2024 AMC 12A 第 8 题

$\log(\sin3\theta)+\log(\cos2\theta)$ $=\log(\sin3\theta\cos2\theta),$ which must equal $0$

A product of two numbers each at most $1$ equals $1$ only when both equal $1$

The equation means $\sin(3\theta)\cos(2\theta)=1$ with both factors positive (for the logs to be defined). Since $\sin(3\theta)\le1$ and $\cos(2\theta)\le1,$ their product is $1$ only if $\sin(3\theta)=1$ and $\cos(2\theta)=1$ simultaneously. But $\cos(2\theta)=1$ forces $\theta\in\{0,\pi,2\pi\},$ where $\sin(3\theta)=0\ne1.$ No angle works. Thus, the correct answer is A.