Answer
The exact value of the trigonometric function $\sin \frac{\theta }{2}$ is $\frac{\sqrt{10}}{10}$.
Work Step by Step
Recall the figure mentioned in the problem.
The figure shows the right-angle triangle. In this triangle, the base is $4$, the perpendicular is $3$, and the hypotenuse is $5$.
Calculate the value of $\sin \frac{\theta }{2}$.
Recall the half angle formula.
$\begin{align}
& \sin \frac{\theta }{2}=\sqrt{\frac{1-\cos \theta }{2}} \\
& =\sqrt{\frac{1-\left( \frac{\text{base}}{\text{hypotenuse}} \right)}{2}}
\end{align}$
Substitute $4$ for the base and $5$ for the hypotenuse.
$\begin{align}
& \sin \frac{\theta }{2}=\sqrt{\frac{1-\left( \frac{\text{4}}{\text{5}} \right)}{2}} \\
& =\sqrt{\frac{1}{10}} \\
& =\frac{1}{\sqrt{10}}\times \frac{\sqrt{10}}{\sqrt{10}} \\
& =\frac{\sqrt{10}}{10}
\end{align}$
Therefore, the exact value of the trigonometric function $\sin \frac{\theta }{2}$ is $\frac{\sqrt{10}}{10}$.
