18. Identities without variables

The curious identity known as Morrie's law

${\displaystyle \cos 20^{\circ }\cdot \cos 40^{\circ }\cdot \cos 80^{\circ }={\frac {1}{8}}}$

is a special case of an identity that contains one variable:

${\displaystyle \prod _{j=0}^{k-1}\cos(2^{j}x)={\frac {\sin(2^{k}x)}{2^{k}\sin x}}.}$

The same cosine identity in radians is

${\displaystyle \cos {\frac {\pi }{9}}\cos {\frac {2\pi }{9}}\cos {\frac {4\pi }{9}}={\frac {1}{8}}.}$

Similarly:

${\displaystyle \sin 20^{\circ }\cdot \sin 40^{\circ }\cdot \sin 80^{\circ }={\frac {\sqrt {3}}{8}}}$

is a special case of an identity with the case x = 20:

${\displaystyle \sin x\cdot \sin(60^{\circ }-x)\cdot \sin(60^{\circ }+x)={\frac {\sin 3x}{4}}.}$

For the case x = 15:

${\displaystyle \sin 15^{\circ }\cdot \sin 45^{\circ }\cdot \sin 75^{\circ }={\frac {\sqrt {2}}{8}},}$

${\displaystyle \sin 15^{\circ }\cdot \sin 75^{\circ }={\frac {1}{4}}.}$

For the case x = 10:

${\displaystyle \sin 10^{\circ }\cdot \sin 50^{\circ }\cdot \sin 70^{\circ }={\frac {1}{8}}.}$

The same cosine identity is

${\displaystyle \cos x\cdot \cos(60^{\circ }-x)\cdot \cos(60^{\circ }+x)={\frac {\cos 3x}{4}}.}$

Similary:

${\displaystyle \cos 10^{\circ }\cdot \cos 50^{\circ }\cdot \cos 70^{\circ }={\frac {\sqrt {3}}{8}}.}$

${\displaystyle \cos 15^{\circ }\cdot \cos 45^{\circ }\cdot \cos 75^{\circ }={\frac {\sqrt {2}}{8}},}$

${\displaystyle \cos 15^{\circ }\cdot \cos 75^{\circ }={\frac {1}{4}}.}$

Similarly:

${\displaystyle \tan 50^{\circ }\cdot \tan 60^{\circ }\cdot \tan 70^{\circ }=\tan 80^{\circ }.}$

${\displaystyle \tan 40^{\circ }\cdot \tan 30^{\circ }\cdot \tan 20^{\circ }=\tan 10^{\circ }.}$

The following is perhaps not as readily generalized to an identity containing variables (but see explanation below):

${\displaystyle \cos 24^{\circ }+\cos 48^{\circ }+\cos 96^{\circ }+\cos 168^{\circ }={\frac {1}{2}}.}$

Degree measure ceases to be more felicitous than radian measure when we consider this identity with 21 in the denominators:

${\displaystyle {\begin{aligned}&\cos {\frac {2\pi }{21}}+\cos \left(2\cdot {\frac {2\pi }{21}}\right)+\cos \left(4\cdot {\frac {2\pi }{21}}\right)\\[10pt]&{}\qquad {}+\cos \left(5\cdot {\frac {2\pi }{21}}\right)+\cos \left(8\cdot {\frac {2\pi }{21}}\right)+\cos \left(10\cdot {\frac {2\pi }{21}}\right)={\frac {1}{2}}.\end{aligned}}}$

The factors 1, 2, 4, 5, 8, 10 may start to make the pattern clear: they are those integers less than 21/2 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.

Other cosine identities include:

${\displaystyle 2\cos {\frac {\pi }{3}}=1,}$

${\displaystyle 2\cos {\frac {\pi }{5}}\times 2\cos {\frac {2\pi }{5}}=1,}$

${\displaystyle 2\cos {\frac {\pi }{7}}\times 2\cos {\frac {2\pi }{7}}\times 2\cos {\frac {3\pi }{7}}=1,}$

and so forth for all odd numbers, and hence

${\displaystyle \cos {\frac {\pi }{3}}+\cos {\frac {\pi }{5}}\times \cos {\frac {2\pi }{5}}+\cos {\frac {\pi }{7}}\times \cos {\frac {2\pi }{7}}\times \cos {\frac {3\pi }{7}}+\dots =1.}$

