Practice Inverse Trigonometric Functions in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

Worksheet PDF Free Answer-key PDF Family

Quick Recap

Functions that reverse the trigonometric functions: given a ratio, they return the corresponding angle. arcsin⁡\arcsin, arccos⁡\arccos, and arctan⁡\arctan are the inverses of sin⁡\sin, cos⁡\cos, and tan⁡\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⁡\sin and cos⁡\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 0.50.5, arcsin⁡(0.5)=30°\arcsin(0.5) = 30° tells you the angle.

Showing a random 20 of 50 problems.

Example 1

easy
What is the domain of arccos⁡(x)\arccos(x)?

Example 2

medium
Evaluate sin⁡(arctan⁡(2))\sin(\arctan(2)).

Example 3

hard
Find ddxarctan⁡(x)\dfrac{d}{dx}\arctan(x).

Example 4

medium
Find arccos⁡(−12)\arccos\left(-\frac{1}{2}\right).

Example 5

challenge
Evaluate arcsin⁡(sin⁡(5π6))\arcsin\left(\sin\left(\frac{5\pi}{6}\right)\right).

Example 6

easy
Find arcsin⁡(−12)\arcsin\left(-\dfrac{1}{2}\right).

Example 7

medium
Find arctan⁡(3)\arctan(\sqrt{3}).

Example 8

easy
Find arcsin⁡(0)\arcsin(0).

Example 9

easy
A right triangle has opposite side 55 and hypotenuse 1313. Find the angle θ\theta opposite the side of length 55 using arcsin⁡\arcsin.

Example 10

challenge
Simplify tan⁡(arcsin⁡x)\tan(\arcsin x) for x∈(−1,1)x \in (-1, 1), expressing in terms of xx.

Example 11

easy
Find arccos⁡(−1)\arccos(-1).

Example 12

easy
What is the domain of arctan⁡(x)\arctan(x)?

Example 13

easy
Find arcsin⁡(32)\arcsin\left(\dfrac{\sqrt{3}}{2}\right).

Example 14

easy
Evaluate cos⁡(arccos⁡(0.4))\cos(\arccos(0.4)).

Example 15

medium
Evaluate cos⁡(arcsin⁡(35))\cos\left(\arcsin\left(\frac{3}{5}\right)\right).

Example 16

easy
Find arctan⁡(1)\arctan(1).

Example 17

hard
Find the exact value of arctan⁡(1)+arctan⁡(2)+arctan⁡(3)\arctan(1) + \arctan(2) + \arctan(3).

Example 18

medium
Evaluate arccos⁡(cos⁡(4π3))\arccos\left(\cos\left(\frac{4\pi}{3}\right)\right).

Example 19

easy
Does sin⁡−1(x)\sin^{-1}(x) mean 1sin⁡x\frac{1}{\sin x}? Answer yes or no.

Example 20

easy
State the range of arctan⁡(x)\arctan(x).
← Back to Inverse Trigonometric Functions Worked Examples Formula Explained