Answer
$\displaystyle \cos 2\theta=-\frac{19}{25}$
$\displaystyle \sin 2\theta=\frac{2\sqrt{66}}{25}$
Work Step by Step
Plan: work out $\sin\theta$, then apply the double-angle identities.
Pythagorean Identity ($\sin\theta$ is positive):
$\sin\theta=+\sqrt{1-\cos^{2}\theta}=\sqrt{1-\dfrac{3}{25}}$
$=\displaystyle \sqrt{\frac{22}{25}}=\frac{\sqrt{22}}{5}$
Double-Angle Identities:
$\displaystyle \cos 2\theta=2\cos^{2}\theta-1=2(\frac{\sqrt{3}}{5})^{2}-1$
$=2\displaystyle \cdot\frac{3}{25}-1=\frac{6}{25}-1=-\frac{19}{25}$
$\sin 2\theta=2\sin\theta\cos\theta$
$=2(\displaystyle \frac{\sqrt{22}}{5})(\frac{\sqrt{3}}{5}$)$=\displaystyle \frac{2\sqrt{66}}{25}$
