The angle between the hour and minute hands at any given time can be determined using the formula: \[ \theta = \left| 30H - \frac{11}{2}M \right| \] where \(H\) represents the hour and \(M\) represents the minutes. For \(H = 3\) and \(M = 15\), the calculation is as follows: \[ \theta = |30 \times 3 - \frac{11}{2} \times 15| = |90 - 82.5| = 7.5^\circ \] Therefore, the angle between the hour and minute hand at 3:15 is \(7.5^\circ\).
Final answer Answer: \(\boxed{7.5^\circ}\)