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 223: 80c

Answer

$V = 50~sin~(120\pi t+0.927295)$

Work Step by Step

We can use the following identity: $a~sin~x+b~cos~x = \sqrt{a^2+b^2}~sin(x+y)$ where $y = tan^{-1}(\frac{b}{a})$ $V_1 = 30~sin~120\pi t$ $V_2 = 40~cos~120\pi t$ We can find $y$: $y = tan^{-1}(\frac{b}{a})$ $y = tan^{-1}(\frac{40}{30})$ $y = 0.927295$ We can find an expression for $V$: $V = V_1+V_2$ $V = 30~sin~120\pi t+40~cos~120\pi t$ $V = \sqrt{30^2+40^2}~sin~(120\pi t+0.927295)$ $V = 50~sin~(120\pi t+0.927295)$
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