Answer
$-2$
Work Step by Step
Step 1. Given $\theta$ in Quadrant III ($\pi\leq\theta\leq\frac{3\pi}{2}$) and $sin\theta=-\frac{4}{5}$, we have $\pi/2\leq\theta/2\leq\frac{3\pi}{4}$ and $cos\theta=-\frac{3}{5}$
Step 2. Use Half-Angle Formula, we have $tan\frac{\theta}{2}=-\sqrt {\frac{1-cos\theta}{1+cos\theta}}
=-\sqrt {\frac{1+3/5}{1-3/5}}=-\sqrt {\frac{5+3}{5-3}}=-2$
