Precalculus (6th Edition)

Published by Pearson
ISBN 10: 013421742X
ISBN 13: 978-0-13421-742-0

Chapter 7 - Trigonometric Identities and Equations - 7.4 Double-Angle and Half-Angle Identities - 7.4 Exercises: 66

Answer

$-\displaystyle \frac{\sqrt{6}}{6}$

Work Step by Step

Letting $A=2x, $ we express $x=\displaystyle \frac{A}{2},\quad \displaystyle \sin x=\sin\frac{A}{2}$ $( \sin x$ is negative, as $x$ terminates in Q.III.) Half-Angle identity: $\displaystyle \sin\frac{A}{2}=-\sqrt{\frac{1-\cos A}{2}}=-\sqrt{\frac{1-(\frac{2}{3})}{2}}$ $=-\displaystyle \sqrt{\frac{3-2}{6}}=-\frac{1}{\sqrt{6}}\cdot\frac{\sqrt{6}}{\sqrt{6}}$ $=-\displaystyle \frac{\sqrt{6}}{6}$
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