Test the graph of the following equation for symmetry with respect to $\theta = \frac { \pi } { 2 }$, the polar axis, and the pole.
$$r = 2 - 2 \sin ( 4 \theta )$$
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

To test for symmetry in polar coordinates:- Symmetry with respect to the polar axis ($\theta = 0$) is tested by replacing $\theta$ with $-\theta$. If the equation remains unchanged, the graph is symmetric with respect to the polar axis. For $r = 2 - 2 \sin(4\theta)$, replacing $\theta$ with $-\theta$ gives $r = 2 - 2 \sin(-4\theta) = 2 - 2(-\sin(4\theta)) = 2 + 2\sin(4\theta)$, which is not the same as the original equation, so it is not symmetric with respect to the polar axis.- Symmetry with respect to the pole is tested by replacing $r$ with $-r$. If the equation remains unchanged or can be manipulated back to the original form, the graph is symmetric with respect to the pole. For $r = 2 - 2 \sin(4\theta)$, replacing $r$ with $-r$ gives $-r = 2 - 2 \sin(4\theta)$, which can be rewritten as $r = 2\sin(4\theta) - 2$, showing that the graph is indeed symmetric with respect to the pole because the manipulation of the equation reflects a symmetry in the radial direction.- Symmetry with respect to $\theta = \frac{\pi}{2}$ is tested by replacing $\theta$ with $\pi - \theta$. For $r = 2 - 2 \sin(4\theta)$, replacing $\theta$ with $\pi - \theta$ gives $r = 2 - 2 \sin(4(\pi - \theta)) = 2 - 2 \sin(4\pi - 4\theta)$, which simplifies to $2 - 2 \sin(-4\theta)$ due to the periodicity and symmetry properties of the sine function, which does not return the graph to its original form, indicating it is not symmetric with respect to $\theta = \frac{\pi}{2}$.Therefore, the graph is symmetric with respect to the pole only.

Test the Graph of the Following Equation for Symmetry with Respect