Precalculus: Mathematics for Calculus, 7th Edition

Published by Brooks Cole
ISBN 10: 1305071751
ISBN 13: 978-1-30507-175-9

Chapter 7 - Section 7.3 - Double-Angle, Half-Angle, and Product-Sum Formulas - 7.3 Exercises: 68

Answer

$\frac{3(\sqrt{2}+\sqrt{3})}{4}$

Work Step by Step

Use the third Product-to-Sum Formula, $\cos u\cos v=\frac{1}{2}[\cos(u+v)+\cos(u-v)]$. $3\cos37.5^\circ\cos7.5^\circ$ $=3\times\frac{1}{2}[\cos(37.5^\circ+7.5^\circ)+\cos(37.5^\circ-7.5^\circ)]$ $=\frac{3}{2}(\cos 45^\circ+\cos 30^\circ)$ $=\frac{3}{2}(\frac{\sqrt{2}}{2}+\frac{\sqrt{3}}{2})$ $=\frac{3}{2}(\frac{\sqrt{2}+\sqrt{3}}{2})$ $=\frac{3(\sqrt{2}+\sqrt{3})}{4}$
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