Trigonometry (11th Edition) Clone

Published by Pearson
ISBN 10: 978-0-13-421743-7
ISBN 13: 978-0-13421-743-7

Chapter 5 - Trigonometric Identities - Section 5.4 Sum and Difference Identities for Sine and Tangent - 5.4 Exercises - Page 227: 58

Answer

$\sin{\left(\dfrac{3\pi}{2}+\theta\right)}=-\cos{\theta}$

Work Step by Step

Use a graphing utility to graph the given expression. (Refer to the graph below.) Note that the graph is identical to the graph of $-\cos{\theta}$. This means that $\sin{(\frac{3\pi}{2}+\theta)}=-\cos{\theta}$. RECALL: $\sin{(A+B)}=\sin{A}\cos{B}+\cos{A}\sin{B}$ Use the identity above with $A=\frac{3\pi}{2}$ and $B=\theta$ to obtain: \begin{align*} \sin{\left(\frac{3\pi}{2}+\theta\right)}&=\sin{\frac{3\pi}{2}}\cos{\theta}+\cos{\frac{3\pi}{2}}\sin{\theta}\\\\ &=-1\cdot \cos{\theta}+0\cdot\sin{\theta}\\\\ &=-\cos{\theta}+0\\\\ &=-\cos{\theta} \end{align*} Therefore, $$\sin{\left(\dfrac{3\pi}{2}+\theta\right)}=-\cos{\theta}$$
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