Answer
$\displaystyle \cos 2\theta=\frac{39}{49}$
$\displaystyle \sin 2\theta=-\frac{4\sqrt{55}}{49}$
Work Step by Step
Plan: work out $\cos\theta$, then apply the double-angle identities.
Pythagorean Identity ($\cos \theta$ is positive):
$\cos\theta=+\sqrt{1-\sin^{2}\theta}=\sqrt{1-\dfrac{5}{49}}$
$=\displaystyle \sqrt{\frac{44}{49}}=\frac{2\sqrt{11}}{7}$
Double-Angle Identities:
$\displaystyle \cos 2\theta=1-2\sin^{2}\theta=1-2(-\frac{\sqrt{5}}{7})^{2}$
$=1-2\displaystyle \cdot\frac{5}{49}=1-\frac{10}{49}=\frac{39}{49}$
$\displaystyle \sin 2\theta=2\sin\theta\cos\theta=2(-\frac{\sqrt{5}}{7})(\frac{2\sqrt{11}}{7})$
$=-\displaystyle \frac{4\sqrt{55}}{49}$
