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 683: 108


The value is $-\frac{7}{25}$.

Work Step by Step

In order to solve the value, apply the following steps given below: $\theta ={{\tan }^{-1}}\left( -\frac{4}{3} \right)$ The value of tan is computed by dividing the perpendicular by the base. The perpendicular (y) is -4 and the base (x) is 3. By taking the help of the Pythagorian Theorem and computing the value of r, we get: $\begin{align} & {{r}^{2}}={{x}^{2}}+{{y}^{2}} \\ & ={{\left( 3 \right)}^{2}}+{{\left( -4 \right)}^{2}} \\ & =9+16 \end{align}$ Further simplify the equation: $\begin{align} & r=\sqrt{25} \\ & =5 \end{align}$ So, the value of $\theta ={{\tan }^{-1}}\left( -\frac{4}{3} \right)$. So, it will become $\cos 2\theta $, Solve it as follows: $\begin{align} & \cos \left[ 2{{\tan }^{-1}}\left( -\frac{4}{3} \right) \right]=\cos 2\theta \\ & =1-2{{\sin }^{2}}\theta \\ & =1-2{{\left( \frac{y}{r} \right)}^{2}} \\ & =1-2{{\left( \frac{-4}{5} \right)}^{2}} \end{align}$ And solve the equation: $\begin{align} & \cos \left[ 2{{\tan }^{-1}}\left( -\frac{4}{3} \right) \right]=1-2\times \frac{16}{25} \\ & =-\frac{7}{25} \end{align}$ So, the value will be $-\frac{7}{25}$. Thus, the value of $\cos \left[ 2{{\tan }^{-1}}\left( -\frac{4}{3} \right) \right]$ is $-\frac{7}{25}$.
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