Given a random variable $X$ with probability density function $P(X)$, and given a transformation function $f(x)$, how does one determine the new resultant probability density function: $P(f(X))$?
For example:
Given random variable $X$ which is evenly distributed over the range $[0,2\pi ]$ such that $P(X) = \dfrac{1}{(2\pi)}$, what would be the probability density function of random variable $Y$ where $Y = \sin(X)$?
This blog post, explains how to get the pdf for $\sin(X)$, but I'd like to know if there is a way to solve this problem in the general case for a transformation of $f(X)$.