Converting Polar Equations into Cartesian equations

Question

I have a question I would like to ask about sketching polar equations.

Firstly, my teacher gave us the following identities to help support us in sketching polar curves.

If $r = f(\theta)$, $$ f(-\theta) = f(\theta) $$ this implies that $r$ is symmetric about the Polar Axis.

If $$ f(\pi - \theta) = f(\theta) $$ this implies that $r$ is symmetric about $\theta = \frac{\pi}{2} $.

If $$f(\theta + \pi) = -f(\theta) $$

this implies that $r$ is symmetric about the pole.

I'm trying to build up an intuition as to why these hold. The first one is simple to see why it holds, but the last two I have no clue why they hold. Any explanations would be great thanks.

Answer 1

The second can be understood in 2 ways. You may consider the rhs as evaluating $f$ at $\theta$ measured anti-clockwise from $\theta=0$ and the lhs as $f$ evaluated at $\theta$ measured clockwise from $\theta=\pi$ giving symmetry about $\theta=\pi/2$. Or let $\theta=\pi/2-\alpha$ and you will find $f (\pi/2+\alpha)=f(\pi/2-\alpha)$ i.e. the required symmetry. The last one does not make sense as it appears to imply r can be negative.

Answer 2

Those features certainly belong to some specific functions. Look at the following polar function and the respective graph.

$$r(\theta)=\theta^2\mid \sin(\theta)\mid.$$