Trigonometric formulas: 15. Inverse trigonometric functions

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}}