distribution of cosine of uniformly random variables

Question

let us consider following data

and here is its chart

generally cosine is not linear function,therefore cosine of uniform variables should not be uniform as well,because if

$x=cos(y)$

$y=\arccos{x}$

according to this site

https://stats.stackexchange.com/questions/56040/cosine-of-a-uniform-random-variable

as i understood cosine of uniform variables distributed in $|0, 2*\pi|$ is not again uniform variables right? then what kind of distribution it is supposed to be? thanks in advance

Answer 1

The uniform distribution on $[-\pi, \pi]$ has constant probability density function $f_Y(y) = 2/\pi$ for all $y$. However if $x$ has uniform distribution on $[-\pi, \pi]$ and $y = \cos (x)$, then $y$ has probability density $f_Y(y) = (1/\pi)(1/\sqrt{1-y^2})$ which integrates to $1$ so it is a probability density, but it is unbounded because values close to $1$ become more and more likely. And so no, $\cos x$ is not uniformly distributed if $x$ is.