Inverse Trigonometric Functions

Because trigonometric functions are periodic, they are not one-to-one on their full domains. Restricting their domains creates invertible functions ( $\arcsin$ , $\arccos$ , $\arctan$ ) essential for trigonometric substitution and integration.

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Function	Notation	Restricted Domain	Range (Principal Values)
Inverse Sine	$y = \arcsin x$	$[- 1, 1]$	$[- \frac{π}{2}, \frac{π}{2}]$
Inverse Cosine	$y = \arccos x$	$[- 1, 1]$	$[0, π]$
Inverse Tangent	$y = \arctan x$	$(- \infty, + \infty)$	$(- \frac{π}{2}, \frac{π}{2})$

Definitions of Inverse Trigonometric Functions

The principal inverse trigonometric functions are defined by restricting the domain of the parent trigonometric functions:

\begin{aligned} y = \arcsin x &\iff \sin y = x \quad\text{and}\quad y \in \left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \\[4pt] y = \arccos x &\iff \cos y = x \quad\text{and}\quad y \in [0, \pi] \\[4pt] y = \arctan x &\iff \tan y = x \quad\text{and}\quad y \in \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \end{aligned}

Graph of arcsin x reflected across line y = x — Graph of $y = \arcsin x$ with domain $[- 1, 1]$ and range $[- π / 2, π / 2]$ .

Graph of arccos x reflected across line y = x — Graph of $y = \arccos x$ with domain $[- 1, 1]$ and range $[0, π]$ .

Graph of arctan x showing horizontal asymptotes at y = plus/minus pi/2 — Graph of $y = \arctan x$ with domain $(- \infty, \infty)$ and horizontal asymptotes $y = \pm \frac{π}{2}$ .

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Definitions of Inverse Trigonometric Functions

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