Answer
$\frac{7}{25}$
Work Step by Step
Step 1. As $\theta$ is in Quadrant III, we have $\pi\lt\theta\lt\frac{3\pi}{2}$ and $2\pi\lt2\theta\lt3\pi$ which means that $cos2\theta$ can be both positive or negative.
Step 2. Given $sin\theta=-\frac{3}{5}$, use the Double-Angle Formula, we have $cos2\theta=1-2sin^2\theta=1-2\times(-\frac{3}{5})^2=1-\frac{2\times9}{25}=\frac{7}{25}$