Chapter 9, Problem 13RE

To determine

To graph: The polar equation.

Explanation of Solution

Given information:

The polar equation is, $r=4-\cos \theta$

Graph:

The polar equation is,

  $r=4-\cos \theta$

Check the symmetry about the polar axis.

For this let us replace $\theta \text{ by -}\theta$ .

We know that cosine function is evan.

Therefore,

  $\begin{array}{c}4-\cos \left(-\theta \right)=4-\cos \theta \\ =r\end{array}$

This shows that the function is symmetry about the polar axis.

Now we will check for the symmetry about the origin.

For this let us replace $r$ by $-r$ .

  $\left(-r\right)=4-\cos \theta$

Therefore, the test is fails.

Therefore, the function may not symmetry about the origin.

Now check for the symmetry about the line $\theta =\frac{\pi }{2}$ .

Replace $\theta$ by $\left(\pi -\theta \right)$

Therefore,

  $4-\cos \left(\pi -\theta \right)=4+\cos \theta$

Therefore, the test is fails.

Therefore, the polar equation is symmetric about the line $\theta =\frac{\pi }{2}$ .

The graph for the function is,

  

Interpretation: Graph for the function shows in figure.