If $\tan \theta=\frac{1}{2},-\frac{\pi}{2}< \theta< \frac{\pi}{2}$, then $\sin \theta=$

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

Step 2 ：We know that \(\tan \theta = \frac{\sin \theta}{\cos \theta}\). So, we can express \(\sin \theta\) in terms of \(\tan \theta\) and \(\cos \theta\).

Step 3 ：We don't know the value of \(\cos \theta\) directly. But we do know that \(\cos^2 \theta = 1 - \sin^2 \theta\) (from the Pythagorean identity).

Step 4 ：We can express \(\sin \theta\) in terms of \(\tan \theta\) alone by substituting \(\cos \theta = \sqrt{1 - \sin^2 \theta}\) into the equation \(\tan \theta = \frac{\sin \theta}{\cos \theta}\).

Step 5 ：This gives us a quadratic equation in \(\sin \theta\) which we can solve to find the value of \(\sin \theta\).

Step 6 ：Solving the equation gives us two possible values for \(\sin \theta\), which are approximately 0.447213595499958 and -0.447213595499958.

Step 7 ：However, since the given condition is \(-\frac{\pi}{2}<\theta<\frac{\pi}{2}\), \(\sin \theta\) should be positive.

Step 8 ：Therefore, the value of \(\sin \theta\) is approximately 0.447213595499958.

Step 9 ：Final Answer: \(\boxed{0.447213595499958}\)