Let's say we want to check whether a polar graph is symmetric about the origin.
Then
- if $(-r,\theta)$ satifies the polar equation for the graph whenever $(r,\theta)$ satisfies the graph, it is symmetric about the origin
- or if $(r,\theta+\pi)$ satifies the polar equation for the graph whenever $(r,\theta)$ satisfies the graph, it is symmetric about the origin.
But $r=3\sin(2\theta)$ is symmetric about the origin and $(r,\theta+\pi)$ satisfies the euqation but $(-r,\theta)$ does not satisfy the equation.
I was wondering if there is an example where $(-r,\theta)$ satisfies the euqation but $(r,\theta+\pi)$ does not satisfy the equation.