Inverse Trigonometric Functions

Function	Domain	Range
$y=\arcsin x$	$[-1,1]$	$[-{\displaystyle \frac{\pi }{2}},{\displaystyle \frac{\pi }{2}}]$
$y=\arccos x$	$[-1,1]$	$[0,\pi ]$
$y=\arctan x$	$(-\mathrm{\infty },\mathrm{\infty })$	$(-{\displaystyle \frac{\pi }{2}},{\displaystyle \frac{\pi }{2}})$

Why Domain Restriction Is Necessary

Trigonometric functions are periodic, so they are not one-to-one on all of $ℝ$. For example, $\sin 0=\sin (\pm \pi )=\sin (\pm 2\pi )=0$. Without a restricted domain, the "inverse" would be multi-valued and not a function.

• We make the sine function one-to-one by restricting its domain to $[-{\displaystyle \frac{\pi }{2}},{\displaystyle \frac{\pi }{2}}]$.
• We make the cosine function one-to-one by restricting its domain to $[0,\pi ]$.
• We make the tangent function one-to-one by restricting its domain to $(-{\displaystyle \frac{\pi }{2}},{\displaystyle \frac{\pi }{2}})$.

The inverses of the restricted sine, cosine, and tangent functions are denoted ${\sin }^{-1}x$, ${\cos }^{-1}x$, ${\tan }^{-1}x$, or equivalently $\arcsin x$, $\arccos x$, $\arctan x$.

Important: ${\sin }^{-1}x$, ${\cos }^{-1}x$, ${\tan }^{-1}x$ are the inverse functions, not the reciprocals. The reciprocal of $\sin x$ is $\csc x$, not ${\sin }^{-1}x$. (This is the exception to the usual convention where ${\sin }^{n}x$ means $(\sin x{)}^n$.)

Definitions and Cancellation Equations

$$\arcsin x=y⟺\sin y=x\text{and}-\frac{\pi }{2}\le y\le \frac{\pi }{2}$$$$\arccos x=y⟺\cos y=x\text{and}0\le y\le \pi$$

Graphs of Inverse Trig Functions

The graphs are obtained by reflecting the restricted trigonometric functions in the line $y=x$.

The graph of $y=\arcsin x$ is obtained by reflecting the graph of $y=\sin x$ restricted to $[-{\displaystyle \frac{\pi }{2}},{\displaystyle \frac{\pi }{2}}]$ in the line $y=x$.

The graph of $y=\arccos x$ is obtained by reflecting the graph of $y=\cos x$ restricted to $[0,\pi ]$ in the line $y=x$.

The graph of $y=\arctan x$ is obtained by reflecting the graph of $y=\tan x$ restricted to $(-{\displaystyle \frac{\pi }{2}},{\displaystyle \frac{\pi }{2}})$ in the line $y=x$.

Properties of the Inverse Trig Functions

Recall: if $f$ and $g$ are inverses, then $f(g(x))=x$ and $g(f(x))=x$ for $x$ in the domain of the inner function. The table below summarizes the cancellation equations (note: these hold only on the restricted domains).

Function	Domain	Range	Cancellation equations
$y=\arcsin x$	$[-1,1]$	$[-{\displaystyle \frac{\pi }{2}},{\displaystyle \frac{\pi }{2}}]$	$\sin (\arcsin x)=x$ for $-1\le x\le 1$; $\arcsin (\sin x)=x$ for $-{\displaystyle \frac{\pi }{2}}\le x\le {\displaystyle \frac{\pi }{2}}$
$y=\arccos x$	$[-1,1]$	$[0,\pi ]$	$\cos (\arccos x)=x$ for $-1\le x\le 1$; $\arccos (\cos x)=x$ for $0\le x\le \pi$
$y=\arctan x$	$(-\mathrm{\infty },\mathrm{\infty })$	$(-{\displaystyle \frac{\pi }{2}},{\displaystyle \frac{\pi }{2}})$	$\tan (\arctan x)=x$ for all $x$; $\arctan (\tan x)=x$ for

Frequently Asked Questions

What is $\arcsin (\sin (3\pi /4))$?

Be careful: $3\pi /4$ is not in the restricted domain $[-\pi /2,\pi /2]$ of arcsine. So $\arcsin (\sin (3\pi /4))\ne 3\pi /4$. Instead, compute $\sin (3\pi /4)=\sin (\pi -\pi /4)=\sin (\pi /4)=\sqrt{2}/2$. Then $\arcsin (\sqrt{2}/2)=\pi /4$ (since $\pi /4\in [-\pi /2,\pi /2]$). Answer: $\pi /4$.

What are the derivatives of the inverse trig functions?

These are some of the most important results in differential calculus: $$\frac{d}{dx}\arcsin x=\frac{1}{\sqrt{1-x^2}},\frac{d}{dx}\arccos x=-\frac{1}{\sqrt{1-x^2}},\frac{d}{dx}\arctan x=\frac{1}{1+x^2}.$$ Note that arctan is defined on all of $ℝ$, so its derivative $\frac{1}{1+x^2}$ is defined everywhere.

Why does arccosine have range $[0,\pi ]$ rather than $[-\pi /2,\pi /2]$?

We need cosine to be one-to-one on the chosen interval. Cosine is not one-to-one on $[-\pi /2,\pi /2]$ (for example, $\cos (-\pi /4)=\cos (\pi /4)$). It is one-to-one on $[0,\pi ]$, which is the standard choice. The range $[0,\pi ]$ also means that $\arccos x$ always returns a non-negative angle, which is consistent with the way angles are usually measured.