Difficult conversion from polar equation to rectangular equation. How do we convert this into rectangular equation?
$r=5\theta$

I get this easily, check that if i am wrong:
$r = a\theta$, so in terms of Cartesian coordinates…
$$x = a\theta\cos\theta,\qquad y =a\theta\sin\theta.$$
We can also solve for $x$ and $y$ through simple algebraic manipulation
We know:  $r^2=x ^2+ y^2$. Let's solve for $x$ first:
$$\begin{align*}
r &= a\theta\\
r^2&=a^2\theta^2\\              
x^2+y^2&=a^2\theta^2\\
x^2&=a^2\theta^2-y^2
\end{align*}$$
Square both sides and substitute   $r^2=x^2+ y^2$, $y=r\sin\theta$ (hence $y^2=r^2\sin^2\theta$),  $x^2=a^2\theta^2-r^2\sin^2\theta$ (hence $x^2=a^2\theta^2- a^2\theta^2\sin^2\theta$) we get
$$x^2=a^2\theta^2(1- \sin^2\theta).$$
$y$ follows easily
