Answer
$$\sin2\theta=\frac{2\sqrt{66}}{25}$$
$$\cos2\theta=-\frac{19}{25}$$
Work Step by Step
$$\cos\theta=\frac{\sqrt3}{5} \hspace{2cm}\sin\theta\gt0$$
$$\sin 2\theta=?\hspace{2cm}\cos2\theta=?$$
1) First, we need to figure out the unknown $\sin\theta$ as they are essential to the calculations of $\sin 2\theta$ and $\cos 2\theta$.
According to Pythagorean Identities:
$$\sin^2\theta=1-\cos^2\theta=1-\Big(\frac{\sqrt3}{5}\Big)^2=1-\frac{3}{25}=\frac{22}{25}$$
$$\sin\theta=\frac{\sqrt{22}}{5}\hspace{1cm}(\sin\theta\gt0)$$
2) Now we can calculate $\sin2\theta$ and $\cos2\theta$ using Double-Angle Identities, which states
$$\sin2\theta=2\sin\theta\cos\theta$$
$$\cos2\theta=2\cos^2\theta-1$$
Thus,
$$\sin2\theta=2\times\frac{\sqrt{22}}{5}\times\frac{\sqrt{3}}{5}=\frac{2\sqrt{66}}{25}$$
$$\cos 2\theta=2\times\Big(\frac{\sqrt{3}}{5}\Big)^2-1=2\times\frac{3}{25}-1=\frac{6}{25}-1=-\frac{19}{25}$$
