Question

test for symmetry and graph the polar equation.
r = 5 - 5 sinθ
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 = (\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 θ = (\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 θ = (\frac{pi}{2}).
c. is the polar equation symmetrical with respect to the pole?
a. yes.
b. the polar equation failed the test for symmetry which means that the graph is not symmetric with respect to the pole.
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 $-\theta$.
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 test for symmetry about the polar axis fails. But a failed test does not guarantee no - symmetry.

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: $r = 5-5\sin(\pi - \theta)=5 - 5\sin\theta$
Since $r = 5-5\sin\theta$ (the same as the original equation), the graph is symmetric about the line $\theta=\frac{\pi}{2}$

Step3: Test for symmetry about the pole ($r = 0$)

Replace $r$ with $-r$
Original equation: $r = 5-5\sin\theta$
After replacement: $-r=5 - 5\sin\theta\Rightarrow r=-5 + 5\sin\theta$
Since $-5 + 5\sin\theta
eq5 - 5\sin\theta$ (except for some specific values of $\theta$), the test for symmetry about the pole fails. But a failed test does not guarantee no - symmetry.

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. 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.