\\\{\mathrm {d} \over \mathrm {d} x}\cos x&=-\sin x,&{\mathrm {d} \over \mathrm {d} x}\arccos x&={-1 \over {\sqrt {1-x^{2}}}}\\\\{\mathrm {d} \over \mathrm {d} x}\tan x&=\sec ^{2}x,&{\mathrm {d} \over \mathrm {d} x}\arctan x&={1 \over 1+x^{2}}\\\\{\mathrm {d} \over \mathrm {d} x}\cot x&=-\csc ^{2}x,&{\mathrm {d} \over \mathrm {d} x}\operatorname {arccot} x&={-1 \over 1+x^{2}}\\\\{\mathrm {d} \over \mathrm {d} x}\sec x&=\tan x\sec x,&{\mathrm {d} \over \mathrm {d} x}\operatorname {arcsec} x&={1 \over |x|{\sqrt {x^{2}-1}}}\\\\{\mathrm {d} \over \mathrm {d} x}\csc x&=-\csc x\cot x,&{\mathrm {d} \over \mathrm {d} x}\operatorname {arccsc} x&={-1 \over |x|{\sqrt {x^{2}-1}}}\end{aligned}}}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a75c64663f2b36cc23cc86c857ae17e24355040" style="border: none; display: inline-block; height: 53.176ex; vertical-align: -26.005ex; width: 57.432ex;" /></dd></dl>
<div style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; line-height: inherit; margin-bottom: 0.5em; margin-top: 0.5em;">
The integral identities can be found in List of integrals of trigonometric functions. Some generic forms are listed below.</div>
<dl style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 20.65px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle \int {\frac {\mathrm {d} u}{\sqrt {a^{2}-u^{2}}}}=\sin ^{-1}\left({\frac {u}{a}}\right)+C}</annotation></semantics></math></span><img alt="\int {\frac {\mathrm {d} u}{\sqrt {a^{2}-u^{2}}}}=\sin ^{-1}\left({\frac {u}{a}}\right)+C" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0bafc1967ae06af3cab39826e7673623de219370" style="border: none; display: inline-block; height: 6.343ex; vertical-align: -2.838ex; width: 30.872ex;" /></dd></dl>
<dl style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 20.65px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle \int {\frac {\mathrm {d} u}{a^{2}+u^{2}}}={\frac {1}{a}}\tan ^{-1}\left({\frac {u}{a}}\right)+C}</annotation></semantics></math></span><img alt="\int {\frac {\mathrm {d} u}{a^{2}+u^{2}}}={\frac {1}{a}}\tan ^{-1}\left({\frac {u}{a}}\right)+C" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/08954077ed44b9efd955bce48f6096d897861b0a" style="border: none; display: inline-block; height: 5.843ex; vertical-align: -2.338ex; width: 31.893ex;" /></dd><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 20.65px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle \int {\frac {\mathrm {d} u}{u{\sqrt {u^{2}-a^{2}}}}}={\frac {1}{a}}\sec ^{-1}\left|{\frac {u}{a}}\right|+C}</annotation></semantics></math></span><img alt="\int {\frac {\mathrm {d} u}{u{\sqrt {u^{2}-a^{2}}}}}={\frac {1}{a}}\sec ^{-1}\left|{\frac {u}{a}}\right|+C" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65bfeaf1fe64d33762d3190c0156900dc87c6d4a" style="border: none; display: inline-block; height: 6.343ex; vertical-align: -2.838ex; width: 33.319ex;" /></dd></dl>
<h3 style="background: none rgb(255, 255, 255); border-bottom: none; font-family: sans-serif; font-size: 1.2em; line-height: 1.6; margin: 0.3em 0px 0px; overflow: hidden; padding-bottom: 0px; padding-top: 0.5em;">
<span class="mw-headline" id="Implications">Implications</span></h3>
<div style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; line-height: inherit; margin-bottom: 0.5em; margin-top: 0.5em;">
The fact that the differentiation of trigonometric functions (sine and cosine) results in linear combinations of the same two functions is of fundamental importance to many fields of mathematics, including differential equations and Fourier transforms.</div>
<h3 style="background: none rgb(255, 255, 255); border-bottom: none; font-family: sans-serif; font-size: 1.2em; line-height: 1.6; margin: 0.3em 0px 0px; overflow: hidden; padding-bottom: 0px; padding-top: 0.5em;">
<span class="mw-headline" id="Some_differential_equations_satisfied_by_the_sine_function">Some differential equations satisfied by the sine function</span></h3>
<div style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; line-height: inherit; margin-bottom: 0.5em; margin-top: 0.5em;">
