## Precalculus (6th Edition) Blitzer

The exact value of the trigonometric function $2\sin \frac{\theta }{2}\cos \frac{\theta }{2}$ is $\frac{3}{5}$.
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 $\frac{3}{5}$.