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⁡\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.

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⁡\arcsin, arccos⁡\arccos, arctan⁡\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\sin^{-1} as 1sin⁡\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⁡ ⁣(12)\arcsin\!\left(\frac{1}{2}\right), arccos⁡ ⁣(−22)\arccos\!\left(-\frac{\sqrt{2}}{2}\right), and arctan⁡(1)\arctan(1). State the range of each inverse trig function.

Answer

arcsin⁡(12)=π6\arcsin(\frac{1}{2})=\frac{\pi}{6}; arccos⁡(−22)=3π4\arccos(-\frac{\sqrt{2}}{2})=\frac{3\pi}{4}; arctan⁡(1)=π4\arctan(1)=\frac{\pi}{4}

First step

1
arcsin⁡(12)\arcsin(\frac{1}{2}): find θ∈[−π/2,π/2]\theta\in[-\pi/2,\pi/2] with sin⁡θ=12\sin\theta=\frac{1}{2}. Answer: θ=π6\theta=\frac{\pi}{6}.

Full solution

  1. 2
    arccos⁡(−22)\arccos(-\frac{\sqrt{2}}{2}): find θ∈[0,π]\theta\in[0,\pi] with cos⁡θ=−22\cos\theta=-\frac{\sqrt{2}}{2}. Answer: θ=3π4\theta=\frac{3\pi}{4} (Q2).
  2. 3
    arctan⁡(1)\arctan(1): find θ∈(−π/2,π/2)\theta\in(-\pi/2,\pi/2) with tan⁡θ=1\tan\theta=1. Answer: θ=π4\theta=\frac{\pi}{4}.
Inverse trig functions return angles in their restricted ranges: arcsin⁡\arcsin in [−π/2,π/2][-\pi/2,\pi/2]; arccos⁡\arccos in [0,π][0,\pi]; arctan⁡\arctan in (−π/2,π/2)(-\pi/2,\pi/2). These restrictions ensure the inverse is a function (one-to-one).

Example 2

hard
Simplify sin⁡(arccos⁡(x))\sin(\arccos(x)) for x∈[−1,1]x\in[-1,1] without trigonometric functions in the final answer.

Example 3

easy
A right triangle has opposite side 11 and adjacent side 3\sqrt{3}. Find the angle opposite to the side of length 11.

Example 4

medium
Solve cos⁡x=−12\cos x = -\dfrac{1}{2} on [0,2π][0, 2\pi] using arccos⁡\arccos for the principal solution.

Example 5

medium
A 2020-ft ladder leans against a wall and reaches 1616 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)\arctan(1) + \arctan(2) + \arctan(3).

Example 7

hard
From a point 5050 m from a tower, the angle of elevation to the top is arctan⁡(2)\arctan(2). Find the tower's height and the angle in degrees (to 11 d.p.).

Example 8

challenge
Show that arctan⁡(12)+arctan⁡(13)=π4\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 55 and hypotenuse 1313. Find the angle θ\theta opposite the side of length 55 using arcsin⁡\arcsin.

Example 2

medium
Solve 2sin⁡(x)−1=02\sin(x)-1=0 on [0,2π][0,2\pi] using arcsin⁡\arcsin, and explain why there are two solutions.

Example 3

easy
Find arcsin⁡(1)\arcsin(1).

Example 4

easy
Find arccos⁡(0)\arccos(0).

Example 5

easy
Find arctan⁡(1)\arctan(1).

Example 6

easy
Find arcsin⁡(0)\arcsin(0).

Example 7

easy
Find arccos⁡(1)\arccos(1).

Example 8

easy
Find arctan⁡(0)\arctan(0).

Example 9

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

Example 10

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

Example 11

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

Example 12

medium
Find arcsin⁡(−22)\arcsin\left(-\frac{\sqrt{2}}{2}\right).

Example 13

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

Example 14

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

Example 15

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

Example 16

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

Example 17

medium
A ramp rises 3 ft over a horizontal run of 4 ft. Find its angle of inclination (to the nearest degree).

Example 18

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

Example 19

medium
Evaluate sin⁡(arccos⁡(1213))\sin\left(\arccos\left(\frac{12}{13}\right)\right).

Example 20

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

Example 21

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

Example 22

challenge
Show that arcsin⁡x+arccos⁡x=π2\arcsin x + \arccos x = \frac{\pi}{2} for all x∈[−1,1]x \in [-1, 1], and use it to find arccos⁡x\arccos x if arcsin⁡x=0.3\arcsin x = 0.3.

Example 23

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

Example 24

easy
Find arccos⁡(12)\arccos\left(\dfrac{1}{2}\right).

Example 25

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

Example 26

easy
Find arccos⁡(−1)\arccos(-1).

Example 27

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

Example 28

medium
Evaluate arctan⁡(−13)\arctan\left(-\dfrac{1}{\sqrt{3}}\right).

Example 29

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

Example 30

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

Example 31

medium
Evaluate arcsin⁡(sin⁡(2π3))\arcsin\left(\sin\left(\dfrac{2\pi}{3}\right)\right).

Example 32

medium
Evaluate tan⁡(arccos⁡(35))\tan\left(\arccos\left(\dfrac{3}{5}\right)\right).

Example 33

medium
Solve tan⁡x=1\tan x = 1 on (−π,π](-\pi, \pi] using arctan⁡\arctan.

Example 34

hard
Simplify sin⁡(2arctan⁡(x))\sin(2\arctan(x)) for x>0x > 0.

Example 35

hard
Simplify cos⁡(2arcsin⁡(x))\cos(2\arcsin(x)) for x∈[−1,1]x \in [-1, 1].

Example 36

hard
Find ddxarcsin⁡(x)\dfrac{d}{dx}\arcsin(x).

Example 37

hard
Find arccos⁡(cos⁡(7π6))\arccos\left(\cos\left(\dfrac{7\pi}{6}\right)\right).

Example 38

challenge
Solve arcsin⁡x+arccos⁡(2x)=π2\arcsin x + \arccos(2x) = \dfrac{\pi}{2}.
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Background Knowledge

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

trigonometric functionsinverse functiondomain