In your math class one morning you finish a quiz early. While you are waiting, you watch the clock as it ticks off five minutes. The time on the clock reads 9:00. Your recent lessons have taught you that one way to measure the position of something on a circle is to use an angle. Suddenly it occurs to you that this can be applied to clocks. Can you determine the angle between the two hands of the clock?

### Rotations in Radians

A lot of interesting information about rotations and how to measure them can come from looking at clocks. We are so familiar with clocks in our daily lives that we don't often stop to think about these little devices, with hands continually rotating. Let's take a few minutes in this lesson for a closer look at these examples of rotational motion.

For the following problems, let's express in radians

1. The hands of a clock show 11:20. Express the obtuse angle formed by the hour and minute hands in radian measure.

The following diagram shows the location of the hands at the specified time.

Because there are 12 increments on a clock, the angle between each hour marking on the clock is

2. Express in radians

The hands of a clock show 4:15. Express the acute angle formed by the hour and minute hands in radian measure.

Because there are 12 increments on a clock, the angle between each hour marking on the clock is

3. The hands of a clock show 2:30. Express the acute angle formed by the hour and minute hands in radian measure.

Because there are 12 increments on a clock, the angle between each hour marking on the clock is

### Examples

#### Example 1

Earlier, you were asked if you can determine the angle between the two hands of a clock.

Since you now know that the angle between the hours on a clock is

So there are

#### The following image shows a 24-hour clock.

#### Example 2

What is the angle between each number of the clock expressed in exact radian measure in terms of

Since there are

#### Example 3

What is the angle between each number of the clock expressed to the nearest tenth of a radian? What about in degree measure?

Since there are

#### Example 4

Estimate the measure of the angle between the hands at the time shown to the nearest whole degree. And then in radian measure in terms of

### Review

Use the clock below to help you find the angle between the hour hand and minute hand at each of the following times. Express your answer in degrees less than

- 3:30
- 5:15
- 4:45
- 6:30
- 6:15
- 2:30
- 12:30
- 9:30
- 10:15
- 11:30
- 3:45
- 2:15
- 7:15
- How many times in 12 hours will the hour and minute hands overlap?
- When is the first time after 12:00 that the hour and minute hands will overlap exactly?

### Review (Answers)

To see the Review answers, open this PDF file and look for section 2.4.