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 = \frac{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 = \frac{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 = \frac{π}{2}.

Explanation:

For symmetry about the polar axis (\(\theta = 0\)), we replace \(\theta\) with \(-\theta\). So \(r = 8\sin(3(-\theta))=8\sin(- 3\theta)=-8\sin(3\theta)\). 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. A failed test for symmetry does not guarantee lack of symmetry, just that we cannot confirm it from the test.

For symmetry about the line \(\theta=\frac{\pi}{2}\), we replace \((r,\theta)\) with \((-r,-\theta)\). So \(-r = 8\sin(3(-\theta))=-8\sin(3\theta)\), or \(r = 8\sin(3\theta)\). This is the same as the original equation. So the polar equation is symmetric about the line \(\theta=\frac{\pi}{2}\)

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.