Lesson 1

Lesson 1: Radians and Degrees in Standard Position

 

Focus

 

When 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? Why not use 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, and that there were 60 days in each season. Or maybe 360 was chosen because it is so easily divided by 24 numbers.

 

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
• determine a relationship between degrees and radians
• determine a method for converting between degrees and radians and between radians and degrees
• sketch angles given in degrees and radians, positive and negative, in standard position

Lesson Questions

 

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.

 

Time

 

Each lesson in Mathematics 30-1 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 in standard position from 0° to 360°
• understand an arc subtended by a central angle
  

Are You Ready?

 

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

1. Draw the following angles in standard position.
   1. 120° Answer
   2. 315° Answer
   3. 60° Answer
   4. 250° Answer
2. In which quadrant does the terminal arm lie for each of the following angles drawn in standard position?
   1. 150° Answer
   2. 20° Answer
   3. 295° Answer
   4. 200° Answer
3. Draw a diagram of a circle with a central angle of approximately 60°, and label the arc subtended by the central angle. Answer

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

Recall that angles in standard position

• have their vertex at the origin (point (0, 0) on the Cartesian plane)
• are formed by an initial arm on the positive x-axis and a terminal arm
• are “swept out” in a counterclockwise direction from the initial arm to the terminal arm
  
  
   
  

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Open Cartesian Plane. Review the four quadrants in the Cartesian plane.

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Review the relationships between the terms to subtend, central angle, and arc in Subtend.

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

Discover

 

Try This 1

Explore

 

In Try This 1 you created a central angle where the radius, AO, of the circle is the same as the length of the subtended arc, AB. This situation defines radian measure. The measure of ∠AOB is 1 rad (radian).

 

You may have determined that the radius of the circle would fit about 6 times around the circumference of a circle. This would mean that one full rotation, which is 360°, would be the same as about 6 rad. In Try This 2 you will try to find a more precise value for one full rotation of a circle measured in radians.

 

Try This 2

 

You will test your new understanding of 1 rad by investigating whether the radius of a circle changes the measure of the central angle. You will then explore the relationship between radians and degrees.

 

Step 1: Open Angles in Degrees and Radians.

 

 

Step 2: Slide the angle slider to change the angle to 60°. Use the arrow keys on your keyboard to change the angle by 1° at a time. Note where 1 rad is positioned.

 

Step 3: Slide the radius slider to change the radius to a larger value and then to a smaller value. Did the radian measure change? Why do you think this happens?

1. Use Angles in Degrees and Radians to complete a chart similar to the one that follows. You will notice that the radian measure is given in red as an exact measure in terms of π first and is then rounded to three decimal places.
   
   

   Degree	Portion of Whole Circle	Radian Measure (in terms of π)	Radian Measure (round to three decimal places)	Sketch or Screen Capture
   360°	1
   180°
   90°
   45°
   −360°	−1
   −180°
   −90°
   −45°

2. If the angle is positive, in which direction does the terminal arm rotate?

3. If the angle is negative, in which direction does the terminal arm rotate?

4. What relationship can you see between degrees and radians? Describe this relationship in terms of π rad.
   

Use the relationship you found in question 4 to answer the following questions.

5. How are degrees related to radians?
6. Complete a chart similar to the one that follows. Leave radians as exact measures. You can check your answers when you are finished by using Angles in Degrees and Radians.
   
   

   Degree	Radian (exact value, fraction of π)	Sketch	Quadrant
   30°
   60°
   −120°

Save your responses in your course folder.

 

Share 2

 

With a partner or in a group, discuss the following questions:

• How did you determine the answer to questions 5 and 6?
• Are your methods for changing degrees to radians and radians to degrees the same or different than your partner’s methods?

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

In Try This 1 you explored how a radian is defined by looking at both the radius and the arc length of a circle. In Try This 2 you made the connection that one rotation of a circle is 360°, and this is equal to 2π rad. You know that

• 360° = 2π rad
• 180° = π rad

In this next example you will see how the relationship between degrees and radians can be used to convert between radians and degrees.

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. In Try This 1 you discovered how to describe an angle measure of 1 rad. You used the ratio of the radius and arc length to determine measurements in radians.

 

In Try This 2, Angles in Degrees and Radians helped you find the relationship 360° = 2π rad. Understanding this relationship allows you to convert between radians and degrees using any method.

 

You were able to visualize the following:

• Angles in standard position are positive when the terminal arm rotates counterclockwise.
• Angles in standard position are negative when the terminal arm rotates clockwise.

You applied this understanding in Module 4 Project: The Ferris Wheel to describe the direction of motion of the Ferris wheel.

 

Throughout the rest of Module 4 you will continue to look at objects that move in circular paths, such as amusement park rides. In Lesson 2 you will use the relationship between radians and degrees to explain the distance travelled by an object moving in a circular path.