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 \(\frac{\pi}{4}\).