# Phase shifts for sine and cosine

Lesson

Phase shift for trigonometric functions means moving the graph of the function to the right or to the left. This transformation occurs when a constant is added to (or subtracted from) the angle to which the function is applied.

For example, the following functions include a phase shift transformation.

$\sin\left(\theta+45^\circ\right)$sin(θ+45°)
$\cos(x-37.5^\circ)$cos(x37.5°)
$\tan\left(\alpha+180^\circ\right)$tan(α+180°)

#### Example 1

The following graph shows the functions $\cos\theta$cosθ and $\cos(\theta+23^\circ)$cos(θ+23°) on the same axes.

The graph of $\cos\theta$cosθ is shown in black. It can be seen that the graph of $\cos(\theta+23^\circ)$cos(θ+23°) is the graph of $\cos\theta$cosθ shifted to the left by the amount $23^\circ$23°

The dotted lines drawn on the diagram are intended to show that the function $\cos(\theta+23^\circ)$cos(θ+23°) when $\theta=20^\circ$θ=20° attains the same value reached by $\cos\theta$cosθ when $\theta=43^\circ$θ=43°. Thus, the shift is to the left.

#### Example 2

The following graph looks like the graph of $\sin\theta$sinθ with a phase shift of $60^\circ$60° to the right.

The graph must belong to the function given by $\sin(\theta-60^\circ)$sin(θ60°). The phase shift to the right has been brought about by adding $-60^\circ$60° to $\theta$θ.

#### Worked examples

##### Question 1

Consider the function $f\left(x\right)=\sin x$f(x)=sinx and $g\left(x\right)=\sin\left(x-90^\circ\right)$g(x)=sin(x90°).

1. Complete the table of values for both functions.

 $x$x $f\left(x\right)$f(x) $g\left(x\right)$g(x) $0$0 $90^\circ$90° $180^\circ$180° $270^\circ$270° $360^\circ$360° $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$
2. Using the table of values, what transformation of the graph of $f\left(x\right)$f(x) results in the graph of $g\left(x\right)$g(x)?

Horizontal translation $90^\circ$90° to the right

A

Horizontal translation $90^\circ$90° to the left

B

Vertical translation $90^\circ$90° upwards

C

Vertical translation $90^\circ$90° downwards

D

Horizontal translation $90^\circ$90° to the right

A

Horizontal translation $90^\circ$90° to the left

B

Vertical translation $90^\circ$90° upwards

C

Vertical translation $90^\circ$90° downwards

D
3. The graph of $f\left(x\right)$f(x) has been provided below.

By moving the points, graph $g\left(x\right)$g(x).

Loading Graph...

##### Question 2

The functions $f\left(x\right)$f(x) and $g\left(x\right)=f\left(x+k\right)$g(x)=f(x+k) have been graphed on the same set of axes in grey and black respectively.

Loading Graph...

1. What transformation has occurred from $f\left(x\right)$f(x) to $g\left(x\right)$g(x)?

Horizontal translation of $135^\circ$135° left.

A

Horizontal stretching by a factor of $135^\circ$135°.

B

Horizontal translation of $135^\circ$135° right.

C

Vertical translation of $135^\circ$135° up.

D

Horizontal translation of $135^\circ$135° left.

A

Horizontal stretching by a factor of $135^\circ$135°.

B

Horizontal translation of $135^\circ$135° right.

C

Vertical translation of $135^\circ$135° up.

D
2. Determine the smallest positive value of $k$k.

##### Question 3

The graph of $y=\sin x$y=sinx is translated $60^\circ$60° to the left.

1. What is the equation of the new curve?

2. What is the amplitude of the new curve?

3. What is the period of the new curve?