I am stuck in a derivation where, in one of the steps, I have to find the distribution of the following function: $$h = z_1e^{i\theta_1} + z_2e^{i\theta_2} + \cdots + z_ne^{i\theta_n},$$ where $\theta_1, \cdots, \theta_n$ are all i.i.d distributed $\text{Unif}[-\pi, \pi]$ and $z_1, \cdots,z_n$ are complex constants.
I tried writing $e^{i\theta} = \cos\theta + i\sin\theta$ and then find the distributions of $\cos \theta$ and $\sin \theta$ seperately. If $X=\cos\theta$, I got the pdf of $X$ to be $f_X(x) = \frac{1}{\pi\sqrt{1-x^2}}$.
Q1) I have difficulty deriving the distribution of $\sin\theta$. I don't know how to go on.
Q2) Let us say, I get some expression for the pdf of $\sin \theta$, how do I go on further?
Q3) Can someone suggest some other method to directly get the distribution of $e^{i\theta}$, without getting into $\cos \theta$ and $\sin \theta$ business?