Trigonometry

Inverse Trigonometric Functions

The trigonometric functions we've considered take an angle and produce the corresponding number. For instance, the function $\sin:\mathbb{R} \longrightarrow \left[-1,1\right]$ takes an angle $x \in \mathbb{R}$ and produces the corresponding number $\sin\left(x\right) \in \left[-1,1\right]$.

How about going in the opposite direction: given a number, say $\frac{1}{2} \in \left[-1,1\right]$, how do we find an angle $x$ such that $\sin\left(x\right)=\frac{1}{2}$. Namely, we are interested in defining the inverse function $f^{-1}\left(x\right)=\sin^{-1}\left(x\right)$ of $f\left(x\right)=\sin\left(x\right)$.

Unfortunately, $\sin$ is not a one-to-one function and so to define an inverse, we need to restrict its domain. (See Functions and Their Inverses for more information.) A standard choice is restricting the domain to $\left[\frac{-\pi}{2},\frac{\pi}{2}\right]$. Then, we have the inverse function: $$\arcsin: \left[-1,1\right] \longrightarrow \left[\frac{-\pi}{2},\frac{\pi}{2}\right]$$ Similarly for the other inverse trigonometric functions, we have:

	Domain	Range
$\arcsin$	$[-1,1]$	$\left[\frac{-\pi}{2},\frac{\pi}{2}\right]$
$\arccos$	$[-1,1]$	$[0,\pi]$
$\arctan$	$\mathbb{R}$	$\left(\frac{-\pi}{2},\frac{\pi}{2}\right)$

Example. Find $\arcsin\left(\frac{\sqrt{3}}{2}\right)$, $\arccos\left(\frac{1}{\sqrt{2}}\right)$, and $\arctan\left(1\right)$.

The only angle $\theta \in \left[\frac{-\pi}{2},\frac{\pi}{2}\right]$ for which $\sin\left(\theta\right)=\frac{\sqrt{3}}{2}$ is $\theta=\frac{\pi}{3}$, so: $$\arcsin\left(\frac{\sqrt{3}}{2}\right)=\frac{\pi}{3}$$ The only angle $\theta \in \left[0,\pi \right]$ for which $\cos\left(\theta\right)=\frac{1}{\sqrt{2}}$ is $\theta=\frac{\pi}{4}$, so: $$\arccos\left(\frac{1}{\sqrt{2}}\right)=\frac{\pi}{4}$$ The only angle $\theta \in \left(\frac{-\pi}{2},\frac{\pi}{2}\right)$ for which $\tan\left(\theta\right)= 1$ is $\theta=\frac{\pi}{4}$, so: $$\arctan\left(1\right)=\frac{\pi}{4}$$

The graphs of the inverse trigonometric functions are included below: