Algebra 2 (1st Edition)

Published by McDougal Littell
ISBN 10: 0618595414
ISBN 13: 978-0-61859-541-9

Chapter 14 Trigonometric Graphs, Identities, and Equations - 14.7 Apply Double-Angle and Half-Angle Formulas - 14.7 Exercises - Skill Practice - Page 959: 5


$2-\sqrt 3$

Work Step by Step

Double -angle Theorem can be defined as: $\tan \dfrac{\theta}{2}=\pm \sqrt {\dfrac{1- \cos \theta}{1+ \cos \theta}}$ $\tan (-165 ^{\circ})=\tan \dfrac{(-330^{\circ})}{2}$ Since $\tan$ is negative in the second quadrant. Thus, $- \sqrt {\dfrac{1- \cos (-330^{\circ})}{1+ \cos (-330^{\circ})}}= \sqrt {\dfrac{1- \cos (360-330^{\circ})}{1+ \cos (360^{\circ}+330 ^{\circ})}}=\sqrt {\dfrac{1- \cos 30 ^{\circ}}{1+ \cos 30^{\circ}}}$ and $\sqrt {\dfrac{1- (\sqrt 3/2)}{1+(\sqrt 3/2)}}=\sqrt {\dfrac{2-\sqrt 3}{2+\sqrt 3}}=\sqrt {\dfrac{2-\sqrt 3}{2+\sqrt 3}} \times \sqrt {\dfrac{2-\sqrt 3}{2-\sqrt 3}}$ Hence, $\tan (-165 ^{\circ})=2-\sqrt 3$
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