Let <span class="texhtml" style="font-family: "nimbus roman no9 l" , "times new roman" , "times" , serif; font-size: 20.65px; line-height: 1; white-space: nowrap;"><i>i</i> = <span class="nowrap">√<span style="border-top: 1px solid; padding: 0px 0.1em;">−1</span></span></span> be the imaginary unit and let ∘ denote composition of differential operators. Then for every <b>odd</b> positive integer <span class="texhtml mvar" style="font-family: "nimbus roman no9 l" , "times new roman" , "times" , serif; font-size: 20.65px; font-style: italic; line-height: 1; white-space: nowrap;">n</span>,</div>
<dl style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 20.65px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\sum _{k=0}^{n}{\binom {n}{k}}&\left({\frac {\mathrm {d} }{\mathrm {d} x}}-\sin x\right)\circ \left({\frac {\mathrm {d} }{\mathrm {d} x}}-\sin x+i\right)\circ \cdots \\&\qquad \cdots \circ \left({\frac {\mathrm {d} }{\mathrm {d} x}}-\sin x+(k-1)i\right)(\sin x)^{n-k}=0.\end{aligned}}}</annotation></semantics></math></span><img alt="{\displaystyle {\begin{aligned}\sum _{k=0}^{n}{\binom {n}{k}}&\left({\frac {\mathrm {d} }{\mathrm {d} x}}-\sin x\right)\circ \left({\frac {\mathrm {d} }{\mathrm {d} x}}-\sin x+i\right)\circ \cdots \\&\qquad \cdots \circ \left({\frac {\mathrm {d} }{\mathrm {d} x}}-\sin x+(k-1)i\right)(\sin x)^{n-k}=0.\end{aligned}}}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee7b01e281996b79a84ff8151904ec970e6df888" style="border: none; display: inline-block; height: 13.509ex; margin-bottom: -0.311ex; vertical-align: -5.86ex; width: 59.594ex;" /></dd></dl>
<div style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; line-height: inherit; margin-bottom: 0.5em; margin-top: 0.5em;">
(When <span class="texhtml mvar" style="font-family: "nimbus roman no9 l" , "times new roman" , "times" , serif; font-size: 20.65px; font-style: italic; line-height: 1; white-space: nowrap;">k</span> = 0, then the number of differential operators being composed is 0, so the corresponding term in the sum above is just <span class="texhtml" style="font-family: "nimbus roman no9 l" , "times new roman" , "times" , serif; font-size: 20.65px; line-height: 1; white-space: nowrap;">(sin <i>x</i>)<sup style="font-size: 16.52px; line-height: 1;"><i>n</i></sup></span>.) This identity was discovered as a by-product of research in medical imaging</div>
Thanh Ahttp://www.blogger.com/profile/16564310847356090654noreply@blogger.comtag:blogger.com,1999:blog-574185785118190739.post-84808333285028826492016-11-11T02:36:00.000-08:002016-12-02T02:42:57.279-08:0021. Exponential definitions<table class="wikitable" style="background-color: white; border-collapse: collapse; border: 1px solid rgb(170, 170, 170); color: black; font-family: sans-serif; font-size: 17.5px; margin: 1em 0px;"><tbody>
<tr><th style="background-color: #f2f2f2; border: 1px solid rgb(170, 170, 170); padding: 0.2em 0.4em; text-align: center;">Function</th><th style="background-color: #f2f2f2; border: 1px solid rgb(170, 170, 170); padding: 0.2em 0.4em; text-align: center;">Inverse function<sup class="reference" id="cite_ref-46" style="font-size: 14px; font-weight: normal; line-height: 1; unicode-bidi: isolate; white-space: nowrap;"><a href="https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-46" style="background: none; color: #0b0080; text-decoration: none;">[46]</a></sup></th></tr>
<tr><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em 0.4em;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 20.65px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle \sin \theta ={\frac {e^{i\theta }-e^{-i\theta }}{2i}}\,}</annotation></semantics></math></span><img alt="\sin \theta ={\frac {e^{i\theta }-e^{-i\theta }}{2i}}\," aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67b0d0fcc1c0e6c76ffe716740ff5ee5cc7f2aaa" style="border: none; display: inline-block; height: 5.676ex; vertical-align: -1.838ex; width: 18.203ex;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em 0.4em;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 20.65px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle \arcsin x=-i\ln \left(ix+{\sqrt {1-x^{2}}}\right)\,}</annotation></semantics></math></span><img alt="\arcsin x=-i\ln \left(ix+{\sqrt {1-x^{2}}}\right)\," aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c4593a02be07f3153918eac5f48fb5aa4d6d195" style="border: none; display: inline-block; height: 4.843ex; vertical-align: -1.838ex; width: 32.787ex;" /></td></tr>
