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: $\boxed{{\displaystyle 0.447213595499958}}$