Question

test for symmetry and then identify the graph of the polar equation.
r = 2 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 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.

Explanation:

Step1: Test for symmetry about the polar axis

Replace \(\theta\) with \(-\theta\) in the equation \(r = 2\sin\theta\).
We get \(r=2\sin(-\theta)\).
Since \(\sin(-\theta)=-\sin\theta\), the equation becomes \(r = - 2\sin\theta\), which is not the same as the original equation \(r = 2\sin\theta\).

Step2: Analyze the result of the symmetry test

A failed test for symmetry about the polar axis does not guarantee that the graph is not symmetric about the polar axis. There are graphs that fail the symmetry test but are still symmetric. For example, \(r=\sin(2\theta)\) fails the test for symmetry about the polar axis when we substitute \(\theta\) with \(-\theta\) (\(r=\sin(- 2\theta)=-\sin(2\theta)
eq\sin(2\theta)\)), but it has symmetry about the polar axis due to its overall properties as a rose - curve.

Answer:

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.