<tr><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em 0.4em;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 20.65px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle \cos \theta ={\frac {e^{i\theta }+e^{-i\theta }}{2}}\,}</annotation></semantics></math></span><img alt="\cos \theta ={\frac {e^{i\theta }+e^{-i\theta }}{2}}\," aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8a4d9f18979bc491518fc76adbea3065252fe56" style="border: none; display: inline-block; height: 5.676ex; vertical-align: -1.838ex; width: 18.458ex;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em 0.4em;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 20.65px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle \arccos x=i\,\ln \left(x-i\,{\sqrt {1-x^{2}}}\right)\,}</annotation></semantics></math></span><img alt="\arccos x=i\,\ln \left(x-i\,{\sqrt {1-x^{2}}}\right)\," aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/85f9f1367687a4a77b0b07ac1fc19b70eb6ac1db" style="border: none; display: inline-block; height: 4.843ex; vertical-align: -1.838ex; width: 31.998ex;" /></td></tr>
<tr><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em 0.4em;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 20.65px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle \tan \theta ={\frac {e^{i\theta }-e^{-i\theta }}{i(e^{i\theta }+e^{-i\theta })}}\,}</annotation></semantics></math></span><img alt="\tan \theta ={\frac {e^{i\theta }-e^{-i\theta }}{i(e^{i\theta }+e^{-i\theta })}}\," aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2bd262cc7b5aca060e581600bb3cc913f543028e" style="border: none; display: inline-block; height: 6.509ex; vertical-align: -2.671ex; width: 21.35ex;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em 0.4em;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 20.65px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle \arctan x={\frac {i}{2}}\ln \left({\frac {i+x}{i-x}}\right)\,}</annotation></semantics></math></span><img alt="\arctan x={\frac {i}{2}}\ln \left({\frac {i+x}{i-x}}\right)\," aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d69f970aaeeead200698cef57b8abcb91477bde8" style="border: none; display: inline-block; height: 6.176ex; vertical-align: -2.505ex; width: 25.391ex;" /></td></tr>
<tr><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em 0.4em;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 20.65px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle \csc \theta ={\frac {2i}{e^{i\theta }-e^{-i\theta }}}\,}</annotation></semantics></math></span><img alt="\csc \theta ={\frac {2i}{e^{i\theta }-e^{-i\theta }}}\," aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f4b65f76cc32456efb7ee96f0d53792a7fde2849" style="border: none; display: inline-block; height: 5.509ex; vertical-align: -2.171ex; width: 18.328ex;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em 0.4em;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 20.65px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle \operatorname {arccsc} x=-i\ln \left({\frac {i}{x}}+{\sqrt {1-{\frac {1}{x^{2}}}}}\right)\,}</annotation></semantics></math></span><img alt="{\displaystyle \operatorname {arccsc} x=-i\ln \left({\frac {i}{x}}+{\sqrt {1-{\frac {1}{x^{2}}}}}\right)\,}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5429848f0799045577bc83fd1cd1b3e8d105ee5" style="border: none; display: inline-block; height: 6.343ex; vertical-align: -2.505ex; width: 34.417ex;" /></td></tr>
<tr><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em 0.4em;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 20.65px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle \sec \theta ={\frac {2}{e^{i\theta }+e^{-i\theta }}}\,}</annotation></semantics></math></span><img alt="\sec \theta ={\frac {2}{e^{i\theta }+e^{-i\theta }}}\," aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40a2dd9c067ad991eb6257566ebbe1b439ea9ffc" style="border: none; display: inline-block; height: 5.676ex; vertical-align: -2.338ex; width: 18.328ex;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em 0.4em;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 20.65px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle \operatorname {arcsec} x=-i\ln \left({\frac {1}{x}}+{\sqrt {1-{\frac {i}{x^{2}}}}}\right)\,}</annotation></semantics></math></span><img alt="{\displaystyle \operatorname {arcsec} x=-i\ln \left({\frac {1}{x}}+{\sqrt {1-{\frac {i}{x^{2}}}}}\right)\,}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c0b1a71e4749016e93e0dd9fc62842e99fde50c0" style="border: none; display: inline-block; height: 6.343ex; vertical-align: -2.505ex; width: 34.417ex;" /></td></tr>
