Answer
1. the function is not symmetric with respect to the polar axis.
2. the function is symmetric with respect to the pole.
3. the function is not symmetric with respect to the line $\theta=\frac{\pi}{2}$
Work Step by Step
Step 1. Test the symmetry with respect to the polar axis: replace $\theta$ with $-\theta$, we have
$r=5cos(-\theta)csc(-\theta)=-5cos\theta csc\theta$ which is different from the original, so the function is not symmetric with respect to the polar axis.
Step 2. Test the symmetry with respect to the pole: replace $\theta$ with $\pi+\theta$, we have
$r=5cos(\pi+\theta) csc(\pi+\theta)=5cos\theta csc\theta$ which is the same as the original, so the function is symmetric with respect to the pole.
Step 3. Test the symmetry with respect to the line $\theta=\frac{\pi}{2}$: replace $\theta$ with $\pi-\theta$, we have $r=5cos(\pi-\theta)csc(\pi-\theta)=-5cos\theta csc\theta$ which is different from the original, so the function is not symmetric with respect to the line $\theta=\frac{\pi}{2}$.
