Lesson 1

Lesson 1: Radian Measure

Focus

When you were asked to measure angles in circles or triangles in previous math courses, you likely gave your response using the unit of degrees. Why degrees? Did you know that there are other units you can use to measure angles? Have you ever heard of radians, gradients, or arc minutes?

Do you know why there are 360° in a circle? One theory gives credit to the ancient Babylonians. The ancient Babylonians, who lived from approximately 2000 BC, used a sexagesimal (base-60) number system instead of the base-10 number system used today.

Another theory is that the ancient Babylonians believed the Sun revolved around Earth in approximately 360 days. Or maybe 360 was chosen because it has so many factors, and so is easily divided into parts.

In this lesson you will learn about measuring angles in a unit other than degrees.

Lesson Outcomes

At the end of this lesson you will be able to

• define radian measure
• estimate conversions between radian and degree measures
• sketch an angle given in radians

Lesson Question

You will investigate the following question: What relationship exists between degree and radian measures?

Assessment

Your assessment may be based on a combination of the following tasks:

• completion of the Lesson 1 Assignment (Download the Lesson 1 Assignment and save it in your course folder now.)
  
• course folder submissions from Try This and Share activities
  
• additions to Glossary Terms
  
• work under Project Connection

Self-Check activities are for your own use. You can compare your answers to suggested answers to see if you are on track. If you have difficulty with concepts or calculations, contact your teacher.

Remember that the questions and activities you will encounter provide you with the practice and feedback you need to successfully complete this course. You should complete all questions and place your responses in your course folder. Your teacher may wish to view your work to check on your progress and to see if you need help.

Materials and Equipment

• protractor (You may find it helpful to use Printable Protractor.)
• a pipe cleaner or some string

Time

Each lesson in Mathematics 30-2 Learn EveryWare is designed to be completed in approximately two hours. You may find that you require more or less time to complete individual lessons. It is important that you progress at your own pace, based on your individual learning requirements.

This time estimation does not include time required to complete Going Beyond activities or the Module Project.

Launch

Do you have the background knowledge and skills you need to complete this lesson successfully? Launch will help you find out.

Before beginning this lesson, you should be able to

• draw and identify angles from 0° to 180°
• measure angles with a protractor
• calculate the circumference of a circle

Are You Ready?

Complete these questions. If you experience difficulty and need help, visit Refresher or contact your teacher.

1. Use a protractor to measure the following angles to the nearest degree.
   
   

   a.	b.	c.

   
   Answer
2. Use a protractor to draw angles of the following sizes.
   1. 34°
   2. 168°
   
   Answers
3.  
   1. Determine the circumference of a circle with a radius of 7 cm.
   2. Determine the radius of a circle with a circumference of 28 m.
   
   Answers

If you answered the Are You Ready? questions without difficulty, move to Discover.

If you found the Are You Ready? questions difficult, complete Refresher.

Refresher

Check how to measure and draw angles by viewing Protractor.

Review how to find different circle measures by using “Circles: Radius, Diameter and Circumference.”

Go back to the Are You Ready? section and try the questions again. If you are still having difficulty, contact your teacher.

Discover

In Try This 1 you will investigate the relationship between the radius and circumference of a circle.

Try This 1

Complete either Investigation A or Investigation B. You will need a protractor and a pipe cleaner or a string to complete Investigation A. Investigation B does not require any materials.

Investigation A

Explore

In Try This 1 you created a central angle similar to the one in the diagram shown here. Here, the radius of the circle, AO, is equal to the length of the subtended arc, AB. Remember that an arc can be defined as a part of the circumference of a circle. This situation defines radian measure. The measure of ∠AOB is 1 rad (radian) since the measure of arc AB is equal to the radius of the circle.

When an angle measurement is given and there are no units written after the measurement, you can assume the units are in radians. When you’re writing an angle measurement in degrees, the degree symbol must be included to indicate the measurement is in degrees. For example,

• 75° indicates 75 degrees
• 110 indicates 110 radians (rad)

In Try This 1 you may have determined that the radius of the circle would fit about six times around its circumference. This would mean that one full rotation, which is 360°, would be approximately the same as 6 rad.

