Test the graph of the following equation for symmetry with respect to $\theta = \frac { \pi } { 2 }$, the polar axis, and the pole.
$$r = - 6 + \cos ( 5 \theta )$$
A) The graph is symmetric with respect to $\theta = \frac { \pi } { 2 }$ only.
B) The graph is symmetric with respect to the polar axis only.
C) The graph is symmetric with respect to the pole only.
D) The graph is symmetric with respect to all three.
E) The graph has no symmetries.

Question

Quizplus · Accepted Answer

To test for symmetry with respect to $\theta = \frac{\pi}{2}$, we check if $r(\theta) = r\left(\frac{\pi}{2} - \theta\right)$. However, we have:
$$r(\theta) \neq r\left(\frac{\pi}{2} - \theta\right)$$
Therefore, the graph is not symmetric with respect to $\theta = \frac{\pi}{2}$.

To test for symmetry with respect to the pole, we check if $r(\theta) = r(-\theta)$. We have:
$$r(\theta) = -6 + \cos(5\theta)$$
$$r(-\theta) = -6 + \cos(-5\theta) = -6 + \cos(5\theta)$$
Therefore, the graph is symmetric with respect to the pole.

To test for symmetry with respect to the polar axis, we check if $r(\theta) = r(\pi - \theta)$. We have:
$$r(\theta) = -6 + \cos(5\theta)$$
$$r(\pi - \theta) = -6 + \cos(5(\pi - \theta)) = -6 - \cos(5\theta)$$
Therefore, the graph is not symmetric with respect to the polar axis.

Thus, the only symmetry that the graph exhibits is symmetry with respect to the pole, so the answer is B.

Test the Graph of the Following Equation for Symmetry with Respect