## Precalculus (6th Edition) Blitzer

To test the symmetry with respect to the line, replace $\left( r,\theta \right)$ with $\left( -r,-\theta \right)$, if the resultant equation is equivalent to the given equation, then the graph is symmetric with respect to $\theta =\frac{\pi }{2}$.
For example: We will replace $\left( r,\theta \right)\,\text{with}\,\left( -r,-\theta \right)$ in the polar equation $r=1-2\cos \theta$ to perform the test for the line $\theta =\frac{\pi }{2}$, \begin{align} & r=1-2\cos \theta \\ & -r=1-2\cos \left( -\theta \right) \\ & -r=1-2\cos \theta \end{align} The resultant equation changes and thus it fails this symmetry test.