Precalculus (6th Edition) Blitzer

Published by Pearson
ISBN 10: 0-13446-914-3
ISBN 13: 978-0-13446-914-0

Chapter 5 - Section 5.3 - Double-Angle, Power-Reducing, and Half-Angle Formulas - Exercise Set - Page 680: 53


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 $\frac{3}{5}$.
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