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: 38


$sin~67°~30'=\frac{\sqrt {2+\sqrt 2}}{2}$ $cos~67°~30'=\frac{\sqrt {2-\sqrt 2}}{2}$ $tan~67°~30'=\sqrt 2+1$

Work Step by Step

$67°~30'$ lies in the first quadrant. $sin~67°~30'=sin\frac{135°}{2}=+\sqrt {\frac{1-cos~135°}{2}}=\sqrt {\frac{1-(-\frac{\sqrt 2}{2})}{2}}=\sqrt {\frac{2+\sqrt 2}{4}}=\frac{\sqrt {2+\sqrt 2}}{2}$ $cos~67°~30'=cos\frac{135°}{2}=+\sqrt {\frac{1+cos~135°}{2}}=\sqrt {\frac{1-\frac{\sqrt 2}{2}}{2}}=\sqrt {\frac{2-\sqrt 2}{4}}=\frac{\sqrt {2-\sqrt 2}}{2}$ $tan~67°~30'=tan\frac{135°}{2}=\frac{1-cos~135°}{sin~135°}=\frac{1-(-\frac{\sqrt 2}{2})}{\frac{\sqrt 2}{2}}=\frac{2+\sqrt 2}{\sqrt 2}=\sqrt 2+1$
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