<tr><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em 0.4em;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 20.65px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle \cot \theta ={\frac {i(e^{i\theta }+e^{-i\theta })}{e^{i\theta }-e^{-i\theta }}}\,}</annotation></semantics></math></span><img alt="\cot \theta ={\frac {i(e^{i\theta }+e^{-i\theta })}{e^{i\theta }-e^{-i\theta }}}\," aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8f0a1b1150a36637696f7a816399762f738a10b3" style="border: none; display: inline-block; height: 6.176ex; vertical-align: -2.171ex; width: 21.09ex;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em 0.4em;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 20.65px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle \operatorname {arccot} x={\frac {i}{2}}\ln \left({\frac {x-i}{x+i}}\right)\,}</annotation></semantics></math></span><img alt="\operatorname {arccot} x={\frac {i}{2}}\ln \left({\frac {x-i}{x+i}}\right)\," aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2171210b8eeb801e9b2c77d8131dc98066e085ea" style="border: none; display: inline-block; height: 6.176ex; vertical-align: -2.505ex; width: 25.13ex;" /></td></tr>
<tr><th style="background-color: #f2f2f2; border: 1px solid rgb(170, 170, 170); padding: 0.2em 0.4em; text-align: center;"></th><th style="background-color: #f2f2f2; border: 1px solid rgb(170, 170, 170); padding: 0.2em 0.4em; text-align: center;"></th></tr>
<tr><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em 0.4em;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 20.65px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle \operatorname {cis} \,\theta =e^{i\theta }\,}</annotation></semantics></math></span><img alt="\operatorname {cis} \,\theta =e^{i\theta }\," aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bfce9be3c5d4f04dc1a66f0684f813dfbc052871" style="border: none; display: inline-block; height: 2.676ex; vertical-align: -0.338ex; width: 10.677ex;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em 0.4em;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 20.65px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle \operatorname {arccis} \,x={\frac {\ln x}{i}}=-i\ln x=\operatorname {arg} \,x\,}</annotation></semantics></math></span><img alt="\operatorname {arccis} \,x={\frac {\ln x}{i}}=-i\ln x=\operatorname {arg} \,x\," aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53b57cb30d2ee5b8588508813e7a38883155d1df" style="border: none; display: inline-block; height: 5.343ex; vertical-align: -1.838ex; width: 34.205ex;" /></td></tr>
</tbody></table>
Thanh Ahttp://www.blogger.com/profile/16564310847356090654noreply@blogger.comtag:blogger.com,1999:blog-574185785118190739.post-70710912739161707242016-11-10T02:37:00.000-08:002018-10-19T19:26:55.032-07:0022. Miscellaneous<div dir="ltr" style="text-align: left;" trbidi="on">
<h3 style="background: none rgb(255, 255, 255); border-bottom: none; font-family: sans-serif; font-size: 1.2em; line-height: 1.6; margin: 0.3em 0px 0px; overflow: hidden; padding-bottom: 0px; padding-top: 0.5em;">
<span class="mw-headline" id="Dirichlet_kernel">Dirichlet kernel</span></h3>
<div class="hatnote" role="note" style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; font-style: italic; margin-bottom: 0.5em; padding-left: 1.6em;">
<br /></div>
<div style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; line-height: inherit; margin-bottom: 0.5em; margin-top: 0.5em;">
The <b>Dirichlet kernel</b> <span class="texhtml" style="font-family: "nimbus roman no9 l" , "times new roman" , "times" , serif; font-size: 20.65px; line-height: 1; white-space: nowrap;"><i>D<sub style="font-size: 16.52px; line-height: 1;">n</sub></i>(<i>x</i>)</span> is the function occurring on both sides of the next identity:</div>
