6.31 Try to find the domain of the function: $y=\lg \left(\sin \frac{x}{2}-\frac{1}{2}\right)+\sqrt{\frac{1}{x-2}}$.
[Solution] From the domain of $\lg \left(\sin \frac{x}{2}-\frac{1}{2}\right)$, we get $\sin \frac{x}{2}-\frac{1}{2}>0$, which means $\sin \frac{x}{2}>\frac{1}{2}$.
Therefore, $\quad \frac{\pi}{6}+2 k \pi < \frac{x}{2} < \frac{5 \pi}{6}+2 k \pi$, i.e., $\frac{\pi}{3}+4 k \pi < x < \frac{5 \pi}{3}+4 k \pi$.
And $x-2>0$, i.e., $x>2$.
Thus, the domain of $y$ is $2<x<\frac{5 \pi}{3}$.
And $\frac{\pi}{3}+4 k \pi < x < \frac{5 \pi}{3}+4 k \pi \cdot(k=1,2, \cdots)$