Trigonometry (10th Edition)

Published by Pearson
ISBN 10: 0321671775
ISBN 13: 978-0-32167-177-6

Chapter 5 - Trigonometric Identities - Section 5.4 Sum and Difference Identities for Sine and Tangent - 5.4 Exercises - Page 221: 52

Answer

$\sin{255^o}=-\dfrac{\sqrt6+\sqrt2}{4}$

Work Step by Step

Note that $255^o=210^o+45^o$. Hence, the given expression is equivalent to $\sin(210^o+45^o)$. RECALL: $\sin{(A+B)}=\sin{A}\cos{B}+\cos{A}\sin{B}$ Use the identity above with $A=210^o$ and $B=45^o$ to obtain: \begin{align*} \sin{255^o}&=\sin{(210^o+45^o)}\\\\ &=\sin{210^o}\cos{45^o}+\cos210^o\sin45^o\\\\ &=\left(-\frac{1}{2}\right)\left(\frac{\sqrt2}{2}\right)+\left(-\frac{\sqrt3}{2}\right)\left(\frac{\sqrt2}{2}\right)\\\\ &=-\frac{\sqrt2}{4}+\left(-\frac{\sqrt6}{4}\right)\\\\ &=-\dfrac{\sqrt6+\sqrt2}{4} \end{align*}
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