Frequency Examples in Math: 68 Problems with Answers

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

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

Concept Recap

The number of complete cycles a periodic function completes per unit of its input variable; the reciprocal of the period, $f = \frac{1}{T}$ .

Frequency counts how many complete cycles occur per unit of the horizontal axis — higher frequency means the wave oscillates more rapidly in the same space or time.

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: Frequency counts complete oscillations per unit of time (or space) — the reciprocal of the period.

Common stuck point: The procedure for frequency is the easy part; the trap is reporting frequency as the period. Asking "Am I counting how many complete cycles happen per unit (not the length of one cycle or its height)?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Am I counting how many complete cycles happen per unit (not the length of one cycle or its height)?

Worked Examples

Example 1

easy

Find the period and frequency of

f(x) = \sin(4x)

Answer

T = \frac{π}{2}, f = \frac{2}{π}

First step

For

f(x) = \sin(Bx)

, the period is

T = \frac{2\pi}{|B|}

Full solution

2
Here $B = 4$ , so $T = \frac{2\pi}{4} = \frac{\pi}{2}$ .
3
Frequency is the reciprocal of the period: $f = \frac{1}{T} = \frac{2}{\pi}$ cycles per unit.

Frequency measures how many complete cycles occur per unit of the independent variable. A higher

B

value compresses the wave horizontally, increasing the frequency and decreasing the period. Frequency and period are always reciprocals of each other.

Example 2

medium

A sound wave completes

440

cycles per second. Find the period and write a sine function modeling this wave with amplitude

1

Example 3

medium

Write a sine function with amplitude

3

and frequency

4

Hz, where

t

is in seconds.

Example 4

medium

A wheel rotates at

90

rpm. Find its frequency in Hz and angular frequency in rad/s.

Example 5

hard

A Ferris wheel completes a full turn every

120

s. A rider's height in meters above ground is modeled as

h(t) = 15 - 14\cos(\omega t)

. Find

\omega

and the frequency.

Example 6

medium

A water wave travels at

3 \text{ m/s}

with wavelength

0.6 \text{ m}

. Find its frequency.

Example 7

hard

A wave travels

480 \text{ m}

4 \text{ s}

and has wavelength

60 \text{ cm}

. Find its frequency.

Practice Problems

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

Example 1

medium

Find the period of

g(t) = 3\cos\left(\frac{\pi t}{5}\right) + 2

Example 2

hard

Two tuning forks produce tones modeled by

y_1 = \sin(880\pi t)

and

y_2 = \sin(884\pi t)

. Find the beat frequency when both are played together.

Example 3

easy

Find the period of

y = \sin(2x)

Example 4

easy

Find the period of

y = \cos(x)

Example 5

easy

Find the period of

y = \sin(4x)

Example 6

easy

The period of a wave is

\frac{1}{2}

second. Find its frequency in Hz.

Example 7

easy

A wave completes

5

cycles per second. What is its frequency?

Example 8

easy

A wave has frequency

4

Hz. Find its period.

Example 9

easy

Find the period of

y = \cos(\pi x)

Example 10

easy

Which has a shorter period:

y = \sin(x)

y = \sin(3x)

Example 11

medium

Find the period of

y = \sin\left(\frac{x}{3}\right)

Example 12

medium

A wave is

y = 3\cos(6x)

. Find its period.

Example 13

medium

A wave repeats every

8

units. Write the inside coefficient

B

for

\sin(Bx)

Example 14

medium

A pendulum swings with period

0.25

s. Find its frequency.

Example 15

medium

Find the period of

y = \tan(2x)

Example 16

medium

Two waves have frequencies

3

Hz and

6

Hz. How do their periods compare?

Example 17

medium

Find the period of

y = \sin(\pi x) + \cos(\pi x)

Example 18

medium

A sound wave has period

\frac{1}{440}

s. Find its frequency.

Example 19

challenge

A wave is

y = \sin(\frac{2\pi}{5}x)

. Find both its period and frequency (cycles per unit).

Example 20

challenge

Two notes have frequencies

200

Hz and

300

Hz played together. After how many seconds does the combined pattern repeat?

Example 21

medium

Find the period of

y = \cos(\frac{\pi}{2}x)

Example 22

challenge

A wheel rotates at

120

revolutions per minute. Find its frequency in Hz.

Example 23

easy

A wave has period

T = 0.1

s. Find its frequency.

