Trigonometric Functions Examples: 46 Problems with Answers

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

Trigonometric functions (sin, cos, tan, etc.) relate angles in right triangles to side ratios and extend to periodic functions of real numbers via the unit circle.

Angles have numbers associated with them—sin, cos, tan capture different ratios.

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: Trig functions convert an angle into a fixed ratio of triangle sides or a coordinate on the unit circle.

Common stuck point: The procedure for trigonometric functions is the easy part; the trap is mixing up which sides belong to sin versus cos. Asking "Am I linking an angle to a ratio of sides (or a point on the unit circle)?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Am I linking an angle to a ratio of sides (or a point on the unit circle)?

Worked Examples

Example 1

easy

Evaluate

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

and

\cos\left(\frac{\pi}{6}\right)

Answer

\sin (\frac{π}{6}) = \frac{1}{2}, \cos (\frac{π}{6}) = \frac{\sqrt{3}}{2}

First step

Convert the angle mentally:

\frac{\pi}{6}

radians equals

30^\circ

Full solution

2
Recall the special-angle values from the unit circle or a $30$ - $60$ - $90$ triangle: $\sin(30^\circ) = \frac{1}{2}$ .
3
Using the same reference triangle, $\cos(30^\circ) = \frac{\sqrt{3}}{2}$ .

The special angles (

30°, 45°, 60°

) and their radian equivalents appear frequently. Memorizing the unit circle values or using the 30-60-90 and 45-45-90 triangle ratios is essential.

Example 2

medium

Find the exact value of

\tan\left(\frac{5\pi}{4}\right)

Example 3

medium

In right triangle

ABC

with right angle at

C

AB = 13

BC = 5

. Find

\sin A

and

\cos A

Example 4

medium

A ramp

20

ft long rises

4

ft. Find the angle of elevation

\theta

to the nearest tenth of a degree.

Example 5

hard

From a point

50

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

32°

. Find the tower's height to the nearest meter.

Example 6

challenge

In triangle

ABC

\angle A = 30°

\angle B = 105°

a = 8

. Find

b

using the Law of Sines.

Practice Problems

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

Example 1

easy

Evaluate

\cos\left(\frac{\pi}{3}\right)

Example 2

hard

\sin\theta = \frac{3}{5}

and

\theta

is in Quadrant II, find

medium

\sin\theta=\frac{3}{5}

and

\theta

is acute, find

\cos\theta

Example 13

medium

Convert

180^\circ

to radians.

Example 14

medium

Find the period of

f(x)=\sin(2x)

Example 15

medium

Evaluate

\tan 60^\circ

Example 16

medium

Find the amplitude and midline of

f(x)=3\sin x+2

Example 17

medium

Solve

\sin x=\frac12

for

0^\circ\le x<360^\circ

Example 18

medium

\cos\theta=\frac{5}{13}

and

\theta

acute, find

\tan\theta

Example 19

challenge

Find the maximum value of

f(x)=4\sin x-3\cos x

Example 20

challenge

Solve

2\sin^2 x-\sin x-1=0

for

0^\circ\le x<360^\circ

Example 21

challenge

A point starts at

(1,0)

and rotates

120^\circ

counterclockwise on the unit circle. Find its coordinates.

Example 22

medium

Find the period of

f(x)=\cos\left(\frac{x}{2}\right)

Example 23

easy

Evaluate

\sin 45°

Example 24

easy

Evaluate

\cos 90°

Example 25

easy

In a right triangle with legs

3

and

4

and hypotenuse

5

\sin\theta

for the angle opposite the side of length

3

is what?

Example 26

easy

Convert

\dfrac{\pi}{4}

radians to degrees.

Example 27

easy

What is the range of

\cos x

Example 28

medium

Evaluate

\sin\!\left(\dfrac{5\pi}{6}\right)

Example 29

medium

\cos\theta = -\tfrac{\sqrt{3}}{2}

and

\theta

is in QIII, find

\sin\theta

Example 30

medium

Convert

210°

to radians.

Example 31

medium

Find

\tan 225°

Example 32

medium

Evaluate

\cos(-60°)

Example 33

medium

Evaluate

\sin(-45°)

Example 34

medium

Solve

\cos x = 0

for

0 \le x < 2\pi

Example 35

medium

State the sign of

\tan\theta

in Quadrant II.

Example 36

hard

\tan\theta = \tfrac{3}{4}

and

\theta

is in QIII, find

\sin\theta

and

\cos\theta

Example 37

hard

Solve

2\cos x + 1 = 0

[0, 2\pi)

Example 38

hard

Solve

\sin x = \cos x

[0, 2\pi)

Example 39

hard

In triangle

ABC

a = 7

b = 9

\angle C = 60°

. Find side

c

using the Law of Cosines.

Example 40

challenge

Find all

x \in [0, 2\pi)

with

\sin x = -\tfrac{\sqrt{3}}{2}

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Related Concepts

Triangles Ratios Unit Circle Periodic Functions

Background Knowledge

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

trianglesratios