Lesson 2

Lesson 2: Applications of Radians

Focus

The Reichstag dome is a glass dome built on top of the Reichstag building in Berlin, Germany. Inside the Reichstag dome are circular paths.

If you walked around one of these circular paths once, could you determine how far you had walked? How about three times? What would be different if you only hiked part of the circle? In each scenario, what would you need to know in order to determine how far you had walked?

In this lesson you will expand your knowledge of degrees and radians and determine how these units can be used in real-life situations.

Lesson Outcomes

At the end of this lesson you will be able to

• determine angles that are coterminal with a given angle
  
• determine the general form of all angles that are coterminal with a given angle
  
• explain the relationship between arc lengths, central angles, and the radius of a circle and solve problems using this relationship

Lesson Questions

You will investigate the following questions:

• How can angles with the same terminal arms be described?
  
• How can the relationship between the central angle, arc length, and radius in a circle be used to solve real-life problems?

Assessment

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

• completion of the Lesson 2 Assignment (Download the Lesson 2 Assignment and save it in your course folder now.)
• course folder submissions from Try This and Share activities
• additions to Glossary Terms and Formula Sheet
• work under Project Connection

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

• determine the circumference of a circle
• add and subtract fractions

Are You Ready?

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

1. A circle has a radius of 5 cm. Determine the circumference of the circle to the nearest tenth of a centimetre. Answer
2. Add or subtract the following fractions.
   
   1. Answer
   2. Answer
   3.  (Leave your answer as an exact value.) Answer
3. What is the difference between natural numbers and integers? 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

Review the circumference of a circle in Circumference.

Watch Fractions—Add and Subtract.

Review natural numbers and integers in Real Numbers.

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

Open Coterminal Angles and answer the following questions.

1. Drag the slider at the bottom of the screen so that the blue terminal arm rotates counterclockwise to 150°. You can use the arrow keys on your keyboard to move the slider in smaller increments. The terminal arm should line up with the existing black terminal arm. This angle is recorded for you in the table that follows.
2. From 150°, move the slider so that the blue terminal arm rotates counterclockwise until the blue terminal arm lines back up with the black terminal arm. Record this new angle, in degrees and radians, and sketch a diagram of the angle in the next row of a table like the one shown.
3. Repeat the step you completed in question 2. Record the new angle and add a diagram to your table.
4. Predict the next counterclockwise angle where the blue terminal arm would line up with the black terminal arm. Note this angle and add a diagram.
5. Return the slider to 0°. Move the slider so that the blue terminal arm rotates clockwise until it lines up with the black terminal arm. Record this angle and add a diagram to your table (in the row marked Clockwise Angle).
6. From this angle, move the slider so that the blue terminal arm rotates clockwise one more rotation to line up with the black terminal arm. Add the angle and diagram to your table.
7. Predict the next counterclockwise angle where the blue terminal arm would line up with the black terminal arm. Record your prediction in your table.
   
   

   	Degrees	Radians	Diagram
   Counterclockwise Angle	150°
   Angle After One Counterclockwise Rotation
   Angle After Two Counterclockwise Rotations
   Angle Predicted After Three Counterclockwise Rotations
   Clockwise Angle
   Angle After One Clockwise Rotation
   Angle Predicted After Two Clockwise Rotations

 Save your responses in your course folder.

Share 1

Discuss your responses to Try This 1 with a classmate. In your discussion, respond to the following questions.

1. What pattern do you see that could help determine the angle after multiple clockwise rotations? How is this different from the pattern for counterclockwise rotations?  
2. Coterminal Angles did not let you rotate the terminal arm more than 3 rotations. How could you determine the angle after 4 counterclockwise rotations? How about after 4 clockwise rotations?
3. How is your process different depending on whether you are working in degrees or radians?

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

Explore

In Discover you saw that each counterclockwise rotation of the terminal arm increases the angle measurement by 2π, or 360°. For each clockwise rotation of the terminal arm, the angle measurement decreases by 2π, or 360°. In Try This 2 you will apply this understanding to determine the distance travelled by a horse.

