Chapter 6.5, Problem 6CFU

To determine

To find: the phase shift of the function and then graph that function.

Answer to Problem 6CFU

  $\frac{\pi }{2}$

Explanation of Solution

Given information:

  $y=3\cos \left(\theta -\frac{\pi }{2}\right)$

Calculation:

The equation of cosine function is as follows:

  $y=A\cos \left(k\theta +c\right)$...........(1)

Here $A$ is an amplitude which is an absolute value of the coefficient and $k=\frac{2\pi }{\text{Period}}$ .

On comparing given equation with equation (1) to get,

  $\begin{array}{l}c=-\frac{\pi }{2}\\ k=1\end{array}$

The phase shift can be evaluated as follows:

  $\begin{array}{c}\text{Phase shift}=-\frac{c}{k}\\ =-\frac{\left(-\frac{\pi }{2}\right)}{1}\left[\text{Put the value of }c\text{ and }k\right]\\ =\frac{\pi }{2}\end{array}$

Thus, the phase shift is $\frac{\pi }{2}$ .

Since $c<0$ , then the shift is towards right. The graph of the function is shown below.

  

The green line represents the graph of $y=3\cos \theta$ and the blue line denotes the graph of $y=3\cos \left(\theta -\frac{\pi }{2}\right)$ . The graph after phase shift moves to right side of graph $y=3\cos \theta$ .