Example 24

easy

Find the period of

y = \sin(6x)

Example 25

easy

Find the period of

y = \cos(5x)

Example 26

easy

What is the period of

y = \sin(\pi x / 2)

Example 27

easy

Find the frequency (in cycles per unit) of

y = \sin(2\pi x)

Example 28

medium

Find the period of

y = 5\sin(\frac{2\pi}{7}x)

Example 29

medium

A pendulum makes

30

swings in

15

s. Find its frequency and period.

Example 30

medium

Find the period of

y = \tan(5x)

Example 31

medium

Find the period of

y = \sin(3x) + \cos(3x)

Example 32

medium

Frequency is

f = 0.25

Hz. Find the period.

Example 33

medium

Write a cosine model with amplitude

2

and period

6

, where

t

is the variable.

Example 34

hard

Find the period of

y = \sin(2x) + \cos(3x)

Example 35

hard

A wave model

y = 5\sin(2\pi(120) t)

describes a tone. Find the frequency in Hz.

Example 36

hard

Two tones

\sin(2\pi \cdot 200 t)

and

\sin(2\pi \cdot 203 t)

are heard together. What is the beat frequency?

Example 37

hard

A wave is described by

y = 4\sin(0.5\pi t - \pi/3)

. Find its period.

Example 38

hard

Find the period of

y = \sin^2(x)

Example 39

challenge

Find the smallest positive period of

y = \sin(3x) \cos(5x)

Example 40

challenge

A musical note A4 is

440

Hz. Find its angular frequency and write a sine model for the pressure wave with amplitude

1

Example 41

easy

A wave has period

0.5 \text{ s}

. Find its frequency.

Example 42

easy

A speaker vibrates 1000 times per second. What is its frequency?

Example 43

easy

A wave has frequency

5 \text{ Hz}

. What is its period?

Example 44

easy

A pendulum makes 30 complete swings in 1 minute. Find its frequency.

Example 45

medium

A sound wave in air has speed

340 \text{ m/s}

and frequency

170 \text{ Hz}

. Find its wavelength.

Example 46

medium

A radio wave has frequency

500 \text{ kHz}

. Find its wavelength. Use

c = 3 \times 10^8 \text{ m/s}

Example 47

medium

A wave passes 240 wave crests in 1 minute. Find its frequency in Hz.

Example 48

medium

Light has speed

3\times 10^8 \text{ m/s}

and a green wavelength of

5\times 10^{-7} \text{ m}

. Find its frequency.

Example 49

medium

Two tuning forks vibrate at

440 \text{ Hz}

and

441 \text{ Hz}

. They are heard together. What is the beat frequency?

Example 50

medium

A wave has wavelength

2.5 \text{ m}

and frequency

120 \text{ Hz}

. Find its speed.

Example 51

medium

A wave of frequency

50 \text{ Hz}

and wavelength

4 \text{ m}

enters a new medium where its speed doubles. What is its new wavelength?

Example 52

medium

An oscillator runs at

60 \text{ MHz}

. Find the period in nanoseconds.

Example 53

hard

A guitar string fixed at both ends has length

0.5 \text{ m}

and wave speed

400 \text{ m/s}

. Find the fundamental frequency.

Example 54

hard

An open-ended pipe of length

0.85 \text{ m}

has speed of sound

340 \text{ m/s}

. Find its fundamental frequency.

Example 55

hard

A microwave oven operates at

2.45 \text{ GHz}

. Find the wavelength. Use

c = 3 \times 10^8 \text{ m/s}

Example 56

hard

A simple pendulum of length

1 \text{ m}

swings with period

T = 2\pi\sqrt{L/g}

. Find its frequency. Use

g = 9.8 \text{ m/s}^2

Example 57

hard

A stationary observer hears a

500 \text{ Hz}

siren on an ambulance approaching at

34 \text{ m/s}

. With sound speed

340 \text{ m/s}

, what frequency does the observer hear?

Example 58

hard

A guitar string has fundamental frequency

200 \text{ Hz}

. What are the frequencies of its 2nd, 3rd, and 4th harmonics?

Example 59

hard

An electric guitar string's fundamental is

330 \text{ Hz}

. To raise the pitch by an octave (double the frequency), by what factor must the wave speed change if length is constant?

Example 60

challenge

Two waves of frequencies

498 \text{ Hz}

and

502 \text{ Hz}

are superimposed. How many beats per second are heard, and what is the average tone frequency?

Example 61

challenge

A wave on a string has frequency

25 \text{ Hz}

, wavelength

0.4 \text{ m}

, and travels along a

4 \text{ m}

string. How long does the wave take to travel the full string length?

← Back to Frequency Formula Explained Practice Mode

Related Concepts

Periodic Functions Unit Rate Trigonometric Functions

Background Knowledge

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

periodic functionsunit ratetrigonometric functions