Answer
$sin~\theta=\pm\frac{\sqrt 3}{2}$
$cos~\theta=\frac{1}{2}$
Work Step by Step
$sin^2\theta+cos^2\theta=1$
$sin^2\theta=1-cos^2\theta$
$5\sqrt 3=\sqrt {100-x^2}$
$5\sqrt 3=\sqrt {100-100~cos^2\theta}$
$5\sqrt 3=\sqrt {100(1-cos^2\theta)}$
$5\sqrt 3=10\sqrt {sin^2\theta}~~~~$ (Square both sides)
$75=100~sin^2\theta$
$sin^2\theta=\frac{75}{100}=\frac{3}{4}$
$sin~\theta=\pm\frac{\sqrt 3}{2}$
$sin^2\theta+cos^2\theta=1$
$cos^2\theta=1-\frac{3}{4}=\frac{1}{4}$
$cos~\theta=\pm\frac{1}{2}$
Since $-\frac{\pi}{2}\lt\theta\lt\frac{\pi}{2}$,
$cos~\theta=\frac{1}{2}$
