## Trigonometry (11th Edition) Clone

Published by Pearson

# Chapter 3 - Radian Measure and the Unit Circle - Section 3.2 Applications of Radian Measure - 3.2 Exercises - Page 111: 31

#### Answer

about $38.5^{o}$

#### 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 $s=r\theta$ $\theta=60.0^{o}$ needs to be converted to radians (case 1): $s=r\displaystyle \cdot 60.0\cdot\frac{\pi}{180}=5.23\cdot 60.0\cdot\frac{\pi}{180}$ The bigger gear rotates for the same arc length, so $s=r\theta\qquad$ ( ... solve for $\theta$) $5.23\displaystyle \cdot 60.0\cdot\frac{\pi}{180}=8.16\theta$ $\displaystyle \theta=\frac{5.23\cdot 60.0\pi}{180\cdot 8.16} \qquad$(...radians to degrees ...) $\displaystyle \theta=\frac{5.23\cdot 60.0\pi}{180\cdot 8.16}\cdot\frac{180^{\mathrm{o}}}{\pi}=\frac{5.23\cdot 60.0}{8.16}\approx$38.4558823529 The larger gear rotates for about $38.5^{o}.$

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