Probability densitiy of $\cos x$

Question

If $x$ is of Uniform distribution between $0$ and $2\pi$, what is the probability of $\cos(x)$ having a value between $-0.5$ and $0.5$?

I tried transformation, but I somehow get $1/6$ but it seems to be $1/3$.

Greetings

Answer 1

$\cos x$ is in the range $\left[-\frac12, \frac12\right]$ when $x$ is in the range $\left[\frac\pi3,\frac{2\pi}3\right]$ or $\left[\frac{4\pi}3,\frac{5\pi}3\right]$.

Each of these two intervals has length $\frac\pi3$; the two together have a total length of $\frac{2\pi}3$. This is exactly $\frac13$ of the entire range, which has length $2\pi$.

Is it possible that you forgot to include the $\left[\frac{4\pi}3,\frac{5\pi}3\right]$ interval?