Harmonious Intervals

You may use the fact that for $x \in \left(0, \dfrac{\pi}{2}\right)$,

1. Prove that $\dfrac{\sin x}{x} > \dfrac{1}{2}$ for $x \in \left(0, \dfrac{\pi}{2}\right)$. 2. Let $f(x) = m\sin x$. If an interval $[a, b]$ is such that when $f(x)$ has domain $[a, b]$ its range is also $[a, b]$, call $[a, b]$ a harmonious interval of $f(x)$.

1. For $m = 1$, does $f(x)$ have a harmonious interval? If so, find them all; if not, explain why.
2. For $m = -2$, does $f(x)$ have a harmonious interval? If so, find them all; if not, explain why.