Inverse Trigonometric Functions

The inverse trigonometric functions: eg. ${\sin }^{-1}x,{\cos }^{-1}x,{\tan }^{-1}x$ are actually the functions $\arcsin ,\arccos ,\arctan$

These functions must be range limited, but their domain is limited inately. They are not Injective Functions so the do not follow the Inverse Function theorem.

Arcsin

for example, arcsin is the the inverse of y=sinx but the range is limited to $-\pi /2<x<\pi /2$

Arccos

Same as x = cosy but range is $0<x<2\pi$

Arctan

Arctan is only one of the lines in the inverse x = tany Simple fit into the interval $-\pi /2<x<\pi /2$ Remember, functions cannot have repeated ys. WE MUST LIMIT THE RANGE!!!