Inverse Trigonometric Functions Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Inverse Trigonometric Functions.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

Functions that reverse the trigonometric functions: given a ratio, they return the corresponding angle. $\arcsin$ , $\arccos$ , and $\arctan$ are the inverses of $\sin$ , $\cos$ , and $\tan$ on restricted domains.

Regular trig functions answer: 'Given an angle, what's the ratio?' Inverse trig functions answer the reverse: 'Given a ratio, what's the angle?' Since $\sin$ and $\cos$ are many-to-one (many angles give the same ratio), we must restrict their domains to make the inverse a proper function. Think of it like this: if you know the slope of a ramp is $0.5$ , $\arcsin(0.5) = 30°$ tells you the angle.

Read the full concept explanation →

How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: $\arcsin$ , $\arccos$ , $\arctan$ undo the trig functions on a restricted range so the answer is a single angle.

Common stuck point: The procedure for inverse trigonometric functions is the easy part; the trap is treating $\sin^{-1}$ as $\frac{1}{\sin}$ . Asking "Am I starting from a ratio and asking for the angle, with the answer pinned to one restricted range?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Am I starting from a ratio and asking for the angle, with the answer pinned to one restricted range?

Worked Examples

Example 1

easy

Evaluate

\arcsin\!\left(\frac{1}{2}\right)

\arccos\!\left(-\frac{\sqrt{2}}{2}\right)

, and

\arctan(1)

. State the range of each inverse trig function.

Answer

\arcsin(\frac{1}{2})=\frac{\pi}{6}

;

\arccos(-\frac{\sqrt{2}}{2})=\frac{3\pi}{4}

;

\arctan(1)=\frac{\pi}{4}

First step

\arcsin(\frac{1}{2})

: find

\theta\in[-\pi/2,\pi/2]

with

\sin\theta=\frac{1}{2}

. Answer:

\theta=\frac{\pi}{6}

Full solution

2
$\arccos(-\frac{\sqrt{2}}{2})$ : find $\theta\in[0,\pi]$ with $\cos\theta=-\frac{\sqrt{2}}{2}$ . Answer: $\theta=\frac{3\pi}{4}$ (Q2).
3
$\arctan(1)$ : find $\theta\in(-\pi/2,\pi/2)$ with $\tan\theta=1$ . Answer: $\theta=\frac{\pi}{4}$ .

Inverse trig functions return angles in their restricted ranges:

\arcsin

[-\pi/2,\pi/2]

;

\arccos

[0,\pi]

;

\arctan

(-\pi/2,\pi/2)

. These restrictions ensure the inverse is a function (one-to-one).

Example 2

hard

Simplify

\sin(\arccos(x))

for

x\in[-1,1]

without trigonometric functions in the final answer.

Example 3

easy

A right triangle has opposite side

1

and adjacent side

\sqrt{3}

. Find the angle opposite to the side of length

1

Example 4

medium

Solve

\cos x = -\dfrac{1}{2}

[0, 2\pi]

using

\arccos

for the principal solution.

Example 5

medium

20

-ft ladder leans against a wall and reaches

16

ft up. Find the angle the ladder makes with the ground.

Example 6

hard

Find the exact value of

\arctan(1) + \arctan(2) + \arctan(3)

Example 7

hard

From a point

50

m from a tower, the angle of elevation to the top is

\arctan(2)

. Find the tower's height and the angle in degrees (to

1

d.p.).

Example 8

challenge

Show that

\arctan\left(\dfrac{1}{2}\right) + \arctan\left(\dfrac{1}{3}\right) = \dfrac{\pi}{4}

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy

A right triangle has opposite side

5

and hypotenuse

13

. Find the angle

\theta

opposite the side of length

5

using

\arcsin

Example 2

medium

Solve

2\sin(x)-1=0

[0,2\pi]

using

\arcsin

\arctan(0)

Example 9

easy

Does

\sin^{-1}(x)

mean

\frac{1}{\sin x}

Example 18

medium

Find

\arctan(\sqrt{3})

Example 19

medium

Evaluate

\sin\left(\arccos\left(\frac{12}{13}\right)\right)

Example 20

challenge

Simplify

\tan(\arcsin x)

for

x \in (-1, 1)

, expressing in terms of

x

Example 21

challenge

Evaluate

\arcsin\left(\sin\left(\frac{5\pi}{6}\right)\right)

Example 22

challenge

Show that

\arcsin x + \arccos x = \frac{\pi}{2}

for all

x \in [-1, 1]

medium

Evaluate

\tan\left(\arccos\left(\dfrac{3}{5}\right)\right)

Example 33

medium

Solve

\tan x = 1

(-\pi, \pi]

using

\arctan

Example 34

hard

Simplify

\sin(2\arctan(x))

for

x > 0

Example 35

hard

Simplify

\cos(2\arcsin(x))

for

x \in [-1, 1]

Example 36

hard

Find

\dfrac{d}{dx}\arcsin(x)

Example 37

hard

Find

\arccos\left(\cos\left(\dfrac{7\pi}{6}\right)\right)

Example 38

challenge

Solve

\arcsin x + \arccos(2x) = \dfrac{\pi}{2}

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Related Concepts

Trigonometric Functions Inverse Function Domain

Background Knowledge

These ideas may be useful before you work through the harder examples.

trigonometric functionsinverse functiondomain