What is distribution of $\sin(x)$? If x is exponential distribution I am trying to find analytically the distribution of $\sin(x)$,
If x belongs to an exponential distribution,
 A: The cumulative distribution function (cdf) of a variable $X$ with an exponential distribution can be written
$$F_\lambda(x) = \Pr(X\le x) = 1 - e^{-\lambda x}.$$
Consequently, for any interval determined by $0\le a\le b,$ the chance $X$ lies in this interval is
$$\Pr(a\lt X\le b) = F_\lambda(b)-F_\lambda(a) = e^{-\lambda a} - e^{-\lambda b}.$$
Let $T=\sin(X).$  Computing the CDF of $T$ using the graphical approach recommended at PDF of function of X, we find for $-1\lt t \le 1$ and $x = \arcsin(t)$ (a number between $-\pi/2$ and $\pi/2$) that the event $T\le t$ is the union of disjoint events 
$$\mathcal{E}_i:2\pi j + \pi-x \le X \le 2\pi j + 2\pi + x,\ j=1,2,3,\ldots$$
together with the event
$$\mathcal{E}_0: 0 \le X \le x.$$
(This event is empty when $t\lt 0.$)  Consequently the desired probability is the sum of probabilities of these events, which simplifies because these probabilities form a geometric sequence:
$$\eqalign{\Pr(T\le t) &= \Pr(X\in\mathcal{E}_0) + \sum_{j=1}^\infty\Pr(X\in\mathcal{E}_j) \\
&= 1-e^{-\lambda x} + \sum_{j=1}^\infty \left(e^{-\lambda(2\pi j + \pi -x)} - e^{-\lambda(2\pi j + 2\pi +x}\right) \\
&= 1-e^{-\lambda x} + \left(e^{-\lambda(\pi-x)} - e^{-\lambda(2\pi+x)}\right)\sum_{j=1}^\infty \left(e^{-\lambda 2\pi}\right)^j \\
&= 1-e^{-\lambda x} + \frac{e^{-\lambda(\pi-x)} - e^{-\lambda(2\pi+x)}}{1-e^{-\lambda 2\pi}} \\
&= 1-e^{-\lambda \arcsin(t)} + \frac{e^{-\lambda(\pi-\arcsin(t))} - e^{-\lambda(2\pi+\arcsin(t))}}{1-e^{-2\pi\lambda }},
}$$
with the first two terms $1-e^{-\lambda \arcsin(t)}$ not appearing when $\arcsin(t)\lt 0.$
Here are illustrations of the CDF, superimposed on the empirical distributions of samples of size 10,000.  The theoretical curves (in red) coincide with the empirical distribution functions everywhere, demonstrating the correctness of these results.  Beneath each CDF is a plot of the PDF of the underlying exponential distribution.

