Answer
$$\sin2\theta=-\frac{4\sqrt{55}}{49}$$
$$\cos2\theta=\frac{39}{49}$$
Work Step by Step
$$\sin\theta=-\frac{\sqrt5}{7} \hspace{2cm}\cos\theta\gt0$$
$$\sin 2\theta=?\hspace{2cm}\cos2\theta=?$$
1) First, we need to figure out the unknown $\cos\theta$ as they are essential to the calculations of $\sin 2\theta$ and $\cos 2\theta$.
According to Pythagorean Identities:
$$\cos^2\theta=1-\sin^2\theta=1-\Big(-\frac{\sqrt5}{7}\Big)^2=1-\frac{5}{49}=\frac{44}{49}$$
$$\cos\theta=\frac{\sqrt{44}}{7}=\frac{2\sqrt{11}}{7}\hspace{1cm}(\cos\theta\gt0)$$
2) Now we can calculate $\sin2\theta$ and $\cos2\theta$ using Double-Angle Identities, which states
$$\sin2\theta=2\sin\theta\cos\theta$$
$$\cos2\theta=1-2\sin^2\theta$$
Thus,
$$\sin2\theta=2\times\Big(-\frac{\sqrt5}{7}\Big)\times\Big(\frac{2\sqrt{11}}{7}\Big)=-\frac{4\sqrt{55}}{49}$$
$$\cos 2\theta=1-2\Big(-\frac{\sqrt{5}}{7}\Big)^2=1-2\times\frac{5}{49}=1-\frac{10}{49}=\frac{39}{49}$$
