Problem

The key insight is that $y = \sin(f(x))$ can only be monotonic everywhere if the inner function $f(x)$ stays within an interval where sine itself doesn't change direction. Since sine is monotonic only on intervals of length $\frac{\pi}{2}$ (like $[-\frac{\pi}{2}, \frac{\pi}{2}]$ where it increases), the range of $f(x)$ must fit entirely within one such interval. The maximum spread of $f(x)$ is therefore the length of the longest monotonic piece of sine.

The key insight is that $y = \sin(f(x))$ can only be monotonic everywhere if the inner function $f(x)$ stays within an interval where sine itself doesn't change direction. Since sine is monotonic only on intervals of length $\frac{\pi}{2}$ (like $[-\frac{\pi}{2}, \frac{\pi}{2}]$ where it increases), the range of $f(x)$ must fit entirely within one such interval. The maximum spread of $f(x)$ is therefore the length of the longest monotonic piece of sine.