Question

test for symmetry and graph the polar equation.
r = 8 sin(3θ)
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 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 may or may
not be symmetric with respect to the pole.
b. yes.
c. the polar equation failed the test for symmetry which means that the graph is not
symmetric with respect to the pole.

Explanation:

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

Replace $\theta$ with $-\theta$ in the equation $r = 8\sin(3\theta)$.
We get $r = 8\sin(- 3\theta)=-8\sin(3\theta)$.
Since $r = 8\sin(3\theta)$ and $r=-8\sin(3\theta)$ are not equivalent (except when $\sin(3\theta)=0$), the test for symmetry about the polar axis fails.
But a failed test for symmetry does not necessarily mean the graph is not symmetric.

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

Replace $(r,\theta)$ with $(-r,-\theta)$ in the equation $r = 8\sin(3\theta)$.
We have $-r = 8\sin(-3\theta)=-8\sin(3\theta)$, which simplifies to $r = 8\sin(3\theta)$.
The equation remains the same, so the graph is symmetric about the line $\theta=\frac{\pi}{2}$.

Step3: Test for symmetry about the pole

Replace $r$ with $-r$ in the equation $r = 8\sin(3\theta)$.
We get $-r = 8\sin(3\theta)$, or $r=-8\sin(3\theta)$.
Since $r = 8\sin(3\theta)$ and $r=-8\sin(3\theta)$ are not equivalent (except when $\sin(3\theta)=0$), the test for symmetry about the pole fails.
But a failed test for symmetry does not necessarily mean the graph is not symmetric.

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.
c. A. The polar equation failed the test for symmetry which means that the graph may or may not be symmetric with respect to the pole.