<dl style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 20.65px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle 1+2\cos x+2\cos(2x)+2\cos(3x)+\cdots +2\cos(nx)={\frac {\sin \left[\left(n+{\frac {1}{2}}\right)x\right\rbrack }{\sin \left({\frac {x}{2}}\right)}}.}</annotation></semantics></math></span><img alt="1+2\cos x+2\cos(2x)+2\cos(3x)+\cdots +2\cos(nx)={\frac {\sin \left[\left(n+{\frac {1}{2}}\right)x\right\rbrack }{\sin \left({\frac {x}{2}}\right)}}." aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b6f887bc7c5c30274241513eed7e884f445fbfc" style="border: none; display: inline-block; height: 10.176ex; vertical-align: -4.505ex; width: 72.11ex;" /></dd></dl>
<div style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; line-height: inherit; margin-bottom: 0.5em; margin-top: 0.5em;">
The convolution of any integrable function of period 2<span class="texhtml" style="font-family: "nimbus roman no9 l" , "times new roman" , "times" , serif; font-size: 20.65px; line-height: 1; white-space: nowrap;">π</span> with the Dirichlet kernel coincides with the function's <span class="texhtml mvar" style="font-family: "nimbus roman no9 l" , "times new roman" , "times" , serif; font-size: 20.65px; font-style: italic; line-height: 1; white-space: nowrap;">n</span>th-degree Fourier approximation. The same holds for any measure or generalized function.</div>
<h3 style="background: none rgb(255, 255, 255); border-bottom: none; font-family: sans-serif; font-size: 1.2em; line-height: 1.6; margin: 0.3em 0px 0px; overflow: hidden; padding-bottom: 0px; padding-top: 0.5em;">
<span class="mw-headline" id="Tangent_half-angle_substitution">Tangent half-angle substitution</span></h3>
<div class="hatnote" role="note" style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; font-style: italic; margin-bottom: 0.5em; padding-left: 1.6em;">
<br /></div>
<div style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; line-height: inherit; margin-bottom: 0.5em; margin-top: 0.5em;">
If we set</div>
<dl style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 20.65px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle t=\tan {\frac {x}{2}},}</annotation></semantics></math></span><img alt="t=\tan {\frac {x}{2}}," aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/20f1eea102b9acfa1328eb1926bf29f5de27317f" style="border: none; display: inline-block; height: 4.676ex; vertical-align: -1.838ex; width: 10.571ex;" /></dd></dl>
<div style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; line-height: inherit; margin-bottom: 0.5em; margin-top: 0.5em;">
then</div>
<dl style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 20.65px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle \sin x={\frac {2t}{1+t^{2}}};\qquad \cos x={\frac {1-t^{2}}{1+t^{2}}};\qquad e^{ix}={\frac {1+it}{1-it}}}</annotation></semantics></math></span><img alt="{\displaystyle \sin x={\frac {2t}{1+t^{2}}};\qquad \cos x={\frac {1-t^{2}}{1+t^{2}}};\qquad e^{ix}={\frac {1+it}{1-it}}}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b222a7b633c24ee65b8da739c64656f3f1193ba1" style="border: none; display: inline-block; height: 6.176ex; vertical-align: -2.338ex; width: 53.104ex;" /></dd></dl>
<div style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; line-height: inherit; margin-bottom: 0.5em; margin-top: 0.5em;">
where <span class="texhtml" style="font-family: "nimbus roman no9 l" , "times new roman" , "times" , serif; font-size: 20.65px; line-height: 1; white-space: nowrap;"><i>e</i><sup style="font-size: 16.52px; line-height: 1;"><i>ix</i></sup> = cos <i>x</i> + <i>i</i> sin <i>x</i></span>, sometimes abbreviated to <span class="texhtml" style="font-family: "nimbus roman no9 l" , "times new roman" , "times" , serif; font-size: 20.65px; line-height: 1; white-space: nowrap;">cis <i>x</i></span>.</div>
<div style="background-color: white; color: #252525; font-family: sans-serif; font-size: 17.5px; line-height: inherit; margin-bottom: 0.5em; margin-top: 0.5em;">
