Question

test for symmetry and then identify the graph of the polar equation.
r = 2 sin θ
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 is not symmetric with respect to the the polar axis.
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 polar axis.
b. is the polar equation symmetrical with respect to the line θ = π/2?
a. the polar equation failed the test for symmetry which means that the graph is not symmetric with respect to the line θ = π/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 θ = π/2.
c. is the polar equation symmetrical with respect to the pole?
a. the polar equation failed the test for symmetry which means that the graph is not symmetric with respect to the pole.
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 pole.

Explanation:

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

Replace $\theta$ with $-\theta$. The equation becomes $r = 2\sin(-\theta)=- 2\sin\theta$. Since $r = 2\sin\theta$ and $r=-2\sin\theta$ are not equivalent (except when $\sin\theta = 0$), the test for symmetry about the polar axis fails. But a failed test does not mean the graph is not symmetric.

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

Replace $(r,\theta)$ with $(-r,-\theta)$. We get $-r = 2\sin(-\theta)\Rightarrow -r=-2\sin\theta\Rightarrow r = 2\sin\theta$, which is the original equation. So, the graph is symmetric about the line $\theta=\frac{\pi}{2}$.

Step3: Test for symmetry about the pole

Replace $r$ with $-r$. The equation becomes $-r = 2\sin\theta\Rightarrow r=-2\sin\theta$. Since $r = 2\sin\theta$ and $r=-2\sin\theta$ are not equivalent (except when $\sin\theta = 0$), the test for symmetry about the pole fails. But a failed test does not mean the graph is not symmetric.

Answer:

a. C. The polar equation failed the test for symmetry which means that the graph may or may not be symmetric with respect to the the polar axis.
b. B. Yes.
c. C. The polar equation failed the test for symmetry which means that the graph may or may not be symmetric with respect to the pole.