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

Question

Quizplus · Accepted Answer

The graph is symmetric with respect to the pole because replacing $\theta$ with $-\theta$ or $\theta + \pi$ results in the equation $r = -2 + \sin(-2\theta)$ or $r = -2 + \sin(2\theta + 2\pi)$, both of which simplify back to the original equation due to the periodic and odd properties of the sine function. This indicates pole symmetry but not symmetry with respect to $\theta = \frac{\pi}{2}$ or the polar axis, as those would require different transformations.

Test the Graph of the Following Equation for Symmetry with Respect