TITLE: Distribution of sine of uniform random variable on $[0, 2\pi]$
QUESTION [10 upvotes]: Let $X$ be a continuous random variable having uniform distribution on $[0, 2\pi]$. What distribution has the random variable $Y=\sin X$ ? I think, it is also uniform. Am I right?

REPLY [12 votes]: By the Jacobian formula, $$f_Y(y)=\frac{\mathbf 1_{|y|\lt1}}{\pi\sqrt{1-y^2}}.$$
This is the so-called Arcsine distribution, which famously appears in probability theory and in number theory, as shown by two mathematical giants from the 20th century:
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