Rewrite the Pythagorean Identity.

$\text{Substitue -\frac{3}{5}\ for sin\theta and simplify.}$

$\text{cos}\theta=\pm\sqrt{1-\text{sin}^2\theta}\\$

$\text{Since } \pi \le \theta \le\frac{3\pi}{2},\text{ the angle in the third quadrant.}$

Are sine and cosine negative or positive?

$\text{cos}\theta=-\frac{4}{5}$