Chapter 6.7, Problem 50E

To determine

To graph: the given function of cosine.

Explanation of Solution

Given information:

  $y=2\cos \frac{\theta }{2}$

Calculation:

The equation of cosine function is as follows:

$y=A\cos kx+c$..... (1)

Here $A$ is the coefficient of absolute value of amplitude, $k$ and $c$ are constants.

Compare the given function with equation (1) to get $A=2,k=\frac{1}{2},\theta =x$ and $c=0$ .

The amplitude of the function can be calculated as follows:

  $\begin{array}{c}\text{Amplitude}=\left|A\right|\\ =\left|2\right|\left[\text{Substitute }A\right]\\ =2\end{array}$

The period of the function can be evaluated as follows:

  $\begin{array}{c}\text{Period}=\frac{2\pi }{k}\\ =\frac{2\pi }{\left(\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\right)}\left[\text{Substitute }k\right]\\ =4\pi \end{array}$

Now divide the period into 4 equal parts $\left(i.e.\pi ,2\pi ,3\pi ,4\pi \right)$ and evaluate the value of $y=2\cos \frac{\theta }{2}$ as shown below.

  $\begin{array}{c}y=2\cos \frac{\pi }{2}\left[\text{Put }x=0\right]\\ =2\left(1\right)\\ =2\end{array}$

  $\begin{array}{c}y=2\cos \frac{\pi }{2}\left[\text{Put }x=\pi \right]\\ =0\\ y=2\cos \pi \left[\text{Put }x=2\pi \right]\\ =2\left(-1\right)\\ =-2\\ y=2\cos \frac{3\pi }{2}\left[\text{Put }x=3\pi \right]\\ =2\left(0\right)\\ =0\\ y=2\cos 2\pi \left[\text{Put }x=4\pi \right]\\ =2\left(1\right)\\ =2\end{array}$

Make a table of values as shown below.

$x$	0	$\pi$	$2\pi$	$3\pi$	$4\pi$
$y$	1	0	$-2$	0	2

Plot the points and connect them with a smooth curve to graph $y=2\cos \frac{\theta }{2}$ as shown below.