7. Let $\omega$ be a positive real number. If there exist $a, b(\pi \leqslant a<b \leqslant 2 \pi)$ such that $\sin \omega a+\sin \omega b=2$, then the range of $\omega$ is $\qquad$
According to the problem, there exist $k, l \in \mathbf{Z}$, such that $2 k \pi+\frac{\pi}{2}, 2 l \pi+\frac{\pi}{2} \in[\omega \pi, 2 \omega \pi]$.
Thus, $\left\{\begin{array}{l}\omega \pi \leqslant 2 k \pi+\frac{\pi}{2}, \\ 2 \omega \pi \geqslant 2 k \pi+\frac{5}{2} \pi\end{array} \Rightarrow\left\{\begin{array}{l}\omega \leqslant 2 k+\frac{1}{2}, \\ 2 \omega \geqslant 2 k+\frac{5}{2}\end{array} \Rightarrow k+\frac{5}{4} \leqslant \omega \leqslant 2 k+\frac{1}{2}\right.\right.$.
Therefore, $k \geqslant 1$. When $k=1$, we have $\omega \in\left[\frac{9}{4}, \frac{5}{2}\right]$; when $k=2$, we have $\omega \in\left[\frac{13}{4}, \frac{9}{2}\right]$. When $k \geqslant 2$, $2 k+\frac{1}{2}>k+1+\frac{5}{4} \Rightarrow \bigcup_{k=2}^{\infty}\left[k+\frac{5}{4}, 2 k+\frac{1}{2}\right]=\left[\frac{13}{4},+\infty\right)$. Therefore, the range of $\omega$ is $\left[\frac{9}{4}, \frac{5}{2}\right] \cup\left[\frac{13}{4},+\infty\right)$.