Precalculus (6th Edition) Blitzer

Published by Pearson
ISBN 10: 0-13446-914-3
ISBN 13: 978-0-13446-914-0

Chapter 5 - Section 5.2 - Sum and Difference Formulas - Exercise Set - Page 669: 26

Answer

The expression $\sin 40{}^\circ \cos 20{}^\circ +\cos 40{}^\circ \sin 20{}^\circ $ is written as $\sin 60{}^\circ $ and the exact value of $\sin 60{}^\circ $ is $\frac{\sqrt{3}}{2}$.

Work Step by Step

Use the sum formula of sines and rewrite the expression as the sum of angles to obtain the sine of the angle as, $\begin{align} & \sin \left( 40{}^\circ +20{}^\circ \right)=\sin 40{}^\circ \cos 20{}^\circ +\cos 40{}^\circ \sin 20{}^\circ \\ & \sin \left( 60{}^\circ \right)=\sin 40{}^\circ \cos 20{}^\circ +\cos 40{}^\circ \sin 20{}^\circ \end{align}$ Therefore, the expression $\sin 40{}^\circ \cos 20{}^\circ +\cos 40{}^\circ \sin 20{}^\circ $ is equivalent to $\sin 60{}^\circ $. From the knowledge of trigonometric ratios defined for sine of an angle, the exact value of $\sin 60{}^\circ $ is $\frac{\sqrt{3}}{2}$.
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