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(x−0.5)
$\tan\left(\alpha+\frac{22}{7}\right)$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(x−1.05). 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(x−π3).

Worked examples

Question 1

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

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What is the amplitude of the function?
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

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(x−π2).

Complete the table of values for both functions.

$x$`x`	$0$0	$\frac{\pi}{2}$π2	$\pi$π	$\frac{3\pi}{2}$3π2	$2\pi$2π
$f\left(x\right)$`f`(`x`)	$\editable{}$	$\editable{}$	$\editable{}$	$\editable{}$	$\editable{}$
$g\left(x\right)$`g`(`x`)	$\editable{}$	$\editable{}$	$\editable{}$	$\editable{}$	$\editable{}$

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
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 3

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

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Outcomes

11.SF.TF.1

Positive and negative angles. Measuring angles in radians and in degrees and conversion from one measure to another. Definition of trigonometric functions with the help of unit circle. Truth of the identity sin^2 x + cos^2 x = 1, for all x. Signs of trigonometric functions and sketch of their graphs. Expressing sin (x + y) and cos (x + y) in terms of sin x, sin y, cos x and cos y. Deducing the identities like following: cot(x + or - y), sin x + sin y, cos x + cos y, sin x - sin y, cos x - cos y (see syllabus document)

Phase shifts for sine and cosine