Writing the equation $r = \theta$ in cartesian cordinates

Question

Trying to write the cartesian version of the equation $r = \theta$ which looks like a spiral when graphed. How is the that going to look?

I have: $$ \sqrt{x^2+y^2} = \arctan\left( \frac{y}{x} \right) $$ But this looks nothing like what I expected.

Please explain your reasoning.

Answer 1

This is the picture of $\tan(\sqrt{x^2+y^2})=\frac{y}{x}$

Note that your LHS is always nonnegative.