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 224: 81

Answer

Write $y'$ in terms of $R, y, z$. $$y'=y\cos R-z\sin R$$

Work Step by Step

$$y'=r\cos(\theta+R)\hspace{1cm}y=r\cos\theta\hspace{1cm}z=r\sin\theta$$ We now would analyze $y'$ formula. For $\cos(\theta+R)$, the cosine sum identity can be applied: $$\cos(\theta+R)=\cos\theta\cos R-\sin\theta\sin R$$ Therefore, $$y'=r(\cos\theta\cos R-\sin\theta\sin R)$$ $$y'=r\cos\theta\cos R-r\sin\theta\sin R$$ $$y'=(r\cos\theta)\cos R-(r\sin\theta)\sin R$$ Now recall that $y=r\cos\theta$ and $z=r\sin\theta$. $$y'=y\cos R-z\sin R$$ That is the formula of $y'$ in terms of $R, y, z$.
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