Given $\tan (\theta)=\frac{3}{7}$, find the exact value of $\sin (2 \theta)$

Step 1 ：Given that \(\tan (\theta) = \frac{3}{7}\), we can assume that \(\sin (\theta) = 3k\) and \(\cos (\theta) = 7k\) for some constant \(k\).

Step 2 ：Substituting these into the Pythagorean identity \(\sin^2 (\theta) + \cos^2 (\theta) = 1\) gives us \((3k)^2 + (7k)^2 = 1\), which we can solve for \(k\).

Step 3 ：The solution for \(k\) is \([-\sqrt{58}/58, \sqrt{58}/58]\). We take the positive root \(k = 0.131306432859723\) because \(\sin (\theta)\) and \(\cos (\theta)\) are positive in the first quadrant.

Step 4 ：Substituting \(k\) into \(\sin (\theta) = 3k\) and \(\cos (\theta) = 7k\) gives us \(\sin (\theta) = 0.393919298579168\) and \(\cos (\theta) = 0.919145030018058\).

Step 5 ：Substituting these values into the equation for \(\sin (2 \theta) = 2 \sin (\theta) \cos (\theta)\) gives us \(\sin (2 \theta) = 0.724137931034483\).

Step 6 ：Final Answer: The exact value of \(\sin (2 \theta)\) is \(\boxed{0.724137931034483}\).