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 3.2 Applications of Radian Measure - 3.2 Exercises - Page 104: 28

Answer

about $213^{o}$

Work Step by Step

Arc length s (for central angle $\theta$):$ \quad s=r\theta$, where $\theta$ is in radians Converting between Degrees and Radians 1. Multiply a degree measure by $\displaystyle \frac{\pi}{180}$ radian and simplify to convert to radians. 2. Multiply a radian measure by $\displaystyle \frac{180^{\mathrm{o}}}{\pi}$ and simplify to convert to degrees. ---------------- For the smaller gear, the length of the arc that turns is $ s=r\theta$ $\theta=315^{o}$ needs to be converted to radians (case 1): $ s=r\displaystyle \cdot 315\cdot\frac{\pi}{180}=4.8\cdot 315\cdot\frac{\pi}{180}=8.4\pi$ The bigger gear rotates for the same arc length, so $ s=r\theta\qquad$ ( ... solve for $\theta$) $ 8.4\pi=7.1\theta$ $\displaystyle \theta=\frac{8.4\pi}{7.1} \qquad$(...radians to degrees ...) $\displaystyle \theta=\frac{8.4\pi}{7.1}\cdot\frac{180^{\mathrm{o}}}{\pi}=\frac{8.4\cdot 180}{7.1}\approx$212.957746479 The larger gear rotates for about $213^{o}.$
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