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.
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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 $38.5^{o}.$
