Calculus 8th Edition

Published by Cengage
ISBN 10: 1285740629
ISBN 13: 978-1-28574-062-1

Appendix D - Trigonometry - D Exercises - Page A33: 85


$ cos (\alpha-\beta)= cos \alpha cos\beta+ sin \alpha sin\beta$

Work Step by Step

Apply distance formula for $(cos \alpha, sin\alpha),(cos \beta, sin\beta)$ $c^{2}=(cos \beta-cos \alpha)^{2}+(sin \beta-sin \alpha)^{2}$ $c^{2}=(cos \beta-cos \alpha)^{2}+(sin \beta-sin \alpha)^{2}$ $c^{2}=2-2 cos \alpha cos\beta-2 sin \alpha sin\beta$ Use law of cosines, we have $c^{2}=a^{2}+b^{2}-2ab cos \theta$ Here, $c=c, a=1, b=1, \theta=\alpha-\beta$ Thus, $c^{2}=2-2 cos (\alpha-\beta)$ ...(2) From equations (1) and (2), we have $2-2 cos (\alpha-\beta)=2-2 cos \alpha cos\beta-2 sin \alpha sin\beta$ $2 cos (\alpha-\beta)=2 cos \alpha cos\beta+2 sin \alpha sin\beta$ Hence, $ cos (\alpha-\beta)= cos \alpha cos\beta+ sin \alpha sin\beta$
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