NZ Level 7 (NZC) Level 2 (NCEA)
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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 $0$0 $90^\circ$90° $180^\circ$180° $270^\circ$270° $360^\circ$360°
    $f\left(x\right)$f(x) $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$
    $g\left(x\right)$g(x) $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$
  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?

 

 

 

 

Outcomes

M7-2

Display the graphs of linear and non-linear functions and connect the structure of the functions with their graphs

91257

Apply graphical methods in solving problems

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