Coordinate Form of $r = x/2$

Question

How can I convert a spiral of archimedes into coordinate form from a polar system? $r = x/2$ where $x$ is an angle.

$2r = x$ if I take the tan then $\tan(2r)$ is hard to deal with. Otherwise, $2r^2 = xr$ also is hard to solve.

Answer 1

Just for clarity I will write the equation for the Archimedean spiral as $r=\theta.$ There is a standard way to shift from polar to Cartesian based on the equations $r=\sqrt{x^2+y^2}$ and $\theta=\arctan\frac{y}{x}.$ Inserting these terms into the equation for Archimedes' spiral gives $\sqrt{x^2+y^2}=\arctan\frac{y}{x}.$