Answer
$38.5^\circ$
Work Step by Step
1. We know that the two gears should travel the same arc length.
2. For the smaller gear, we have the arc length traveled as $L_1=5.23\times60^\circ\times\frac{\pi}{180^\circ}\ cm$
3. For the larger gear, we have the arc length traveled as $L_2=8.16\times \theta^\circ\times\frac{\pi}{180^\circ}\ cm$
4. Let $L_2=L1$, we can find $\theta=\frac{5.23\times60}{8.16}\approx38.5^\circ$