When this substitution of <span class="texhtml mvar" style="font-family: "nimbus roman no9 l" , "times new roman" , "times" , serif; font-size: 20.65px; font-style: italic; line-height: 1; white-space: nowrap;">t</span> for <span class="texhtml" style="font-family: "nimbus roman no9 l" , "times new roman" , "times" , serif; font-size: 20.65px; line-height: 1; white-space: nowrap;">tan <span class="sfrac nowrap" style="display: inline-block; font-size: 17.5525px; text-align: center; vertical-align: -0.5em;"><span style="display: block; line-height: 1em; margin: 0px 0.1em;"><i>x</i></span><span class="visualhide" style="height: 1px; left: -10000px; overflow: hidden; position: absolute; top: auto; width: 1px;">/</span><span style="border-top: 1px solid; display: block; line-height: 1em; margin: 0px 0.1em;">2</span></span></span> is used in calculus, it follows that <span class="texhtml" style="font-family: "nimbus roman no9 l" , "times new roman" , "times" , serif; font-size: 20.65px; line-height: 1; white-space: nowrap;">sin <i>x</i></span> is replaced by <span class="texhtml" style="font-family: "nimbus roman no9 l" , "times new roman" , "times" , serif; font-size: 20.65px; line-height: 1; white-space: nowrap;"><span class="sfrac nowrap" style="display: inline-block; font-size: 17.5525px; text-align: center; vertical-align: -0.5em;"><span style="display: block; line-height: 1em; margin: 0px 0.1em;">2<i>t</i></span><span class="visualhide" style="height: 1px; left: -10000px; overflow: hidden; position: absolute; top: auto; width: 1px;">/</span><span style="border-top: 1px solid; display: block; line-height: 1em; margin: 0px 0.1em;">1 + <i>t</i><sup style="font-size: 14.042px; line-height: 1;">2</sup></span></span></span>, <span class="texhtml" style="font-family: "nimbus roman no9 l" , "times new roman" , "times" , serif; font-size: 20.65px; line-height: 1; white-space: nowrap;">cos <i>x</i></span> is replaced by <span class="texhtml" style="font-family: "nimbus roman no9 l" , "times new roman" , "times" , serif; font-size: 20.65px; line-height: 1; white-space: nowrap;"><span class="sfrac nowrap" style="display: inline-block; font-size: 17.5525px; text-align: center; vertical-align: -0.5em;"><span style="display: block; line-height: 1em; margin: 0px 0.1em;">1 − <i>t</i><sup style="font-size: 14.042px; line-height: 1;">2</sup></span><span class="visualhide" style="height: 1px; left: -10000px; overflow: hidden; position: absolute; top: auto; width: 1px;">/</span><span style="border-top: 1px solid; display: block; line-height: 1em; margin: 0px 0.1em;">1 + <i>t</i><sup style="font-size: 14.042px; line-height: 1;">2</sup></span></span></span> and the differential <span class="texhtml" style="font-family: "nimbus roman no9 l" , "times new roman" , "times" , serif; font-size: 20.65px; line-height: 1; white-space: nowrap;">d<i>x</i></span> is replaced by <span class="texhtml" style="font-family: "nimbus roman no9 l" , "times new roman" , "times" , serif; font-size: 20.65px; line-height: 1; white-space: nowrap;"><span class="sfrac nowrap" style="display: inline-block; font-size: 17.5525px; text-align: center; vertical-align: -0.5em;"><span style="display: block; line-height: 1em; margin: 0px 0.1em;">2 d<i>t</i></span><span class="visualhide" style="height: 1px; left: -10000px; overflow: hidden; position: absolute; top: auto; width: 1px;">/</span><span style="border-top: 1px solid; display: block; line-height: 1em; margin: 0px 0.1em;">1 + <i>t</i><sup style="font-size: 14.042px; line-height: 1;">2</sup></span></span></span>. Thereby one converts rational functions of <span class="texhtml" style="font-family: "nimbus roman no9 l" , "times new roman" , "times" , serif; font-size: 20.65px; line-height: 1; white-space: nowrap;">sin <i>x</i></span> and <span class="texhtml" style="font-family: "nimbus roman no9 l" , "times new roman" , "times" , serif; font-size: 20.65px; line-height: 1; white-space: nowrap;">cos <i>x</i></span> to rational functions of <span class="texhtml mvar" style="font-family: "nimbus roman no9 l" , "times new roman" , "times" , serif; font-size: 20.65px; font-style: italic; line-height: 1; white-space: nowrap;">t</span> in order to find their antiderivatives.</div>
</div>
Thanh Ahttp://www.blogger.com/profile/16564310847356090654noreply@blogger.com