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: 4


$-\sqrt 2-1$

Work Step by Step

Double -angle Theorem can be defined as: $\tan \dfrac{\theta}{2}=\pm \sqrt {\dfrac{1- \cos \theta}{1+ \cos \theta}}$ $\tan 112.5 ^{\circ}=\tan \dfrac{225^{\circ}}{2}$ Since $\tan$ is Negative in the second quadrant. Thus, $- \sqrt {\dfrac{1- \cos 225 ^{\circ}}{1+ \cos 225^{\circ}}}= \sqrt {\dfrac{1- \cos (180 ^{\circ}+45 ^{\circ})}{1+ \cos (180 ^{\circ}+45 ^{\circ})}}=- \sqrt {\dfrac{1+ \cos 45 ^{\circ}}{1- \cos 45 ^{\circ}}}$ and $-\sqrt {\dfrac{1+ (1/\sqrt 2)}{1- (1/\sqrt 2)}}=-\sqrt {\dfrac{\sqrt 2+1}{\sqrt 2-1}}$ Hence, $\tan 112.5 ^{\circ}=-\sqrt 2-1$
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