So, I haven't thought about polar coordinates in about 10 years...
In $\mathbb{R}^2$, suppose I have the region $\{x <0, y < kx\}$ where $k \neq 0$ is some given constant. How do I convert this to polar coordinates?
I get that $r \in (0, \infty)$, and the lower bound for $\theta$ should be $\pi$. The upper bound seems to differ depending on whether $k > 0$ or $k < 0$, but I'm not sure how to find the upper bound for $\theta$ explicitly. Because of the way $\tan^{-1}$ is defined, I think if $k < 0$, I can use $\tan^{-1}(k)$ given how $\tan^{-1}$ is defined, but not sure what to do for when…