# Thread: polar to rectangular equation

1. ## polar to rectangular equation

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

$r^2 = sin\Theta$

I know $r^2 = x^2 + y^2$ , but I'm not sure how to eliminate $sin\Theta$. I know $rsin\Theta= y$, but multiplying both sides by r just muddies it further for me.

2. $y = r \sin \theta$
$y = r\, r^2$
$y^2 = r^2r^4$
$y^2 = (x^2 + y^2)^3$

Let $u := y^2$. Let $v:= x^2$.

$u = u^3 + v^3 + 3u^2v + 3 uv^2$
$u^3 +3vu^2 + (3v^2-1)u + v^3 = 0$

Ummm.... here's where I lose it...

Let $\alpha := \sqrt{3} \sqrt{27x^4 - 4} - 9x^2$.

$|y| = \big( \sqrt[3]{\alpha/18}+\sqrt[3]{2/3\alpha} - x^2 \big)^{1/2}$

... I think (I used the cubic formula). There might also be more solutions (pretty sure there are...)