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 ma
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.

Explanation:

Step1: Test symmetry about polar axis

Replace \(\theta\) with \(-\theta\) in \(r = 8\sin(3\theta)\).
We get \(r = 8\sin(- 3\theta)\).
Using the identity \(\sin(-x)=-\sin(x)\), we have \(r=-8\sin(3\theta)\).
This is not the same as the original equation \(r = 8\sin(3\theta)\).

Step2: Analyze the result of symmetry test

A failed symmetry test about the polar axis (using the substitution \(\theta
ightarrow-\theta\)) does not guarantee that the graph is not symmetric about the polar axis.
The test is a sufficient - but - not - necessary condition for symmetry.

Answer:

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.