Precalculus: Concepts Through Functions, A Unit Circle Approach to Trigonometry (3rd Edition)

Published by Pearson
ISBN 10: 0-32193-104-1
ISBN 13: 978-0-32193-104-7

Chapter 5 - Trigonometric Functions - Section 5.3 Properties of the Trigonometric Functions - 5.3 Assess Your Understanding - Page 418: 88



Work Step by Step

Recall: $\cos{\theta} = \cos{(\theta+360^{\circ})}$ This means that $\cos{(-430^{\circ})} = \cos{(-430^{\circ}+360^{\circ})} = \cos{(-70^{\circ})}$ Recall further that: Cosine is an even function so $\cos{(-\theta)} = \cos{\theta}$. Thus, $\cos{(-70^{\circ})} = \cos{70^{\circ}}$ And so $\dfrac{\sin{70^{\circ}}}{\cos{(-430^{\circ})}} = \dfrac{\sin{70^{\circ}}}{\cos{70^{\circ}}}$ Since $\tan{\theta} =\dfrac{\sin{\theta}}{\cos{\theta}}$, then $\dfrac{\sin{70^{\circ}}}{\cos{70^{\circ}}} = \tan{70^{\circ}}$ Recall that $\tan{(-\theta)}= - \tan{\theta}$. Hence, $\tan{(-70^{\circ})} = - \tan{70^{\circ}}$ and $\dfrac{\sin{70^{\circ}}}{\cos{70^{\circ}}} + \tan{(-70^{\circ})} = \tan{70^{\circ}} - \tan{70^{\circ}} = \boxed{0}$
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