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: 2


The exact value of the trigonometric function $\cos 2\theta $ is $\frac{7}{25}$.

Work Step by Step

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 2\theta $. Recall the double angle formula. $\begin{align} & \cos 2\theta ={{\cos }^{2}}\theta -{{\sin }^{2}}\theta \\ & ={{\left( \frac{\text{base}}{\text{hypotenuse}} \right)}^{2}}-{{\left( \frac{\text{perpendicular}}{\text{hypotenuse}} \right)}^{2}} \end{align}$ Substitute $4$ for the base, $3$ for the perpendicular and $5$ for the hypotenuse. $\begin{align} & \cos 2\theta ={{\left( \frac{\text{4}}{\text{5}} \right)}^{2}}-{{\left( \frac{\text{3}}{\text{5}} \right)}^{2}} \\ & =\frac{16}{25}-\frac{9}{25} \\ & =\frac{7}{25} \end{align}$ Therefore, the exact value of the trigonometric function $\cos 2\theta $ is $\frac{7}{25}$.
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