I would like to know how my professor had these two results:
First: Let X be a r.v uniformly distributed in [0, 2\pi], what's the density function of $Y = \sin(X)$.
He gave as an answer $$\frac{1}{\pi\sqrt{1-y^2}}$$ but I would say that the answer should be $$\frac{1}{2\pi\sqrt{1-y^2}}$$ given that the CDF is $$\frac{1}{2\pi}$$ (I used CDF and then derived it to find density function) can someone tell me where did the "2" go ?
Second, question of the same type : Let X be a r.v uniformly distributed in [0, 1], what's the density function of $$Y = a\sin(2\pi nX+\phi)$$ with $a$,$n$ and $\phi$ constants. He gave as an answer $$\frac{1}{\pi\sqrt{a^2-y^2}}$$ but I've found $$\frac{1}{2n\pi\sqrt{a^2-y^2}}$$ and I don't know where did the $2n\pi$ go.
Can someone please help me ?