Answer
The graph is symmetric with respect to the line $\theta =\frac{\pi }{2}$.
Work Step by Step
We look for symmetry by making the following substitutions:
(a) $\theta \to - \theta$ :$$r=\sin (-\theta ) \cos ^2(- \theta ) \quad \Rightarrow \quad r=- \sin \theta \cos ^2 \theta$$Thus, the graph does not necessarily have symmetry with respect to the polar axis.
(b) $r \to -r, \quad \theta \to -\theta$ :$$-r =\sin (-\theta ) \cos ^2 (-\theta ) \quad \Rightarrow \quad r=\sin \theta \cos ^2 \theta$$Thus, the graph is symmetric with respect to the line $\theta=\frac{\pi}{2}$.
(c) $r \to -r$ :$$-r =\sin \theta \cos ^2 \theta \quad \Rightarrow \quad r=-\sin \theta \cos ^2 \theta$$Thus, the graph does not necessarily have symmetry with respect to the pole.
