On which interval is $f(x) = \log(\sin x)$ strictly decreasing?

Explanation

1. 01

   $\log(\sin x)$ is defined where $\sin x > 0$, and $f'(x) = \frac{\cos x}{\sin x} = \cot x$.

2. 02

   $\cot x < 0$ on $\left(\frac{\pi}{2}, \pi\right)$, so $f$ falls there (and rises on $\left(0, \frac{\pi}{2}\right)$).