EZ Math

Trigonometry · Sine and cosine graphs

How to Find the Period of a Cosine Function

Problem

y=cos⁡(4x)y = \cos(4x)

Answer

π2\frac{\pi}{2}

Cosine repeats forever. The period is the length of one full repeat, and the number b inside the function speeds the wave up or slows it down.

Step-by-step solution

  1. Match the function to the form y = cos(bx).

    b=4b = 4

    Why: The number multiplying x is b.

  2. Write the period formula.

    T=2πbT = \frac{2\pi}{b}

    Why: A cosine wave repeats every 2 pi before b stretches it.

  3. Substitute b and simplify.

    T=2π4=π2T = \frac{2\pi}{4} = \frac{\pi}{2}

    Why: Dividing 2 pi by 4 gives pi over 2.

Common mistakes

  • Reading the period as 4. The b value changes the period, it is not the period.
  • Using pi instead of 2 pi. One full cycle of cosine is 2 pi.
  • Taking b from the whole term 4x and forgetting to simplify.

Practice problems

Use the same method. Work each problem on paper, then open the answer to check.

  1. y=cos⁡(2x)y = \cos(2x)

    Show answer

    π\pi

  2. y=cos⁡(3x)y = \cos(3x)

    Show answer

    2π3\frac{2\pi}{3}

  3. y=cos⁡(x2)y = \cos\left(\frac{x}{2}\right)

    Show answer

    4π4\pi

  4. y=cos⁡(5x)y = \cos(5x)

    Show answer

    2π5\frac{2\pi}{5}

  5. y=cos⁡(6x)y = \cos(6x)

    Show answer

    π3\frac{\pi}{3}