Chapter 7.5, Problem 64E

To determine

To graph: the given function.

Explanation of Solution

Given information:

  $y=\frac{2}{3}\cos \theta$

Calculation:

The given function is in the form of $y=A\cos kx$ . Compare the given function with the cosine function to get $x=\theta ,A=\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.$ and $k=1$ . The amplitude and period of function is evaluated as shown below.

  $\begin{array}{c}\text{Amplitude}=\left|A\right|\end{array}$

  $\begin{array}{c}=\left|\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.\right|\left[\text{Put the value of }A\right]\end{array}$

  $\begin{array}{c}=\frac{2}{3}\end{array}$

The period of function is evaluated as shown below.

  $\begin{array}{c}\text{Period}=\frac{2\pi }{k}\end{array}$

  $\begin{array}{c}=\frac{2\pi }{1}\left[\text{Put the value of }k\right]\end{array}$

  $\begin{array}{c}=2\pi \end{array}$

Now divide the period into four equal lengths (i.e. $\frac{\pi }{2},\pi ,\frac{3\pi }{2}$ and $2\pi$ ) to evaluate the points as shown below.

  $\begin{array}{c}y=\frac{2}{3}\cos 0\left[\text{For }\theta =0\right]\end{array}$

  $\begin{array}{c}=\frac{2}{3}\end{array}$

  $\begin{array}{c}y=\frac{2}{3}\cos \frac{\pi }{2}\left[\text{For }\theta =\frac{\pi }{2}\right]\end{array}$

  $\begin{array}{c}=0\end{array}$

  $\begin{array}{c}y=\frac{2}{3}\cos \pi \left[\text{For }\theta =\pi \right]\end{array}$

  $\begin{array}{c}=-\frac{2}{3}\end{array}$

  $\begin{array}{c}y=\frac{2}{3}\cos \frac{3\pi }{2}\left[\text{For }\theta =\frac{3\pi }{2}\right]\end{array}$

  $\begin{array}{c}=0\end{array}$

  $\begin{array}{c}y=\frac{2}{3}\cos 2\pi \left[\text{For }\theta =2\pi \right]\end{array}$

  $\begin{array}{c}=\frac{2}{3}\end{array}$

Create a table of values as shown below.

$x$	0	$\raisebox{1ex}{$\pi $}\!\left/ \!\raisebox{-1ex}{$2$}\right.$	$\pi$	$\raisebox{1ex}{$3\pi $}\!\left/ \!\raisebox{-1ex}{$2$}\right.$	$2\pi$
$y$	$\frac{2}{3}$	0	$-\frac{2}{3}$	0	$\frac{2}{3}$

Plot the coordinates on the graph and connect them with a smooth curve as shown below.