Inverse Trigonometric Functions

Overview

In many different areas of geometry, you will utilize something called an Inverse Trigonometric Function, which is simply the inverse function of the corresponding trigonometric function.

Inverse Functions

Three Core Inverse Trig Functions

Arcsine, or $\arcsin$ is the inverse function of the sine function. It is defined on the domain $[-1, 1]$, and its range (in radians) is $[-\frac{\pi}{2}, \frac{\pi}{2}]$. It is defined as:

$$ \arcsin{\theta} = \sin^{-1}{\theta} $$

Arccosine, or $\arccos$ is the inverse function of the cosine function. It is defined on the domain $[-1, 1]$, and its range (in radians) is $[0, \pi]$. It is defined as:

$$ \arccos{\theta} = \cos^{-1}{\theta} $$

Arctangent, or $\arctan$ is the inverse function of the tangent function. It is defined across all real numbers, and its range (in radians) is $[-\frac{\pi}{2}, \frac{\pi}{2}]$. It is defined as:

$$ \arctan{\theta} = \tan^{-1}{\theta} $$

Expanded Inverse Trig Functions

Arccosecant is the inverse function of the cosecant function. It is defined across $x \leq -1$ and $x \geq 1$, with the range of $[-\frac{\pi}{2}, 0)$ and $(0, \frac{\pi}{2}]$.

Arcsecant is the inverse function of the secant function. It is defined across $x \leq -1$ and $x \geq 1$, with the range of $[0, \frac{\pi}{2})$ and $(\frac{\pi}{2}, \pi]$.

Arccotangant is the inverse function of the cotangent function. It is defined across all real numbers, with the range of $(0, \pi)$.