Algebra and Trigonometry 10th Edition

Published by Cengage Learning
ISBN 10: 9781337271172
ISBN 13: 978-1-33727-117-2

Chapter 7 - 7.5 - Multiple-Angle and Product-to-Sum Formulas - 7.5 Exercises - Page 548: 36


$sin~165°=\frac{\sqrt {2-\sqrt 3}}{2}$ $cos~165°=-\frac{\sqrt {2+\sqrt 3}}{2}$ $tan~165°=-2+\sqrt 3$

Work Step by Step

$165°$ lies in the second quadrant. $sin~165°=sin\frac{330°}{2}=+\sqrt {\frac{1-cos~330°}{2}}=\sqrt {\frac{1-\frac{\sqrt 3}{2}}{2}}=\sqrt {\frac{2-\sqrt 3}{4}}=\frac{\sqrt {2-\sqrt 3}}{2}$ $cos~165°=cos\frac{330°}{2}=-\sqrt {\frac{1+cos~330°}{2}}=-\sqrt {\frac{1+\frac{\sqrt 3}{2}}{2}}=-\sqrt {\frac{2+\sqrt 3}{4}}=-\frac{\sqrt {2+\sqrt 3}}{2}$ $tan~165°=tan\frac{330°}{2}=\frac{1-cos~330°}{sin~330°}=\frac{1-\frac{\sqrt 3}{2}}{-\frac{1}{2}}=-2+\sqrt 3$
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