Precalculus (6th Edition) Blitzer

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

Chapter 6 - Section 6.5 - Complex Numbers in Polar Form; DeMoivre's Theorem - Exercise Set - Page 769: 114


The exact value of $\cos \left( {{\tan }^{-1}}\frac{3}{4} \right)$ is $\frac{4}{5}$.

Work Step by Step

Let $\theta ={{\tan }^{-1}}\frac{3}{4}$ represent the angle in $\left( -\frac{\pi }{2},\frac{\pi }{2} \right)$. $\begin{align} & \theta ={{\tan }^{-1}}\frac{3}{4} \\ & \tan \theta =\frac{3}{4} \end{align}$ As the value of $\tan \theta $ is positive, thus $\theta $ lies in the first quadrant. Hence, the measures of the two sides of the right triangle are $3$ and $4$. The hypotenuse of the triangle is found by applying the Pythagorian Theorem, $\begin{align} & r=\sqrt{{{3}^{2}}+{{4}^{2}}} \\ & =\sqrt{25} \\ & =5 \end{align}$ From above sketch $\begin{align} & \theta ={{\tan }^{-1}}\frac{3}{4} \\ & \cos \theta =\cos \left( {{\tan }^{-1}}\frac{3}{4} \right) \end{align}$ By the basic definition of the cosine function, $\begin{align} & \cos \theta =\frac{\text{side adjecent to angle }\theta }{\text{hypotenuse}} \\ & \cos \left( {{\tan }^{-1}}\frac{4}{3} \right)=\frac{4}{5} \end{align}$ Hence, the value of $\cos \left( {{\tan }^{-1}}\frac{3}{4} \right)$ is $\frac{4}{5}$
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