EZ Math

Trigonometry · Sine and cosine graphs

How to Find the Phase Shift of a Sine Graph

Problem

y=sin⁡(x−π2)y = \sin\left(x - \frac{\pi}{2}\right)

Answer

π2 to the right\frac{\pi}{2} \text{ to the right}

A phase shift slides the wave left or right without changing its shape. The sign inside the brackets points the opposite way from the direction of travel.

Step-by-step solution

  1. Match the function to the form y = sin(x - c).

    c=π2c = \frac{\pi}{2}

    Why: The phase shift is the number subtracted inside the brackets.

  2. Read the direction of the shift.

    x−c⇒shift rightx - c \Rightarrow \text{shift right}

    Why: Subtracting inside the function moves the graph in the positive direction.

  3. Write the shift.

    π2 to the right\frac{\pi}{2} \text{ to the right}

    Why: The whole wave starts pi over 2 later than the plain sine graph.

Common mistakes

  • Shifting left because the sign is minus. Minus inside means right.
  • Treating the number outside the brackets as the phase shift.
  • Confusing a horizontal shift with a vertical shift.

Practice problems

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

  1. y=sin⁡(x−π6)y = \sin\left(x - \frac{\pi}{6}\right)

    Show answer

    π6 to the right\frac{\pi}{6} \text{ to the right}

  2. y=cos⁡(x+π3)y = \cos\left(x + \frac{\pi}{3}\right)

    Show answer

    π3 to the left\frac{\pi}{3} \text{ to the left}

  3. y=sin⁡(x−π)y = \sin(x - \pi)

    Show answer

    π to the right\pi \text{ to the right}

  4. y=cos⁡(x+π4)y = \cos\left(x + \frac{\pi}{4}\right)

    Show answer

    π4 to the left\frac{\pi}{4} \text{ to the left}

  5. y=sin⁡(x+π2)y = \sin\left(x + \frac{\pi}{2}\right)

    Show answer

    π2 to the left\frac{\pi}{2} \text{ to the left}