Answer
The exact value of the trigonometric function $2\sin \frac{\theta }{2}\cos \frac{\theta }{2}$ is $\frac{3}{5}$.
Work Step by Step
Calculate the value of $2\sin \frac{\theta }{2}\cos \frac{\theta }{2}$.
Recall the half angle formula.
$\begin{align}
& 2\sin \frac{\theta }{2}\cos \frac{\theta }{2}=2\cdot \sqrt{\frac{1-\cos \theta }{2}}\cdot \sqrt{\frac{1+\cos \theta }{2}} \\
& =2\cdot \sqrt{\frac{1-\left( \frac{\text{base}}{\text{hypotenuse}} \right)}{2}}\cdot \sqrt{\frac{1+\left( \frac{\text{base}}{\text{hypotenuse}} \right)}{2}}
\end{align}$
Substitute $4$ for the base and $5$ for the hypotenuse.
$\begin{align}
& 2\sin \frac{\theta }{2}\cos \frac{\theta }{2}=2\cdot \sqrt{\frac{1-\left( \frac{\text{4}}{\text{5}} \right)}{2}}\cdot \sqrt{\frac{1+\left( \frac{4}{\text{5}} \right)}{2}} \\
& =2\cdot \left( \sqrt{\frac{1}{10}} \right)\cdot \left( \sqrt{\frac{9}{10}} \right) \\
& =2\cdot \left( \frac{3}{10} \right) \\
& =\frac{3}{5}
\end{align}$
Therefore, the exact value of the trigonometric function $2\sin \frac{\theta }{2}\cos \frac{\theta }{2}$ is $\frac{3}{5}$.
