Question

Graphing Polar Equations

Use a graphing device to graph the polar equation. Choose the domain$\theta$ to make sure you produce the entire graph

$r=\cos (\theta /2)$

Short answer

The graph of the given polar equation is-

Step-by-step solution

Step 1. Given information

A polar equation is given as $r=\cos (\theta /2)$

Step 2. Concept used

For sketching the graph, first, find different values r for different$\theta$. After that, convert them into x and y by using equations-

$\begin{array}{l}x=r\cos \theta \\ y=r\sin \theta \end{array}$

Once we have the points draw a graph by plotting them.

Step 3. Calculation

The polar equation is given as$r=\cos (\theta /2)$,

Let$\theta =0$,

$\begin{array}{l}r=\cos \frac{\theta }{2}\\ r=\cos \frac{0}{2}\\ r=1\end{array}$

Now when we plug the values in the equations $\begin{array}{l}x=r\cos \theta \\ y=r\sin \theta \end{array}$

We get, $\begin{array}{l}x=1\\ y=0\end{array}$

Similarly,

Let$\theta =1$,

$\begin{array}{l}r=\cos \frac{\theta }{2}\\ r=\cos \frac{1}{2}\\ r=0.877\end{array}$

Let$\theta =-0.5$,

$\begin{array}{l}r=\cos \frac{\theta }{2}\\ r=\cos \left(\frac{-0.5}{2}\right)\\ r=0.96\end{array}$

Plot the points on graph-