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