jmc

There are 12 hours on a clock, so each hour mark is $360^{\circ }/12=30^{\circ }$ from its neighbors. At 3:30, the minute hand points at the 6, while the hour hand is mid-way between the 3 and the 4. Therefore, the hour hand is $\frac{1}{2}\cdot 30^{\circ }=15^{\circ }$ from the 4 on the clock, and there are $2\cdot 30^{\circ }=60^{\circ }$ between the 4 and the 6 on the clock. So, the hour and the minute hand are $15^{\circ }+60^{\circ }=\boxed{75^{\circ }}$ apart.