Precalculus (10th Edition)

Published by Pearson
ISBN 10: 0-32197-907-9
ISBN 13: 978-0-32197-907-0

Chapter 7 - Analytic Trigonometry - 7.5 Sum and Difference Formulas - 7.5 Assess Your Understanding - Page 488: 59

Answer

Please refer to the step-by-step part below.

Work Step by Step

Recall: (1) $\sin{(\alpha+\beta)}=\sin{(\alpha)}\cos{(\beta)}+\cos{(\alpha)}\sin{(\beta)}$ (2) $\sin{(\alpha-\beta)}=\sin{(\alpha)}\cos{(\beta)}-\cos{(\alpha)}\sin{(\beta)}$ Hence, $\sin{(\alpha+\beta)}+\sin{(\alpha-\beta)}\\ =[\sin{(\alpha)}\cos{(\beta)}+\cos{(\alpha)}\sin{(\beta)}]+[\sin{(\alpha)}\cos{(\beta)}-\cos{(\alpha)}\sin{(\beta)}]\\ =[\sin{(\alpha)}\cos{(\beta)}+\sin{(\alpha)}\cos{(\beta)}+\cos{(\alpha)}]+[\sin{(\beta)}-\cos{(\alpha)}\sin{(\beta)}]\\ =2\sin{(\alpha)}\cos{(\beta)} +0\\ =2\sin{(\alpha)}\cos{(\beta)}$
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