Test the graph of the following equation for symmetry with respect to $\theta = \frac { \pi } { 2 }$, the polar axis, and the pole.
$$r = 5 + 4 \cos ( \theta )$$
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 substitute $\theta = \frac{\pi}{2} - \alpha$ and see if we obtain the same equation. Substituting, we get:

$$r = 5 + 4\cos\left(\frac{\pi}{2} - \alpha\right) = 5 + 4\sin\alpha$$

This is not the same as our original equation, so the graph is not symmetric with respect to $\theta = \frac{\pi}{2}$.

To test for symmetry with respect to the polar axis, we substitute $\theta = -\theta$ and see if we obtain the same equation. Substituting, we get:

$$r = 5 + 4\cos(-\theta) = 5 + 4\cos\theta$$

This is the same as our original equation, so the graph is symmetric with respect to the polar axis.

To test for symmetry with respect to the pole, we substitute $\theta = \pi - \theta$ and see if we obtain the same equation. Substituting, we get:

$$r = 5 + 4\cos(\pi - \theta) = 5 - 4\cos\theta$$

This is not the same as our original equation, so the graph is not symmetric with respect to the pole.

Therefore, the answer is B: the graph is symmetric with respect to the polar axis only.

Test the Graph of the Following Equation for Symmetry with Respect