I need help converting $x^2 + y^2 = -4y$ into a polar equation

Question

I know the whole $r^2 = x^2 + y^2$ and $x = r \cos \theta$ and $y = r \sin \theta$, but I just can't seem to apply those rules to the equation $x^2 + y^2 = -4y$ to make it a polar one.

Answer 1

Just substitute the equations you have above.

You know $x^2 + y^2 = r^2$, so substituting this in, we get $r^2 = -4y$.

We also know that $y = r \sin \theta$, so substituting that in, we get $r^2 = - 4 r \sin \theta$.

Cancelling the $r$ on both sides, we get $r = -4 \sin \theta$. We're done.