91Ó°ÊÓ

To test for symmetry, replace \(\theta\) by \(-\theta\), \(\pi - \theta\), and \(\pi + \theta\) in the equation.\nFor x-axis symmetry (polar axis symmetry), replace \(\theta\) by \(-\theta\): \(r^2 = 9\sin(2(-\theta)) = 9\sin(-2\theta) = -9\sin(2\theta)\) which is not the same as \(r^2 = 9\sin(2\theta)\). So, there's no symmetry about the x-axis.\nFor y-axis symmetry, replace \(\theta\) by \(\pi - \theta\): \(r^2 = 9\sin(2(\pi - \theta)) = 9\sin(2\pi - 2\theta) = -9\sin(2\theta)\) which is not the same as \(r^2 = 9\sin(2\theta)\). So, there's no symmetry about the y-axis.\nFor origin symmetry, replace \(\theta\) by \(\pi + \theta\): \(r^2 = 9\sin(2(\pi + \theta)) = 9\sin(2\pi + 2\theta) = 9\sin(2\theta)\) which is the equation. So, the polar equation has origin symmetry.

Use a polar plot to graph the polar equation. The graph will show that it has origin symmetry as concluded from step 1.