Phase shifts for sine and cosine
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(x−37.5°)
$\tan\left(\alpha+180^\circ\right)$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(θ−60°). 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(x−90°).
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) $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ 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
AHorizontal translation $90^\circ$90° to the left
BVertical translation $90^\circ$90° upwards
CVertical translation $90^\circ$90° downwards
DThe 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.
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.
AHorizontal stretching by a factor of $135^\circ$135°.
BHorizontal translation of $135^\circ$135° right.
CVertical translation of $135^\circ$135° up.
DDetermine 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.
What is the equation of the new curve?
What is the amplitude of the new curve?
What is the period of the new curve?