Symmetry of polar equations

Question

In your opinion how to show symmetry in polar equations without graphing.

i thought of these methods :-

1. converting to cartesian then test .

2. check the period of the function .

please help me any other method or details would be help . thanks for advance :) .

Answer 1

Let $P$ be a point in the plane with polar coordinates $(r,\theta)$. Let $P_x$ be the reflection of $P$ across the $x$ axis; $P_y$ the reflection of $P$ across the $y$ axis; and $P_O$ be the reflection of $P$ through the origin.

$x$ axis symmetry: $P_x$ has polar coordinates $(r,-\theta)$, and so replacing $\theta$ with $-\theta$ into a polar equation and getting the same equation back again is an $x$-axis symmetry test. Because $P_x$ has many polar representations, other tests are possible. For instance $P_x$ also has polar coordinates $(-r,\pi-\theta)$, which would give another test.

$y$ axis symmetry: Same ideas, but noting $P_y$ has polar coordinates $(r, \pi-\theta)$, or $(-r,-\theta)$.

Origin symmetry: Same ideas. $P_O$ has polar coordinates $(r,\pi+\theta)$ or $(-r,\theta)$.