This is Example 2.1.2, from Casella and Berger's book Statistical Inference, Page 49.
Suppose X has a uniform distribution on the interval $(0,2\pi)$, that is,
$$f_X= \begin{cases} 1/(2\pi) & 0<x<2\pi \\ 0 & \text{otherwise} \end{cases} $$
Consider $Y=sin^2(X)$. Then
$$P(Y \le y) = P(X \le x_1) + P(x_2 \le X \le x_3) + P(X \ge x_4)$$
My question: How did they come up with those cut-offs at $x_1, \dots, x_4$? What happened to the interval between $x_1$ and $x_2$, $x_3$ and $x_4$? The book does a poor job and never explained.
Can someone solve this question step-by-step? Thank you!