Trigonometry (10th Edition)

Published by Pearson
ISBN 10: 0321671775
ISBN 13: 978-0-32167-177-6

Chapter 3 - Radian Measure and the Unit Circle - Section 3.3 The Unit Circle and Circular Functions - 3.3 Exercises - Page 117: 49

Answer

$\tan(6.29)$ is positive

Work Step by Step

The angles (in radians) representing borders between quadrants are 1. for positive angles, that is, radian measures counted counterclockwise from the point (1,0) : $[0]$ ... I ... $[\displaystyle \frac{\pi}{2}\approx 1.57]$ $[\displaystyle \frac{\pi}{2}\approx 1.57]$ ... II ... $[\pi\approx 3.14]$ $[\pi\approx 3.14]$ ... III ...$[\displaystyle \frac{3\pi}{2}\approx 4.71]$ $[\displaystyle \frac{3\pi}{2}\approx 4.71]$... IV... $2\pi\approx 6.28$ 2. for negative (clockwise from the point (1,0)): $[0]$ ... IV ... $[-\displaystyle \frac{\pi}{2}\approx-1.57]$ $[-\displaystyle \frac{\pi}{2}\approx-1.57]$ ... III ... $[-\pi\approx-3.14]$ $[-\pi\approx-3.14]$ ... II ...$[-\displaystyle \frac{3\pi}{2}\approx-4.71]$ $[-\displaystyle \frac{3\pi}{2}\approx-4.71]$... I... $[-2\pi\approx-6.28]$ ------------ For any real number $s$ represented by a directed arc on the unit circle, $\sin s=y\quad \cos s=x \quad \displaystyle \tan s=\frac{y}{x} (x\neq 0)$ ================= A radian measure of $6.29 $ (counterclockwise) is coterminal with an angle with radian measure $6.28+0.1$ which represents an angle in quadrant I, ( radian measure of 6.28 means one whole revolution has been completed) $0 < 0.1 < 1.57$ in quadrant I, both coordinates are positive, which means that $ \displaystyle \frac{y}{x}$ is also positive, so $\tan(6.29)=\tan(0.1)$ is positive
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