NZ Level 8 (NZC) Level 3 (NCEA) [In development]
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 or number to which the function is applied.

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

$\sin\left(\theta+\frac{\pi}{4}\right)$sin(θ+π4)
$\cos(x-0.5)$cos(x0.5)
$\tan\left(\alpha+\frac{22}{7}\right)$tan(α+227)

#### Example 1

The following graph shows the functions $\cos x$cosx and $\cos(x+0.4)$cos(x+0.4) on the same axes.

The graph of $\cos x$cosx is shown in black. It can be seen that the graph of $\cos(x+0.4)$cos(x+0.4) is the graph of $\cos x$cosx shifted to the left by the amount $0.4$0.4

The dotted lines drawn on the diagram are intended to show that the function $\cos(x+0.4)$cos(x+0.4) when $x=0.5$x=0.5 attains the same value reached by $\cos x$cosx when $x=0.9$x=0.9. Thus, the shift is to the left.

#### Example 2

The following graph looks like the graph of $\sin x$sinx with a phase shift of $1.05$1.05 to the right.

The graph must belong to the function given by $\sin(x-1.05)$sin(x1.05). The phase shift to the right has been brought about by adding $-1.05$1.05 to $x$x.

A more precisely drawn horizontal scale might reveal that the graph actually crosses the axis at $1.047$1.047 which is approximately $\frac{\pi}{3}$π3. So, another way of writing the function is with the expression $\sin\left(x-\frac{\pi}{3}\right)$sin(xπ3).

#### Worked examples

##### Question 1

Consider the given graph of $y=\cos\left(x+\frac{\pi}{2}\right)$y=cos(x+π2).

1. What is the amplitude of the function?

2. How can the graph of $y=\cos x$y=cosx be transformed into the graph of $y=\cos\left(x+\frac{\pi}{2}\right)$y=cos(x+π2)?

By reflecting it about the $x$x-axis, and then translating it horizontally $\frac{\pi}{2}$π2 units to the left.

A

By reflecting it about the $x$x-axis, and then translating it horizontally $\frac{\pi}{2}$π2 units to the right.

B

By translating it horizontally $\frac{\pi}{2}$π2 units to the right.

C

By changing the period of the function.

D

By translating it horizontally $\frac{\pi}{2}$π2 units to the left.

E

By reflecting it about the $x$x-axis, and then translating it horizontally $\frac{\pi}{2}$π2 units to the left.

A

By reflecting it about the $x$x-axis, and then translating it horizontally $\frac{\pi}{2}$π2 units to the right.

B

By translating it horizontally $\frac{\pi}{2}$π2 units to the right.

C

By changing the period of the function.

D

By translating it horizontally $\frac{\pi}{2}$π2 units to the left.

E

##### Question 2

Consider the function $f\left(x\right)=\cos x$f(x)=cosx and $g\left(x\right)=\cos\left(x-\frac{\pi}{2}\right)$g(x)=cos(xπ2).

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 $\frac{\pi}{2}$π2​ $\pi$π $\frac{3\pi}{2}$3π2​ $2\pi$2π $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$
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)?

vertical translation $\frac{\pi}{2}$π2 units downwards

A

horizontal translation $\frac{\pi}{2}$π2 units to the left

B

horizontal translation $\frac{\pi}{2}$π2 units to the right

C

vertical translation $\frac{\pi}{2}$π2 units upwards

D

vertical translation $\frac{\pi}{2}$π2 units downwards

A

horizontal translation $\frac{\pi}{2}$π2 units to the left

B

horizontal translation $\frac{\pi}{2}$π2 units to the right

C

vertical translation $\frac{\pi}{2}$π2 units upwards

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).

##### Question 3

Determine the equation of the graphed function given that it is of the form $y=\cos\left(x-c\right)$y=cos(xc), where $c$c is the least positive value.

### Outcomes

#### M8-2

Display and interpret the graphs of functions with the graphs of their inverse and/or reciprocal functions