If $\sin \theta=\frac{2}{5}, 0<\theta<\frac{\pi}{2}$, find the exact value of each of the following. (a) $\sin (2 \theta)$ (b) $\cos (2 \theta)$ (c) $\sin \frac{\theta}{2}$ (d) $\cos \frac{\theta}{2}$ (a) $\sin (2 \theta)=\square$ (Type an exact answer, using radicals as needed.)

Step 1 ：We are given that \(\sin \theta = \frac{2}{5}\) and \(0 < \theta < \frac{\pi}{2}\). We are asked to find the value of \(\sin (2 \theta)\).

Step 2 ：We can use the double angle formula for sine, which is \(\sin (2 \theta) = 2 \sin \theta \cos \theta\). However, we only know the value of \(\sin \theta\), not \(\cos \theta\).

Step 3 ：We can find \(\cos \theta\) using the Pythagorean identity \(\sin^2 \theta + \cos^2 \theta = 1\). Solving for \(\cos \theta\), we get \(\cos \theta = \sqrt{1 - \sin^2 \theta} = \sqrt{1 - \left(\frac{2}{5}\right)^2} = \sqrt{1 - \frac{4}{25}} = \sqrt{\frac{21}{25}} = \frac{\sqrt{21}}{5}\).

Step 4 ：Now that we have the value of \(\cos \theta\), we can substitute it into the double angle formula to find \(\sin (2 \theta)\). Substituting \(\sin \theta = \frac{2}{5}\) and \(\cos \theta = \frac{\sqrt{21}}{5}\) into the formula, we get \(\sin (2 \theta) = 2 \cdot \frac{2}{5} \cdot \frac{\sqrt{21}}{5} = \frac{4\sqrt{21}}{25}\).

Step 5 ：Final Answer: \(\sin (2 \theta) = \boxed{\frac{4\sqrt{21}}{25}}\)