5. In $\triangle A B C$, $A B=1, B C=2$, then the range of $\angle C$ is
A. $\left(0, \frac{\pi}{6}\right]$
B. $\left(\frac{\pi}{4}, \frac{\pi}{2}\right)$
C. $\left(\frac{\pi}{6}, \frac{\pi}{3}\right)$
D. $\left(0, \frac{\pi}{2}\right)$
Answer
As shown in the figure, point $A$ moves on a circle centered at $B$ with a radius of 1. When $AC$ is tangent to circle $B$, $\angle C$ reaches its maximum value. At this time, $\sin C=\frac{1}{2}$, so $C \in\left(0, \frac{\pi}{6}\right]$, hence the answer is $A$.