How to mathematically define a region in polar coordinate?

Question

I want to define a region in polar coordinate, specified only with angle (i.e. radius doesn't matter). The region I want to show is plotted in the photo below and the boundary is $\cos^{-1}(x)$. How should I write this in mathematical format? The region I want to describe

Answer 1

Assuming you take your angle to be $0 \leq \theta < 2 \pi$, the region you have drawn is described by $$ \begin{align} &\theta \leq \cos^{-1}(x) \\\text{ OR } \quad &\pi-\cos^{-1}(x) \leq \theta \leq \pi + \cos^{-1}(x)\quad \\ \text{ OR }\quad &2 \pi - \cos^{-1}(x) < \theta. \end{align} $$ If you are satisfied with an inequality that is not solved for $\theta$, we could describe the region with $$ |\cos \theta| \geq x. $$