若 $\frac{\sqrt{3}}{2} \sin x+\frac{1}{2} \cos x=4-m$, 则实数 $m$ 的取值范围是 (A)
[解析] $\because \frac{\sqrt{3}}{2} \sin x+\frac{1}{2} \cos x=\frac{\sqrt{3}}{2} \sin x+\frac{1}{2} \cos x$

$=\cos x \cos \frac{\pi}{3}+\sin x \sin \frac{\pi}{3}=\cos \left(x-\frac{\pi}{3}\right)=4-m$,

$\therefore \cos \left(x-\frac{\pi}{3}\right)=4-m, \therefore|4-m| \leqslant 1$, 解得 $3 \leqslant m \leqslant 5$.