Answer
symmetric with respect to the line $\theta=\frac{\pi}{2}$, may or may not be symmetric with respect to the pole and the polar axis.
Work Step by Step
1. Given $r=5sin\theta cos^2\theta$, to test symmetry with respect to the pole, replace $r$ with $-r$, we have $-r=5sin\theta cos^2\theta$ which changes the original equation, thus it may or may not be symmetric with respect to the pole.
2. To test symmetry with respect to the polar axis, replace $\theta$ with $-\theta$, we have $r=5sin(-\theta) cos^2(-\theta)$which changes the original equation, thus it may or may not be symmetric with respect to the polar axis.
3. To test symmetry with respect to the line $\theta=\frac{\pi}{2}$, replace $\theta$ with $\pi-\theta$, we have $r=5sin(\pi-\theta) cos^2(\pi-\theta)$ which does not change the original equation, thus it is symmetric with respect to the line $\theta=\frac{\pi}{2}$.
