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: 3



Work Step by Step

Double -angle Theorem can be defined as: $\sin \dfrac{\theta}{2}=\pm \sqrt {\dfrac{1- \cos \theta}{2}}$ $\sin 105 ^{\circ}=\sin \dfrac{210 ^{\circ}}{2}=\pm \sqrt {\dfrac{1- \cos 210 ^{\circ}}{2}}$ Since $\sin$ is positive in the second quadrant: Thus, $ \sqrt {\dfrac{1- \cos 210 ^{\circ}}{2}}= \sqrt {\dfrac{1- \cos (180 ^{\circ}+30 ^{\circ})}{2}}= \sqrt {\dfrac{1+ \cos 30 ^{\circ}}{2}}$ and $\sqrt {\dfrac{1+ \cos 30 ^{\circ}}{2}}=\sqrt {\dfrac{1+ (\sqrt 3/2)}{2}}=\dfrac{\sqrt{2+\sqrt{3}}}{2}$
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