# Inverse trigonometric function

The inverse trigonometric functions are the inverse functions of the trigonometric functions and are either written as $\arcsin x$, $\arccos x$ and so on, or as $\sin^{-1} x$, $\cos^{-1} x$ and so on.

As none of the trigonometric functions are one-to-one (injective), we cannot take their inverses without first restricting their domains. The domains that are normally chosen give rise to the following inverse trigonometric functions.

Function Domain Range
$y=\sin^{-1} x$ $|x|\le 1$ $-\frac{\pi}{2}\le y\le \frac{\pi}{2}$
$y=\cos^{-1} x$ $|x|\le 1$ $0\le y\le\pi$
$y=\tan^{-1} x$ $x\in\mathbb{R}$ $-\frac{\pi}{2}<y<\frac{\pi}{2}$
$y=\cosec^{-1} x$ $|x|\ge 1$ $-\frac{\pi}{2}\le y\le\frac{\pi}{2}$, $y\ne 0$
$y=\sec^{-1} x$ $|x|\ge 1$ $0\le y\le\pi$, $y\ne\frac{\pi}{2}$
$y=\cot^{-1} x$ $x\in\mathbb{R}$ $0<y<\pi$

The range given in the above table is sometimes called the “principal value range” of the inverse trigonometric function.

So while an equation such as $\tan x = 1$ has infinitely many solutions ($\frac{\pi}{4}$, $\frac{5\pi}{4}$, $-\frac{3\pi}{4}$ and so on), when we write $\tan^{-1}1$, we mean the single value which lies in the principal value range, which in this case is $\frac{\pi}{4}$.