Random Variables and Density Function

Question

The following problem is from the Schaum book called "Theory and Problems of Probability, Random Variables, and Random Processes":

Let $Y = \sin X$, where $X$ is uniformly distributed over $(0, 2)$. Find the pdf of $Y$.

Here is my solution: $$ P(Y \le y_0) = P(\sin X \le y_0 ) = P( X \le \arcsin y_0 ) = \int_0^{\arcsin y_0} \frac 1{ 2\pi } dx $$ When I evaluate that integral I get a function with differs from the book's answer by a factor of 2. I am hoping that somebody here can tell me what I am missing.

Bob

Answer 1

Presumably $X$ is uniformly distributed over $(0,2 \pi)$.

Hint: Note that $\sin x$ takes on the same value at $x$ and $\pi - x$.