# Equation in polar coordinates to rectangular coordinates

The equation in polar coordinates is $r=2\sin(\theta)\tan(\theta)$. Show that the equation in rectangular coordinates is $y^2=\frac{x^3}{2-x}$ x cannot equal 2.

$r=2\sin(\theta)\tan(\theta)$

1.) I multiplied both sides by r so. $r^2=2r\sin(\theta)\tan(\theta)$

2.) $r\sin(\theta)=y$ and $\tan(\theta)=\frac{y}{x}$ and $r^2=y^2+x^2$ so I plugged these in and got $y^2+x^2=2y\frac{y}{x}$

3.) Then got $y^2=\frac{2y^2}{x}-x^2$ which equals $y^2=\frac{2y^2-x^3}{x}$ this is close but not the right answer; I don't know how to get it to be $y^2=\frac{x^3}{2-x}$.

Good start! You want to gather the $y^2$ terms on one side, though, so instead, from $$x^2+y^2=\frac{2y^2}x,$$ subtract $y^2$ from both sides and factor, yielding $$x^2=\frac{2y^2}x-y^2=\left(\frac2x-1\right)y^2=\frac{2-x}xy^2.$$ Can you finish it from here?