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.

Explanation:

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

Replace $\theta$ with $-\theta$. The equation becomes $r = 2\sin(-\theta)=- 2\sin\theta$. This is not the same as the original equation $r = 2\sin\theta$. But a failed symmetry test does not necessarily mean the graph is not symmetric. For example, $r=\sin\theta$ is a circle centered at $(0,1)$ in polar coordinates. It is symmetric about the line $\theta=\frac{\pi}{2}$, but not about the polar axis. A failed test for symmetry about the polar axis just means we can't be sure from the test alone.

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

Replace $(r,\theta)$ with $(-r,-\theta)$ or $(r,\pi - \theta)$. Using the second method: $r = 2\sin(\pi-\theta)$. Since $\sin(\pi - x)=\sin x$, we have $r = 2\sin\theta$, which is the same as the original equation.

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 polar axis.
b. B. Yes.