Question

Convert R=(12)/4+8sin the theta to rectangular form

Answer 1

$r=\dfrac{12}{4+8\sin\theta}=\dfrac3{1+2\sin\theta}$

Let $y=2\sin\theta$, and recall that in polar coordinates, $r=\sqrt{x^2+y^2}$. This means you have

$\sqrt{x^2+y^2}=\dfrac3{1+y}$

You can stop there, or try to find something that looks somewhat nicer.

$x^2+y^2=\dfrac9{(1+y)^2}$