Distribution of cosine of a uniform random variable

Question

I am trying to find the pdf of $Y=\cos(X)$ where $X$ is a random variable distributed uniformly in $[-\pi/2,\pi/2]$.

I tried to use the change of variable theorem but there is a detail in $[-\pi/2,\pi/2]$ since the cosine it's not monotone, and since $\cos(\pi/2)=0=\cos(\pi/2)$. I don't have a way to see the interval that corresponds to $Y$.

Thank you for helping!

Answer 1

You can let $Z=\vert X \vert$. Then $Z$ is uniform on $[0, \pi/2].$ And $Y=cos(X)=cos(Z)$. Note the mapping from $Z$ to $Y$ is monotone, and you can apply the change of variable theorem there now.