Computing the Fourier transform of the characteristic function of a cone

Question

I've been trying to compute the Fourier transform of the characteristic function of a cone in two dimensions, e.g., $\{(x,y) \in \mathbb R^2 : |y| > c|x| \}$. My idea was to guess the solution (e.g. with heuristics and formal manipulations) and then prove that it works, but I don't know where to begin.