Many of those curious identities stem from more general facts like the following:

${\displaystyle \prod _{k=1}^{n-1}\sin {\frac {k\pi }{n}}={\frac {n}{2^{n-1}}}}$

and

${\displaystyle \prod _{k=1}^{n-1}\cos {\frac {k\pi }{n}}={\frac {\sin {\frac {\pi n}{2}}}{2^{n-1}}}}$

Combining these gives us

${\displaystyle \prod _{k=1}^{n-1}\tan {\frac {k\pi }{n}}={\frac {n}{\sin {\frac {\pi n}{2}}}}}$

If n is an odd number (n = 2m + 1) we can make use of the symmetries to get

${\displaystyle \prod _{k=1}^{m}\tan {\frac {k\pi }{2m+1}}={\sqrt {2m+1}}}$

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:

${\displaystyle \prod _{k=1}^{n}\sin {\frac {\left(2k-1\right)\pi }{4n}}=\prod _{k=1}^{n}\cos {\frac {\left(2k-1\right)\pi }{4n}}={\frac {\sqrt {2}}{2^{n}}}}$

Computing π

An efficient way to compute π is based on the following identity without variables, due to Machin:

${\displaystyle {\frac {\pi }{4}}=4\arctan {\frac {1}{5}}-\arctan {\frac {1}{239}}}$

or, alternatively, by using an identity of Leonhard Euler:

${\displaystyle {\frac {\pi }{4}}=5\arctan {\frac {1}{7}}+2\arctan {\frac {3}{79}}}$

or by using Pythagorean triples:

${\displaystyle \pi =\arccos {\frac {4}{5}}+\arccos {\frac {5}{13}}+\arccos {\frac {16}{65}}=\arcsin {\frac {3}{5}}+\arcsin {\frac {12}{13}}+\arcsin {\frac {63}{65}}.}$

A useful mnemonic for certain values of sines and cosines

For certain simple angles, the sines and cosines take the form √n/2 for 0 ≤ n ≤ 4, which makes them easy to remember.

${\displaystyle {\begin{matrix}\sin 0&=&\sin 0^{\circ }&=&{\frac {\sqrt {0}}{2}}&=&\cos 90^{\circ }&=&\cos {\frac {\pi }{2}}\\\sin {\frac {\pi }{6}}&=&\sin 30^{\circ }&=&{\frac {\sqrt {1}}{2}}&=&\cos 60^{\circ }&=&\cos {\frac {\pi }{3}}\\\sin {\frac {\pi }{4}}&=&\sin 45^{\circ }&=&{\frac {\sqrt {2}}{2}}&=&\cos 45^{\circ }&=&\cos {\frac {\pi }{4}}\\\sin {\frac {\pi }{3}}&=&\sin 60^{\circ }&=&{\frac {\sqrt {3}}{2}}&=&\cos 30^{\circ }&=&\cos {\frac {\pi }{6}}\\\sin {\frac {\pi }{2}}&=&\sin 90^{\circ }&=&{\frac {\sqrt {4}}{2}}&=&\cos 0^{\circ }&=&\cos 0\\[6pt]&&&&\uparrow \\&&&&{\text{These}}\\&&&&{\text{radicands}}\\&&&&{\text{are}}\\&&&&0,\,1,\,2,\,3,\,4.\end{matrix}}}$

Miscellany

With the golden ratio φ:

${\displaystyle \cos {\frac {\pi }{5}}=\cos 36^{\circ }={\tfrac {1}{4}}({\sqrt {5}}+1)={\tfrac {1}{2}}\varphi }$

${\displaystyle \sin {\frac {\pi }{10}}=\sin 18^{\circ }={\tfrac {1}{4}}({\sqrt {5}}-1)={\tfrac {1}{2}}\varphi ^{-1}}$

Also see trigonometric constants expressed in real radicals.

An identity of Euclid

Euclid showed in Book XIII, Proposition 10 of his Elements 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:

${\displaystyle \sin ^{2}18^{\circ }+\sin ^{2}30^{\circ }=\sin ^{2}36^{\circ }.\,}$

Ptolemy used this proposition to compute some angles in his table of chords.