Question

test for symmetry and graph the polar equation.
r = 6 + 6\cos\theta
a. is the polar equation symmetrical with respect to the polar axis?
a. the polar equation failed the test for symmetry which means that the graph is not symmetric with respect to the polar axis.
b. yes.
c. the polar equation failed the test for symmetry which means that the graph may or may not be symmetric with respect to the polar axis.
b. is the polar equation symmetrical with respect to the line \theta=\frac{\pi}{2}?
a. yes.
b. the polar equation failed the test for symmetry which means that the graph may or may not be symmetric with respect to the line \theta=\frac{\pi}{2}.
c. the polar equation failed the test for symmetry which means that the graph is not symmetric with respect to the line \theta=\frac{\pi}{2}.

Explanation:

Step1: Test for polar - axis symmetry

Replace \(\theta\) with \(-\theta\) in the equation \(r = 6+6\cos\theta\).
Since \(\cos(-\theta)=\cos\theta\), we get \(r = 6 + 6\cos(-\theta)=6 + 6\cos\theta\). The equation remains the same, so it is symmetric about the polar axis.

Step2: Test for symmetry about the line \(\theta=\frac{\pi}{2}\)

Replace \((r,\theta)\) with \((r,\pi - \theta)\) or \((-r,-\theta)\).
If we use \((r,\pi-\theta)\): \(r = 6+6\cos(\pi - \theta)=6-6\cos\theta
eq6 + 6\cos\theta\) (in general).
If we use \((-r,-\theta)\): \(-r=6 + 6\cos(-\theta)=6 + 6\cos\theta\Rightarrow r=-6 - 6\cos\theta
eq6 + 6\cos\theta\) (in general).
The polar equation fails the test for symmetry about the line \(\theta=\frac{\pi}{2}\). But a failed test does not guarantee no symmetry.

Answer:

a. B. Yes.
b. B. The polar equation failed the test for symmetry which means that the graph may or may not be symmetric with respect to the line \(\theta=\frac{\pi}{2}\).