When is the area between two polar curves symmetric?

Question

What are the conditions for the area between two polar curves to be composed of two symmetric areas?

Here is a demonstration of the symmetry in the area between $3\sin{\theta}$ and $3\cos{\theta}$. See how it's symmetric about $\frac{\pi}{4}$:

Answer 1

I'm going to address the special case when the graph of $r = f(\theta)$ is the mirror image of the graph of $r = g(\theta)$ across the line $\theta = c$, for some constant $c$.

This is the case if the (polar) point $(r, c + t)$ is on the graph of $f$ whenever $(r, c - t)$ is on the graph of $g$, and vice versa. In other words, $f$ is the mirror image of $g$ across the line $\theta = c$ if $f(c + t) = g(c - t)$ for all $t$.

In the example you mentioned, we have $3\sin(\pi/4 + t) = 3\cos(\pi/4-t)$ for all $t$, so the graphs are symmetric about the line $\theta = \pi/4$ as claimed.

Answer 2

Convert back to Cartesian:

$$r=3\sin\theta \implies r^2=3r\sin\theta \implies x^2+y^2=3y \implies x^2+\left(y-\frac32\right)^2=\frac94 \\ r=3\cos\theta \implies \left(x-\frac32\right)^2+y^2=\frac94$$

Notice how swapping $x$ and $y$ in one equation produces the other equation, so one curve is a reflection of the other across the line $y=x$ i.e. the ray $\theta=\dfrac\pi4$, and any overlap in the areas bounded by either curve will also be symmetric.