After manipulating the formula C = 2πr, you get When you divide the circumference by the radius measure, you can conclude that 2π, or 6.283…, is the radian measure of a circle. Since you know that there are 360° in a circle, this means 2π is equal to 360°. However, for most of this lesson, the approximation 360° = 6 rad or 60° = 1 rad will be used.

In Try This 2 you will use this information to estimate the size of an angle in radians given an angle measured in degrees.

Try This 2

1. Draw a sketch of each of the following angles. Estimate the size of each angle in radians.
   1. 120°
   2. 210°
   3. 341°

2.  

   1. Open Angles in Degrees and Radians. Move the angle slider to check your estimates for question 1. The left and right arrow keys can be used to change the angle by single degrees. The angles you drew may be of different orientations to those in the program, but they should be a similar size.

       

       

      

   2. Does the approximation that 1 rad is about 60° produce good estimates?

Save your responses in your course folder.

In Try This 2 you may have noticed that a proportion can be used to estimate the size of an angle in radians given an angle in degrees. You could use the proportion if you want exact solutions, but sometimes estimates are sufficient. For example, to estimate the size of 120° in radians, you could do the following, where r is the number of radians and d is the number of degrees:

Notice that no units are used when representing radians.

In Try This 2 you estimated the size of angles in radians. In Try This 3 you will estimate the size of an angle in degrees given an angle in radians.

Try This 3

Estimate the size of each angle in degrees and sketch each angle.

1. 2
2. 5.1
3. 4.4

Save your responses in your course folder.

Share 2

With a partner or in a group, discuss the following questions based on information from Try This 3.

1. How is the strategy you used to estimate a radian measure when given degrees similar to the strategy you used to estimate degrees when given a radian measure?
2. Explain the strategy you used to sketch the angles.

If required, save a record of your discussion in your course folder.

In Try This 3 you may have noticed that it is possible to estimate an angle measure in degrees when you are given radians by multiplying the number of radians by 60°. For example, 4 rad is about 4 × 60 = 240°.

So far, you have looked at angles between 0° and 360° or 0 and 2π. It is also possible to express angles larger than 360°. Open Angles Larger than 360° and move the angle slider to see a representation of larger angles.

Angles Larger than 360° can be used to describe objects that rotate. A spiral is often used to show this in a diagram.

This is an angle larger than 360°.

In Try This 4 you will explore the rotation of the minute hand on a clock.

Try This 4

iStockphoto/Thinkstock

1. Suppose it is 12:00 right now. Explain how many degrees the minute hand has travelled when it is
   1. 12:15, fifteen minutes later
   2. 1:15, an hour and fifteen minutes later
   3. 1:40, an hour and forty minutes later
2. Suppose it is 12:00 right now. Explain what time will it be when the minute hand has travelled
   1. 210°
   2. 510°
   3. 5π rad
3. Describe a procedure that will determine the time, given an angle in degrees, through which the minute hand has moved in a clockwise direction.

Save your responses in your course folder.

protractor: iStockphoto/Thinkstock, clock: Stockbyte/Thinkstock

In Try This 4, you may have noticed that because there are 360° in a circle and 60 min in one rotation of the minute hand, the minute hand passes through per minute or, equivalently, 5 min represents 30°.

In general, angles larger than 360° can be treated the same as smaller angles. Converting between radians and degrees follows the same procedure as the one used in Try This 3. In question 2.c. of Try This 4, you were asked to determine what time it was after the minute hand travelled through 5π. You know that each rotation is 2π; therefore, 5π would be 2.5 rotations, or 150 min.

Connect

Lesson 1 Assignment

Lesson 1 Summary

In this lesson you learned that angles can be described using different measures and units. You focused on the relationship between degrees and radians and used the relationship 1 rad ≈ 60° to estimate the conversion between the two. You also saw that it is possible to have an angle larger than 360°.

This is an angle larger than 360°.

In Lesson 2 you will begin to explore periodic functions. These are functions that repeat themselves regularly.