Trigonometry (11th Edition) Clone

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

Chapter 3 - Radian Measure and the Unit Circle - Section 3.2 Applications of Radian Measure - 3.2 Exercises - Page 111: 32


about $81.6^{o}$

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$ $\theta=150^{o}$ needs to be converted to radians (case 1): $s=r\displaystyle \cdot 150\cdot\frac{\pi}{180}=6.84\cdot 150\cdot\frac{\pi}{180}=\frac{6.84\cdot 5\cdot\pi}{6}$ The bigger gear rotates for the same arc length, so $ s=r\theta\qquad$ ( ... solve for $\theta$) $\displaystyle \frac{6.84\cdot 5\cdot\pi}{6}=12.46\theta$ $\displaystyle \theta=\frac{6.84\cdot 5\cdot\pi}{6\cdot 12.46} \qquad$(...radians to degrees ...) $\displaystyle \theta=\frac{6.84\cdot 5\cdot\pi}{6\cdot 12.46}\cdot\frac{180^{\mathrm{o}}}{\pi}=\frac{6.84\cdot 5\cdot 30^{o}}{12.46}\approx$81.568471337$6^{o}$ The larger gear rotates for about $81.6^{o}.$
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