Symmetry of polar curve $r = 1- \cos{\theta}$

Question

For a curve to be symmetric with respect to the $X$ axis, for every point $(r, \theta)$ there must be a point $(r, -\theta)$ on the curve

When we plug $(r, -\theta)$ in $r = 1- \cos{\theta}$ we get the original curve which means it is symmetric about $X$ axis

The point $(-r, \pi - \theta)$ is the same as $(r, -\theta)$ but doesn't satisfy the curve

I would appreciate if someone could point out what I am doing wrong.

Answer 1

Note that the polar representation of a point is not unique. For example, $ (1,\pi )$ and $(1,-\pi)$ represent the same point geometrically but they do not satisfy the same algebraic equations. For instance the equation $r=\frac {\theta}{\pi}$ is satisfied by $ (1,\pi )$ but not by $(1,-\pi)$

Answer 2

I had the same doubt regarding polar coordinates.. This video cleared it. The crux is that just any one representation of mirror image point in polar coordinates is necessary to test symmetry. The above symmetry conditions are sufficent but not necessary.

https://www.youtube.com/watch?v=iIsWV92Z1bU . Watch from 41:40.