Chapter 9, Problem 90CR

To determine

To identify any symmetry and zeros of $r$ and sketch the graph of the given polar equation and also use graphing utility to verify the results.

Answer to Problem 90CR

The symmetry is at polar axis and the graph of the equation is dimpled limacon.

Explanation of Solution

Given information:

The polar equation is $r=7-7\cos \theta$

Calculation:

Replace $\left(r,\theta \right)$ with $\left(r,-\theta \right)$ produces

$\begin{array}{l}r=7-7\cos \left(-\theta \right)\left[\text{replace }\left(r,\theta \right)\text{ with }\left(r,-\theta \right)\right]\\ r=7-7\cos \theta \left[\cos \left(-\theta \right)=\cos \theta \right]\end{array}$

So, by the use of even trigonometric identity, it is concluded that the curve is symmetric with respect to the polar axis. Plotting the points in the table and using the polar axis symmetry the graph can be obtained.

$\theta$	0	$\frac{\pi }{6}$	$\frac{\pi }{3}$	$\frac{\pi }{2}$	$\frac{2\pi }{3}$	$\frac{5\pi }{6}$	$\pi$
$r$	0	$\frac{14-7\sqrt{3}}{2}$	$\frac{7}{2}$	7	$\frac{21}{2}$	$\frac{61}{5}$	14

Now, the graph is represented to verify the obtained results,

Therefore, the graph is dimpled limacon. Hence the obtained result is verified from the graph as symmetry is at polar axis.