Points of intersection for two polar equations question

Question

Why is it that when I try to find the points of intersection for $r=2$ and $r=4*\cos(2\theta)$, I only get the $\theta$ where the reference angle is $\pi/6$? There is clearly another solution between $0$ and $\pi/2$, as the picture of the graph of the two equations shows above. When I set $-2 = 4*\cos(2\theta)$, I do ultimately find what seems to me to be the other angle (which I believe is $\pi/3$), why is this?.

Answer 1

$2=4\cos(2\theta)$ if $\cos(2\theta)=\frac{1}{2}$ so $2\theta = \frac{\pi}{3}+2\pi n$ where $n$ is an integer. Divide by 2 to get $\theta = \frac{\pi}{6}+\pi n$.

Answer 2

The reason is that for example when $\theta = \frac{\pi}{3}$, we have $4\cos 2\theta = -2$. The resulting point, $\left(-2,\frac{\pi}{3}\right)$, lies on both curves even though here $r=-2$, not $2$. The essential issue here is that there are multiple representations for a point in polar coordinates. To get all of the solutions, you should solve $2 = |4 \cos 2\theta|$.