To find the angle between the hour and minute hands at any given time, we need to break down each component's movement on the clock face. The formula to calculate the angle is as follows:

1. Hour Hand Movement: The hour hand moves $30^\circ$ per hour (since $360^\circ / 12 = 30^\circ$).

   • Additionally, it moves $0.5^\circ$ per minute ($30^\circ / 60$).

2. Minute Hand Movement: The minute hand moves $6^\circ$ per minute (since $360^\circ / 60 = 6^\circ$).

General Formula

For a time given in hours ($h$) and minutes ($m$), the angle $\theta$ between the hands of the clock can be calculated as: $\theta = |(30 \times h + 0.5 \times m) - (6 \times m)|$ If $\theta$ exceeds $180^\circ$, we take $360^\circ - \theta$ for the smallest angle.

Let's apply this formula to each time given.

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1. Time: 3:00

• Hour hand angle = $30 \times 3 + 0.5 \times 0 = 90^\circ$
• Minute hand angle = $6 \times 0 = 0^\circ$
• $\theta = |90 - 0| = 90^\circ$

Answer: $90^\circ$ or $\frac{\pi}{2}$ radians.

2. Time: 5:15

• Hour hand angle = $30 \times 5 + 0.5 \times 15 = 150 + 7.5 = 157.5^\circ$
• Minute hand angle = $6 \times 15 = 90^\circ$
• $\theta = |157.5 - 90| = 67.5^\circ$

Answer: $67.5^\circ$ or $\frac{9\pi}{24}$ radians.

3. Time: 7:10

• Hour hand angle = $30 \times 7 + 0.5 \times 10 = 210 + 5 = 215^\circ$
• Minute hand angle = $6 \times 10 = 60^\circ$
• $\theta = |215 - 60| = 155^\circ$

Answer: $155^\circ$ or approximately $2.705$ radians.

4. Time: 2:35

• Hour hand angle = $30 \times 2 + 0.5 \times 35 = 60 + 17.5 = 77.5^\circ$
• Minute hand angle = $6 \times 35 = 210^\circ$
• $\theta = |77.5 - 210| = 132.5^\circ$

Answer: $132.5^\circ$ or approximately $2.31$ radians.

5. Time: 4:13

• Hour hand angle = $30 \times 4 + 0.5 \times 13 = 120 + 6.5 = 126.5^\circ$
• Minute hand angle = $6 \times 13 = 78^\circ$
• $\theta = |126.5 - 78| = 48.5^\circ$

Answer: $48.5^\circ$ or approximately $0.846$ radians.

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Would you like more details on any of these steps or additional examples?

Related Questions

1. What is the angle at 10:30?
2. How can you calculate the angle for times with seconds?
3. Why does the formula use 0.5 degrees per minute for the hour hand?
4. How do we determine if we need the supplementary angle for times like 9:45?
5. How does the angle calculation change on a 24-hour clock?

Tip: Always check if your angle exceeds 180 degrees to ensure you have the smallest angle.