Calculus: Early Transcendentals 8th Edition

Published by Cengage Learning
ISBN 10: 1285741552
ISBN 13: 978-1-28574-155-0

APPENDIX D - Trigonometry - D Exercises - Page A 33: 75

Answer

$-1 \lt tan ~x \lt 1~~$ when $~~x~~$ is in: $[0,\frac{\pi}{4})\cup (\frac{3\pi}{4},\frac{5\pi}{4}) \cup (\frac{7\pi}{4},2\pi]$

Work Step by Step

Note that $tan~x = 1$ when $x= \frac{\pi}{4}$ and $x = \frac{5\pi}{4}$ Note that $tan~x = -1$ when $x= \frac{3\pi}{4}$ and $x = \frac{7\pi}{4}$ We can use the graph of $tan~x$ to solve this question: On the interval $[0,\frac{\pi}{4})$: $0 \leq tan ~x \lt 1$ On the interval $(\frac{3\pi}{4},\frac{5\pi}{4})$: $-1 \lt tan ~x \lt 1$ On the interval $(\frac{7\pi}{4},2\pi]$: $-1 \lt tan ~x \leq 0$ Therefore, $~~-1 \lt tan ~x \lt 1~~$ when $~~x~~$ is in: $[0,\frac{\pi}{4})\cup (\frac{3\pi}{4},\frac{5\pi}{4}) \cup (\frac{7\pi}{4},2\pi]$
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