Use the Pythagorean identity:

\(\sin^2\theta+\cos^2\theta=1\)

\(\sin^2\theta=1−\cos^2\theta\)

Since \(\theta\) is Quadrant III, then \(\sin\theta<0\) so take the negative root:

\(\sin\theta=−\sqrt{1−\cos2\theta}\)

\(\sin\theta=−\sqrt{1−0.489}^{2}\)

\(\sin\theta\approx−0.872\)