Calculus 8th Edition

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

Appendix D - Trigonometry - D Exercises - Page A33: 60


$cos(x+y)=\frac{1}{15}(8\sqrt 2-3)$

Work Step by Step

Evaluate the expression $cos(x+y)$ Given: $sinx=\frac{1}{3}$ and $secy=\frac{5}{4}$ Since, $cos(x+y)=cosxcosy-sinxsiny$ ...(1) Thus, $sinx=\frac{1}{3}$ gives opp =1 , hyp = 3 and adj $=\sqrt {3^{2}-1^{2}}=2\sqrt 2$ Therefore, $cos x=\frac{2\sqrt 2}{3}$ Now, $secy=\frac{5}{4}$ gives hyp =5 , adj =4 and opp $=\sqrt {5^{2}-3^{2}}=3$ Therefore, $siny=\frac{3}{4}$ and $cosy=\frac{4}{5}$ Equation (1) becomes $cos(x-y)=(\frac{2\sqrt 2}{3})(\frac{4}{5})-(\frac{1}{3})(\frac{3}{5})$ $=\frac{8\sqrt 2}{15}-\frac{3}{15}$ Hence, $cos(x+y)=\frac{1}{15}(8\sqrt 2-3)$
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