Topic 4 of 8

Inverse trig functions

Step 14 · Trigonometry and precalculus topics

Know the restricted ranges that make sine, cosine and tangent invertible, evaluate inverse trig functions and compositions, and use them to find angles.

Restricted ranges

Sine is not one-to-one: many angles share each sine. To build an inverse, we keep only part of it, where each value appears exactly once.

Function	Inputs	Outputs (angles)
$\sin^{-1} x$	−1 ≤ x ≤ 1	$-\frac{\pi}{2} \le y \le \frac{\pi}{2}$
$\cos^{-1} x$	−1 ≤ x ≤ 1	$0 \le y \le \pi$
$\tan^{-1} x$	all real numbers	$-\frac{\pi}{2} < y < \frac{\pi}{2}$

y = sin⁻¹x: inputs from −1 to 1, outputs from −π/2 to π/2 (about −1.57 to 1.57).

Evaluating inverses

To evaluate an inverse, ask which angle in the allowed range has that value.

Example: Inverse sine

Find $\sin^{-1}\frac{1}{2}$ and $\sin^{-1}(-1)$ .

$\sin\frac{\pi}{6} = \frac{1}{2}$ , and $\frac{\pi}{6}$ is in the range.
$\sin\left(-\frac{\pi}{2}\right) = -1$ .

Answer $\frac{\pi}{6}$ and $-\frac{\pi}{2}$

Example: Inverse cosine

Find $\cos^{-1}\left(-\frac{\sqrt{2}}{2}\right)$ .

Cosine is negative in quadrant II.
$\cos\frac{3\pi}{4} = -\frac{\sqrt{2}}{2}$ , and $\frac{3\pi}{4}$ is between 0 and π.

Answer $\frac{3\pi}{4}$

Example: A composition

Find $\sin\left(\cos^{-1}\frac{3}{5}\right)$ .

Let θ = $\cos^{-1}\frac{3}{5}$ : a right triangle with adjacent side 3 and hypotenuse 5.
The opposite side is 4.

Answer $\frac{4}{5}$

Using inverses to find angles

Inverse trig functions turn side lengths into angles.

Example: An angle from sides

A ramp rises 1.2 m over a horizontal run of 8 m. What angle does it make with the ground?

tan θ = 1.2 ÷ 8 = 0.15.
θ = tan⁻¹(0.15), about 8.5°.

AnswerAbout 8.5°

Example: Every solution

Solve sin x = 0.4 for 0 ≤ x < 2π.

sin⁻¹(0.4) gives one answer, about 0.412.
The other is π − 0.412, about 2.730.

Answerx ≈ 0.412 or x ≈ 2.730

Example: A direction

A plane flies 300 km north and 400 km east. On what bearing (angle east of north) is it from its start?

tan θ = 400 ÷ 300.
θ = $\tan^{-1}\frac{4}{3}$ , about 53.1°.

AnswerAbout 53.1° east of north

Try it yourself

Work each one on paper first, then open the answer to check.

Find $\sin^{-1}\frac{\sqrt{3}}{2}$ .
Show answer
$\frac{\pi}{3}$
Find $\cos^{-1} 0$ .
Show answer
$\frac{\pi}{2}$
Find $\tan^{-1}(-1)$ .
Show answer
$-\frac{\pi}{4}$
Find $\cos\left(\sin^{-1}\frac{5}{13}\right)$ .
Show answer
$\frac{12}{13}$
What is $\sin^{-1}\left(\sin\frac{3\pi}{4}\right)$ ?
Show answer
$\frac{\pi}{4}$
A 20 ft ladder reaches 16 ft up a wall. What angle does it make with the ground?
Show answer
About 53.1°