Answer
$$\cos2\theta=\frac{17}{25}$$
$$\sin2\theta=-\frac{4\sqrt{21}}{25}$$
Work Step by Step
$$\sin\theta=\frac{2}{5}\hspace{2cm}\cos\theta\lt0$$
$$\sin2\theta=?\hspace{2cm}\cos2\theta=?$$
To find $\sin2\theta$ and $\cos2\theta$, it would be wise to find the unknown $\cos\theta$.
Using Pythagorean Identities:
$$\cos^2\theta=1-\sin^2\theta=1-\Big(\frac{2}{5}\Big)^2=1-\frac{4}{25}=\frac{21}{25}$$
$$\cos\theta=-\frac{\sqrt{21}}{5}\hspace{1.5cm}(\cos\theta\lt0)$$
Now we apply the Double-Angle Identities for $\cos2\theta$ and $\sin2\theta$:
$$\cos2\theta=\cos^2\theta-\sin^2\theta=\Big(-\frac{\sqrt{21}}{5}\Big)^2-\Big(\frac{2}{5}\Big)^2=\frac{21}{25}-\frac{4}{25}$$
$$\cos2\theta=\frac{17}{25}$$
$$\sin2\theta=2\sin\theta\cos\theta=2\times\frac{2}{5}\times\Big(-\frac{\sqrt{21}}{5}\Big)$$
$$\sin2\theta=-\frac{4\sqrt{21}}{25}$$
