Chapter 9, Problem 22RE

To determine

To sketch:the graph of the given polar equation.

Answer to Problem 22RE

  $r=2+2\cos \theta \left(-\theta \right)=2-2\cos \theta$

Explanation of Solution

Given:

The given polarequation is $r=2+2\cos \theta$ .

Calculation:

As it is known that $x=r\cos \theta y=r\sin \theta$ .

The given equation is $r=2+2\cos \theta$

First check for symmetry:

Polar Axis: replace $\theta \to -\theta$

  $r=2+2\cos \left(-\theta \right)=2+2\cos \theta$

The test is satisfied; therefore, the graph is symmetric with respect to polar axis.

The line $\theta =\frac{\pi }{2}$ : replace $\theta \to \pi -\theta$

  $r=2+2\cos \theta \left(-\theta \right)=2-2\cos \theta$

The test fails; therefore, the graph is not symmetric with respect to the line $\theta =\frac{\pi }{2}$ .

Now, draw the graph of the given polar equation. The graph is: