Determining the angle between the hour and minute hands on an analog clock at a specific time is a classic math problem that involves some simple trigonometry. At 7:45, the hour hand is halfway between the 7 and 8, while the minute hand is at the 45-minute mark. To find the angle between them, we need to understand how the hands on an analog clock work.

## How Clock Hands Work

On an analog clock, the hour hand makes one complete revolution around the clock face every 12 hours. This means it rotates through 360° in 12 hours, or 30° per hour. The minute hand makes one complete revolution every 60 minutes. So it rotates through 360° in 60 minutes, or 6° per minute.

Some key facts:

- The hour hand rotates 1/12 of 360° (30°) per hour
- The minute hand rotates 1/60 of 360° (6°) per minute
- At 12:00, both hands are vertically aligned at the 12

Knowing this, we can figure out the angles of the hands at any time.

## Finding the Angle at 7:45

At 7:45, the hour hand is halfway between the 7 and 8. Since it moves 30° per hour, at 7:30 (halfway to 8) it will have moved:

7 hours * 30° per hour = 210°

Plus half an hour more:

210° + 15° (half of 30°) = 225°

So at 7:45, the hour hand is at 225°.

The minute hand moves 6° per minute. At 45 minutes past the hour, it will have moved:

45 minutes * 6° per minute = 270°

So at 7:45, the minute hand is at 270°.

To find the angle between the hands, we take the absolute difference between their angles:

| Hour hand angle | Minute hand angle | Difference |

|-|-|-|

| 225° | 270° | 270° – 225° = 45° |

## Explaining the Math

Here is a more detailed explanation of the math behind finding the angle:

### Hour Hand Position

– The hour hand moves 30° per hour

– At 7:00, it is at the 7, or 210° (7 * 30°)

– At 7:30, it has moved 15° more, to 225° (halfway to 240°)

– So at 7:45, it is halfway between 7:30 and 8:00, which is 225°

### Minute Hand Position

– The minute hand moves 6° per minute

– At 0 minutes, it is at the 12, or 0°

– At 45 minutes, it has moved 45 * 6 = 270°

– So at 7:45, it is at 270°

### Finding the Angle Between

– Hour hand angle: 225°

– Minute hand angle: 270°

– Absolute difference: |270° – 225°| = 45°

So the angle between the hands at 7:45 is 45°.

## Visual Representation

It can help to visualize the clock hands positions:

Time | Hour Hand Position | Minute Hand Position |
---|---|---|

12:00 | 0° (at 12) | 0° (at 12) |

7:00 | 210° (at 7) | 0° (at 12) |

7:30 | 225° (between 7 & 8) | 90° (at 6) |

7:45 | 225° (between 7 & 8) | 270° (at 9) |

This helps show how the hands move and end up at 225° and 270° respectively at 7:45.

## Solving for Any Time

The process above can be generalized to find the angle between the hands for any time.

For a time of H hours and M minutes:

1. Hour hand angle = (H * 30°) + (M/2 * 30°)

2. Minute hand angle = (M * 6°)

3. Angle between hands = Absolute value of the difference between hour and minute angles

We can summarize this in a formula:

Angle = |(H * 30 + M/2 * 30) – (M * 6)|

Where:

– H is the hour (1-12)

– M is the minutes (0-59)

This allows us to quickly calculate the angle for any time without having to work through each step.

## Other Examples

Let’s try a few more examples:

### 3:30

– H = 3 hours

– M = 30 minutes

– Hour hand angle = (3 * 30) + (30/2 * 30) = 90°

– Minute hand angle = (30 * 6) = 180°

– Angle between hands = |90 – 180| = 90°

### 9:15

– H = 9 hours

– M = 15 minutes

– Hour hand angle = (9 * 30) + (15/2 * 30) = 270°

– Minute hand angle = (15 * 6) = 90°

– Angle between hands = |270 – 90| = 180°

### 12:00

– H = 12 hours

– M = 0 minutes

– Hour hand angle = (12 * 30) = 360° = 0°

– Minute hand angle = (0 * 6) = 0°

– Angle between hands = |0 – 0| = 0°

So the angle is 0° when the hands are perfectly aligned at 12.

## Special Cases

There are a couple special cases to be aware of:

### When M=60

If the number of minutes (M) is 60, the hour hand has moved to the next hour. So we need to increment H by 1 and set M to 0.

For example, at 1:60, the time is really 2:00. So:

– H = 2

– M = 0

– Hour hand angle = (2 * 30) = 60°

– Minute hand angle = (0 * 6) = 0°

– Angle between hands = |60 – 0| = 60°

### When H=0

On clocks showing 1-12 hours, 0 technically refers to 12. So:

– H = 12

– Calculate angles as normal

This allows times like 12:15 and 12:30 to work correctly.

## Dealing with Negative Angles

Due to how the clock hands rotate, the angle between them is always positive, in the range 0° to 180°.

However, if you calculate a negative angle based on the formula, you can make it positive by adding 360°.

For example, at 5:40:

– H = 5 hours

– M = 40 minutes

– Hour hand angle = 150°

– Minute hand angle = 240°

– Angle = |150 – 240| = -90°

Since -90° is invalid, we adjust it:

-90° + 360° = 270°

So the angle at 5:40 is 270°.

## Conclusion

Finding the angle between analog clock hands requires basic trigonometry but follows a straightforward process:

1. Calculate the hour hand angle based on hours elapsed

2. Calculate the minute hand angle based on minutes elapsed

3. Take the absolute difference between the angles

This can be expressed by the formula:

Angle = |(H * 30 + M/2 * 30) – (M * 6)|

Where H is hours and M is minutes.

Visualizing the positions of the hands and trying examples helps build intuition for how the angles change over time. The general process can be applied to find the angle between the hands for any time on a 12-hour analog clock.