Trigonometric formulas: Inverse trigonometric functions

Showing posts with label Inverse trigonometric functions. Show all posts

Showing posts with label Inverse trigonometric functions. Show all posts

Thursday, November 17, 2016

15. Inverse trigonometric functions

Inverse trigonometric functions

{\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}}

Compositions of trig and inverse trig functions

{\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}}

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