Find the degrees and radians the angle between the hour hand and the minute hand of a clock at half past three.

Explanation

To find the angle between the hour hand and the minute hand of a clock at half past three, we can follow these steps:

### Step 1: Determine the position of the hour hand
At 3:00, the hour hand is at 3. By 3:30, the hour hand has moved halfway between 3 and 4.

- Each hour represents 30 degrees (since 360 degrees / 12 hours = 30 degrees per hour).
- Therefore, at 3:00, the hour hand is at \(3 \times 30 = 90\) degrees.
- By 3:30, the hour hand moves an additional half of 30 degrees (since it is halfway to the next hour):
\[ \text{Additional movement} = \frac{30}{2} = 15 \text{ degrees} \]
- Thus, the position of the hour hand at 3:30 is:
\[ 90 + 15 = 105 \text{ degrees} \]

### Step 2: Determine the position of the minute hand
At 3:30, the minute hand is at the 6 on the clock.

- Each minute represents 6 degrees (since 360 degrees / 60 minutes = 6 degrees per minute).
- Therefore, at 30 minutes, the minute hand is at:
\[ 30 \times 6 = 180 \text{ degrees} \]

### Step 3: Calculate the angle between the hour hand and the minute hand
Now, we can find the angle between the hour hand and the minute hand by taking the absolute difference between their positions:

\[ \text{Angle} = |180 - 105| = 75 \text{ degrees} \]

### Step 4: Convert degrees to radians
To convert degrees to radians, we use the conversion factor \(\frac{\pi}{180}\):

\[ \text{Angle in radians} = 75 \times \frac{\pi}{180} = \frac{75\pi}{180} = \frac{5\pi}{12} \]

### Final Answer
Thus, the angle between the hour hand and the minute hand at half past three is:
- **Degrees:** 75 degrees
- **Radians:** \(\frac{5\pi}{12}\)

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