Inverse Trigonometric Functions — AP® Precalculus Notes

You know how to go from an angle to its sine, cosine or tangent. Many problems run the other way: you know a ratio (a rise over a run, a coordinate on a circle) and need the angle. This page builds the functions that do that, $\sin^{-1}$ , $\cos^{-1}$ and $\tan^{-1}$ , and shows how to evaluate them exactly and with a calculator.

Why the trigonometric functions need restricting

A function has an inverse only if each output comes from exactly one input (Topic 2.8). Sine fails badly: $\sin\frac{\pi}{6} = \sin\frac{5\pi}{6} = \frac{1}{2}$ , and every period repeats the pattern. So we keep one piece of each graph that takes every output once and throw the rest away.

Sine on $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$ : it increases from $-1$ to $1$ .
Cosine on $[0, \pi]$ : it decreases from $1$ to $-1$ .
Tangent on $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$ : it increases through every real number.

Swapping inputs and outputs of these pieces gives the inverse functions. Their outputs are angles, in radians.

Function	Domain (the ratio you put in)	Range (the angle you get)	Quadrants of the answer
$\sin^{-1} x$	$[-1, 1]$	$\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$	I or IV
$\cos^{-1} x$	$[-1, 1]$	$[0, \pi]$	I or II
$\tan^{-1} x$	$(-\infty, \infty)$	$\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$	I or IV

The graph of each inverse is the restricted graph reflected over the line $y = x$ . Tangent's vertical asymptotes at $\pm\frac{\pi}{2}$ become horizontal asymptotes of $\tan^{-1}$ : $\lim_{x \to \infty} \tan^{-1} x$ $= \frac{\pi}{2}$ and $\lim_{x \to -\infty} \tan^{-1} x$ $= -\frac{\pi}{2}$ .

Notation. $\sin^{-1} x$ means the inverse function (also written $\arcsin x$ ). It is not $\frac{1}{\sin x}$ . The reciprocal of sine is a different function, which you will meet in Topic 3.11. The exponent $-1$ on a function name means "inverse"; to write a reciprocal, write the fraction.

Exact values

To evaluate $\sin^{-1} a$ , ask: which angle in the range has sine $a$ ? Use the unit circle values from Topic 3.3, then check the range.

Worked example 1. Find $\sin^{-1}\left(-\frac{1}{2}\right)$ , $\cos^{-1}\left(-\frac{\sqrt{2}}{2}\right)$ and $\tan^{-1}\sqrt{3}$ .

$\sin^{-1}\left(-\frac{1}{2}\right)$ : the angle in $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$ with sine $-\frac{1}{2}$ . The reference angle is $\frac{\pi}{6}$ , and a negative sine in that range means Quadrant IV: the answer is $-\frac{\pi}{6}$ (not $\frac{7\pi}{6}$ or $\frac{11\pi}{6}$ , which are outside the range).

$\cos^{-1}\left(-\frac{\sqrt{2}}{2}\right)$ : the angle in $[0, \pi]$ with cosine $-\frac{\sqrt{2}}{2}$ . Reference angle $\frac{\pi}{4}$ ; negative cosine in $[0, \pi]$ means Quadrant II: $\pi - \frac{\pi}{4} = \frac{3\pi}{4}$ .

$\tan^{-1}\sqrt{3}$ : the angle in $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$ with tangent $\sqrt{3}$ , which is $\frac{\pi}{3}$ .

Composites: inside first

An inverse undoes its function, but only on the restricted interval. $\sin\left(\sin^{-1} x\right) = x$ for every $x$ in $[-1, 1]$ . The other order is the trap: $\sin^{-1}(\sin x) = x$ only when $x$ is already in $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$ .

Worked example 2. Find $\cos^{-1}\left(\cos\frac{7\pi}{4}\right)$ and $\tan\left(\cos^{-1}\frac{5}{13}\right)$ .

First one, inside first: $\cos\frac{7\pi}{4} = \frac{\sqrt{2}}{2}$ . Then $\cos^{-1}\left(\frac{\sqrt{2}}{2}\right) = \frac{\pi}{4}$ , the angle in $[0, \pi]$ . The answer is $\frac{\pi}{4}$ , not $\frac{7\pi}{4}$ .

Second one: let $\theta = \cos^{-1}\frac{5}{13}$ , an angle in Quadrant I with adjacent side $5$ and hypotenuse $13$ . The opposite side is $\sqrt{169 - 25} = 12$ , so $\tan\theta = \frac{12}{5}$ .

With a calculator

When the ratio is not a unit circle value, use the $\sin^{-1}$ , $\cos^{-1}$ or $\tan^{-1}$ key in radian mode. Write the setup first, then the value to three decimal places.

Worked example 3. A ramp rises $0.6$ meters over a run of $7.2$ meters. Find its angle with the ground.

$\tan\theta = \frac{0.6}{7.2}$ , so $\theta$ $= \tan^{-1}\left(\frac{0.6}{7.2}\right)$ $\approx 0.08314$ radians, which is $0.083$ to three decimals. Keep the full value stored if you use it again.

The calculator gives only the angle in the range. If a problem needs a different angle with the same ratio, you find it from the unit circle, which is the next topic's job.

Common slips

Reporting an angle outside the range, such as $\frac{5\pi}{3}$ for $\sin^{-1}\left(-\frac{\sqrt{3}}{2}\right)$ . The answer is $-\frac{\pi}{3}$ .
Reading $\tan^{-1} x$ as $\frac{1}{\tan x}$ .
Giving a negative angle for $\cos^{-1}$ of a negative number. $\cos^{-1}$ outputs are between $0$ and $\pi$ .
Leaving the calculator in degree mode.