Frequency Examples in Math

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

Frequency is the number of complete cycles of a periodic process per unit of input (often time).

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: In f(x) = \sin(bx), the frequency is b/(2\pi) and the period is 2\pi/b. Frequency and period are reciprocals: higher frequency means shorter period.

Common stuck point: The coefficient b in \sin(bx) is the angular frequency (radians per unit), not the ordinary frequency (cycles per unit) — divide by 2\pi to convert.

Sense of Study hint: Compute period first, then use frequency as its reciprocal.

Worked Examples

Example 1

easy
Find the period and frequency of f(x) = \sin(4x).

Solution

  1. 1
    For f(x) = \sin(Bx), the period is T = \frac{2\pi}{|B|}.
  2. 2
    Here B = 4, so T = \frac{2\pi}{4} = \frac{\pi}{2}.
  3. 3
    Frequency is the reciprocal of the period: f = \frac{1}{T} = \frac{2}{\pi} cycles per unit.

Answer

T = \frac{\pi}{2}, \quad f = \frac{2}{\pi}
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.

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.
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Background Knowledge

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

periodic functionsunit ratetrigonometric functions