Answer
$-\frac{\sqrt 3}{2}$
Work Step by Step
Step 1. Given $\frac{\pi}{2}\lt\theta\lt\pi$ and $cos\theta=-\frac{\sqrt 3}{2}$, we get $sin^2\theta=1-cos^2\theta=1-\frac{3}{4}=\frac{1}{4}$ which gives $sin\theta=\frac{1}{2}$ (Pythagorean Identity)
Step 2. Use the Double-Angle Formula, $sin2\theta=2sin\theta cos\theta=2\times\frac{1}{2}\times(-\frac{\sqrt 3}{2})=-\frac{\sqrt 3}{2}$
