Question

test for symmetry and graph the polar equation.
r = 5 - 5\sin\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 may or may not be symmetric with respect to the polar axis.
b. yes.
c. the polar equation failed the test for symmetry which means that the graph is not symmetric with respect to the polar axis.
b. is the polar equation symmetrical with respect to the line \theta=\frac{\pi}{2}?
a. 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}.
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 line \theta=\frac{\pi}{2}.

Explanation:

Step1: Test for symmetry about the polar axis ($\theta = 0$)

Replace $\theta$ with $-\theta$.
Original equation: $r = 5-5\sin\theta$.
After replacement: $r = 5 - 5\sin(-\theta)=5 + 5\sin\theta$.
Since $5 + 5\sin\theta
eq5 - 5\sin\theta$ (except for some specific values of $\theta$), the equation fails the symmetry - about - polar - axis test. But failure of the test does not necessarily mean the graph is not symmetric.

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

Replace $\theta$ with $\pi-\theta$.
Original equation: $r = 5-5\sin\theta$.
After replacement: $r = 5-5\sin(\pi - \theta)$.
Using the identity $\sin(\pi - \theta)=\sin\theta$, we get $r = 5-5\sin\theta$.

Answer:

a. A. 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. B. Yes.