[Math] Find the cartesian equation of: $r=2\cos\left(\frac {3\theta}{2}\right)$

Question

I've managed to use identities to simplify it down to:
$$r = 2\left(\cos^3\left({\theta\over2}\right)-3\sin\left({\theta\over2}\right)\cos\left({\theta\over2}\right)\right)$$
using trig identities, but I am stuck now. Does this have a Cartesian equivalent?

Answer 1

Hint. The difficulty is the half angle. So: $$\eqalign{r^2 &=4\cos^2\Bigl(\frac{3\theta}{2}\Bigr)\cr &=2\bigl(1+\cos(3\theta)\bigr)\ .\cr}$$ Now you should be able to write in various ways $\cos(3\theta)$ in terms of $\cos\theta$ or $\sin\theta$ or both, and the rest should not be too difficult. Good luck!