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: 77c

Answer

$F$ would be maximum when $\theta=0^\circ$

Work Step by Step

From part b), we get an approximation formula of $F$: $$F\approx2.9W\cos\theta$$ In this exercise, consider that $W$ is unchanged and the only variable is $\theta$, we need to find at which value of $\theta$ that $F$ is maximum. Here, since $2.9W$ is unchanged, $F$ is maximum when $\cos\theta$ is maximum. We already know the range of $\cos\theta$ is $[-1,1]$. In other words, $$-1\le\cos\theta\le1$$ Therefore, the maximum value of $\cos\theta$ is $1$, occurring when $\theta=k180^\circ$ $(k\in Z)$. However, in this exercise we only consider the capacity of a straight back making an angle of $\theta$ with the horizontal, so $0\le\theta\le180^\circ$. So the maximum value of $\cos\theta$ is $1$ occurring when $\theta=0^\circ$. That means, overall, $F$ would be maximum when $\theta=0^\circ$
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