📂Functions

Inverse Trigonometric Functions

Definition¹

The inverse functions of trigonometric functions are called inverse trigonometric functions, and they are denoted by adding arc- to the name of the trigonometric function.

$$ \begin{align*} \arcsin x &= \sin^{-1} x \qquad & \operatorname{arccsc} x &= \csc^{-1} x \\ \arccos x &= \cos^{-1} x \qquad & \operatorname{arcsec} x &= \sec^{-1} x \\ \arctan x &= \tan^{-1} x \qquad & \operatorname{arccot} x &= \cot^{-1} x \end{align*} $$

Description

Since trigonometric functions themselves are not bijective, their domains are usually restricted to $\displaystyle \left[ -\pi , \pi \right]$ or $\displaystyle \left[ -{{\pi}\over{2}} , {{\pi}\over{2}} \right]$.

In programming, it is common to define the arctangent function as atan() using only a- instead of arc-. This convention has been carried over from FORTRAN.

Domain and Range

$$ \begin{align*} \sin^{-1} &: [-1, 1] \to \left[ -{{\pi}\over{2}} , {{\pi}\over{2}} \right] \qquad & \csc^{-1} &: \mathbb{R} \setminus (-1, 1) \to (0, \pi/2] \cup (\pi, 3\pi/2] \\[1em] \cos^{-1} &: [-1, 1] \to [0, \pi] \qquad & \sec^{-1} &: \mathbb{R} \setminus (-1, 1) \to [0, \pi/2) \cup [\pi, 3\pi/2) \\[1em] \tan^{-1} &: \mathbb{R} \to \left( -{{\pi}\over{2}} , {{\pi}\over{2}} \right) \qquad & \cot^{-1} &: \mathbb{R} \to (0, \pi) \end{align*} $$