Precalculus (6th Edition) Blitzer

Published by Pearson
ISBN 10: 0-13446-914-3
ISBN 13: 978-0-13446-914-0

Chapter 4 - Section 4.4 - Trigonometric Functions of Any Angle - Exercise Set - Page 575: 55



Work Step by Step

$\dfrac{23\pi}{4}$ is close to $\dfrac{24\pi}{4}=6\pi$. This means that this angle is between $5.5\pi$ and $6\pi$. RECALL: An angle $\theta$, where $5.5\pi \lt \theta \lt 6\pi$, is coterminal with: $\theta-4\pi$ Thus, the given angle is coterminal with: $=\dfrac{23\pi}{4}-4\pi=\dfrac{23\pi}{4} - \dfrac{16\pi}{4} = \dfrac{7\pi}{4}$ $\dfrac{7\pi}{4}$ is coterminal with $\dfrac{23\pi}{4}$. $\dfrac{7\pi}{4}$ is in Quadrant IV so $\dfrac{23\pi}{4}$ is also in Quadrant IV. RECALL: The following are the means on how to find the reference angle of an angle $0 \leq \theta \lt 2\pi$ based on its position: (1) Quadrant I: $\theta$ (itself) (2) Quadrant II: $\pi-\theta$ (3) Quadrant III: $\theta - \pi$ (4) Quadrant IV: $2\pi - \theta$ Use formula (4) above to obtain: reference angle of $\dfrac{23\pi}{4}$ = reference angle of $\dfrac{7\pi}{4}$, which is $\\2\pi - \dfrac{7\pi}{4} \\=\dfrac{8\pi}{4} - \dfrac{7\pi}{4} \\=\dfrac{\pi}{4}$
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