What’s the angle between the hour hand and the minute hand at 3:45 pm?

The minute hand is at 9 that is at 270 degrees from 12. The hour hand is between 3 and 4. The hour hand covers 30 degrees in 60 minutes and so in 45 minutes it would have covered 22.5 degrees from 3. Therefore the hour hand is at 30*3+22.5 = 90+22.5 = 112.5 degrees from 12. Hence the angle between the hour hand and the minute hand at 3:45 pm or am, as well, is

270–112.5 = 157.5 degrees. The angle between hours and minutes for a given time x:y(x hours and y minutes) is

|30*x-11*y/2|

This comes from the fact that a hour hand elapses 30 degrees for every hour (360/12) and minute hand elapses 6 degrees (360/60).

For 3:45, the hour hand also elapses extra offset from its original position where it has to be there at 3:00 sharp. The offset is calculated as follows:
1.For every 60 minutes of minutes hand, hour hand elapses 1 hour. So distance travelled by hour hand when the minute hand travels 1 min is 1/60. So the offset is (45/60) * 30 degrees for the hour hand .
Total angle covered by hour hand for 3:45 is 3 * 30 degrees + offset= 90 + 22.5 = 112.5 degrees

2. Angle covered by minute hand is 45 * 6 degrees = 270 degrees