Question

Why is `r=3cos2theta` not symmetric over `theta=pi/2`?

Answer 1

The graph IS symmetric about that line.

Explanation:

You already see the graph, so you were able to observe its symmetry.
One test to determine symmetry about `theta = pi/2` is to substitute
`theta - pi` for `theta`.

`3cos(2(theta -pi)) = 3cos(2theta -2pi)`
`=3cos2thetacos2pi+sin2thetasin2pi`
`=3cos2theta`.

Therefore, the function is symmetric about `theta = pi/2`.