I am trying to find the values of $\theta$ where the following polar curve intersect itself:$$r = 1 + 2\cos(\theta) - 4\cos^2 (\theta).$$ I am studying this function on $[0, 2\pi] $ since it is periodic.
My approach
At intersection, we can say that the points $(r,\theta)$ and $ (r', \theta')$ are the same, for different values of theta, this can happen only if:
$r = r$' and $ \theta = \theta' + 2k\pi$ or $r = -r'$ and $ \theta = \theta' + 2k\pi + \pi $.
So I tried to replace, for the first case I get : $$1 + 2\cos(\theta) - 4\cos^2(\theta) = 1 + 2\cos(\theta' + 2k\pi) - 4\cos^2(\theta' + 2k\pi).$$
But since $\cos$ is $2\pi$-periodic then we get: $$1 + 2\cos(\theta) - 4\cos^2(\theta) = 1 + 2\cos(\theta') - 4\cos^2(\theta' ).$$ Thus $\theta = \theta' + 2k\pi $ and we take $k = 0$ because the study domain is just $[0,2\pi]$ so I get that $\theta = \theta'$ and this cannot be a solution.
For the second case where I use the fact that $\cos(\theta + 2k\pi + \pi ) = -\cos(\theta)$ then I get something like 1 = 1 which is not helpful .
How would you find these points? And what is the general method to use?