Converting the polar equation $r=12-\sin\theta+2\sin3\theta+2\sin5\theta-\sin7\theta+3\cos2\theta-2\cos4\theta$ to rectangular form

Question

How do I convert the following polar equation to rectangular equation?

$$r = 12 - \sin(θ) + 2\sin(3θ) + 2\sin(5θ) - \sin(7θ) +3\cos(2θ) - 2\cos(4θ)$$

Answer 1

Hint #1: Use some of the sum-to-product reduction formulas, for instance $$\sin x + \sin y = 2 \sin \frac {x+y} 2 \cos \frac {x-y} 2$$ to reduce it to at most $4 \theta$.

Hint #2: If you know that $x = r \cos \theta$ and $y = r \sin \theta$, can you establish the other identities? To wit: $x y = r^2 \cos \theta \sin \theta \rightarrow 2 x y = 2 r^2 \cos \theta \sin \theta \rightarrow 2 x y = r^2 \sin 2\theta \rightarrow \frac {2 x y} {r^2} = \sin 2\theta$, etc.