## Precalculus (6th Edition) Blitzer

The exact value of the trigonometric function $\cos \frac{\theta }{2}$ is $\frac{3\sqrt{10}}{10}$.
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 $\cos \frac{\theta }{2}$. Recall the half angle formula, \begin{align} & \cos \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} & \cos \frac{\theta }{2}=\sqrt{\frac{1+\left( \frac{\text{4}}{\text{5}} \right)}{2}} \\ & =\sqrt{\frac{9}{10}} \\ & =\frac{3}{\sqrt{10}}\times \frac{\sqrt{10}}{\sqrt{10}} \\ & =\frac{3\sqrt{10}}{10} \end{align} Therefore, the exact value of the trigonometric function $\cos \frac{\theta }{2}$ is $\frac{3\sqrt{10}}{10}$.