Precalculus (6th Edition) Blitzer

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

Chapter 5 - Section 5.2 - Sum and Difference Formulas - Exercise Set - Page 669: 31


The expression $\frac{\tan \frac{\pi }{5}-\tan \frac{\pi }{30}}{1+\tan \frac{\pi }{5}\tan \frac{\pi }{30}}$ is written as $\tan \frac{\pi }{6}$ and the exact value of $\tan \frac{\pi }{6}$ is $\frac{1}{\sqrt{3}}$.

Work Step by Step

Use the difference formula of tangent and rewrite the expression as the difference of angles to obtain the tangent of the angle as, $\begin{align} & \tan \left( \frac{\pi }{5}-\frac{\pi }{30} \right)=\frac{\tan \frac{\pi }{5}-\tan \frac{\pi }{30}}{1+\tan \frac{\pi }{5}\tan \frac{\pi }{30}} \\ & \tan \left( \frac{\pi }{6} \right)=\frac{\tan \frac{\pi }{5}-\tan \frac{\pi }{30}}{1+\tan \frac{\pi }{5}\tan \frac{\pi }{30}} \end{align}$ Therefore, the expression $\frac{\tan \frac{\pi }{5}-\tan \frac{\pi }{30}}{1+\tan \frac{\pi }{5}\tan \frac{\pi }{30}}$ is equivalent to $\tan \frac{\pi }{6}$. From the knowledge of trigonometric ratios defined for tangent of an angle, the exact value of $\tan \frac{\pi }{6}$ is $\frac{1}{\sqrt{3}}$.
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