Try This 2

Sometimes a horse walker is used for horses that need rehabilitation. The horses are tied to a pole at the centre, and they walk at a constant rate around the circular horse walker.

A horse walker has a radius of 6 m. The horses take 12 s to walk 1 revolution.

1. What distance does a horse walk after 1 revolution? Place this information in a table similar to the one shown.
2. What distance, and through what central angle, will the white horse travel after the times specified in the table?
3. Complete the rest of the table.
   
   

   	12 s	24 s	36 s
   Number of Revolutions	1
   Distance Travelled by White Horse to the Nearest Tenth of a Metre	37.7
   Diagram of Rotational Angle in Standard Position
   Angle of Rotation in Degrees	360°
   Angle of Rotation in Radians	2π

4. How did you determine the distance travelled by the horse? Explain your process.

5. Explain how you found the rotational angle.
6. How do the terminal arms for each revolution compare?

 Save your responses in your course folder.

All of the angles you used in Try This 1 and Try This 2 had terminal arms in the same position. Angles with the same terminal arm are called coterminal angles. The following angles are coterminal angles.

In Try This 1 you found patterns for coterminal angles. Since you can keep rotating the terminal arm counterclockwise or clockwise, there is an infinite number of coterminal angles. This means you can keep adding or subtracting multiples of 360° or 2π. You can use both of the following expressions to describe all possible coterminal angles.
 

θ ± (360°)n or θ ± 2πn, n ∈ N

θ + (360°)n or θ + 2πn, n ∈ I

To this point in the lesson you have only been dealing with coterminal angles—angles where the terminal arms coincide. In Try This 3 you will explore angles formed by incomplete revolutions. You will explore how the distance a horse moves in a circle relates to the radian measure of the angle the horse travels.

Try This 3

In Try This 2 you looked at a horse walker with a radius of 6 m. The horses took 12 s to walk 1 revolution. You found the distance of 1 revolution to be

1. What distance, and through what central angle, will the white horse travel after each specified amount of time that is less than 12 s? Complete a table similar to the one shown. The first column has been completed for you.
   
   

   	6 s	3 s	4 s
   Fraction of 1 Revolution	of a revolution
   Distance Travelled by White Horse to the Nearest Tenth of a Metre, or Arc Length	circumference × amount of a revolution
   Diagram of Rotational Angle in Standard Position
   Angle of Rotation in Degrees	360° × = 180°
   Angle of Rotation in Radians	2π × = π

2. Explain how you determined the distance travelled by the horse.

3. Explain how you determined the angle of rotation.

 Save your responses in your course folder.

Share 2

Discuss your responses to Try This 3 and the following questions with a classmate.

1. How do your strategies compare for finding the distance travelled and the angle of rotation? What are the similarities and differences between your strategies and your partner’s strategies?
2. What is the relationship between the distance around a circle (arc length) and the radian measure of the angle?

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

Different ways can be used to determine the distance the horse travelled in Try This 3. You may have seen that if you know what proportion of the circle was travelled and the circumference of the circle, you can calculate the distance travelled by the horse. A completed portion of the table from Try This 3 shows example calculations that could be used to find the distance and angle travelled.

	4 s
Fraction of 1 Revolution	of a revolution
Distance Travelled by White Horse to the Nearest Tenth of a Metre, or Arc Length	circumference × amount of a revolution
Diagram of Rotational Angle in Standard Position
Angle of Rotation in Degrees
Angle of Rotation in Radians

In Try This 4 you will see another way to look at the arc length around a circle. You will look at the relationship between arc length, radius, and central angle.

Try This 4

Step 1: Open Unwrapping a Circle and click on Show radius scale and Show radian measure.

Step 2: Change each arc length to match the value in the table that follows. Use your keyboard’s arrow keys to move the slider in smaller increments.

Step 3: Slide the Unwrap slider to the right to see the arc length of the circle stretched out, and then measure the length by the number of radii using the radius scale.

Complete a table like the one shown.

 Save a copy of your table in your course folder.

Share 3

Discuss your responses with a classmate. In your discussion, address the following points.

1. Based on the patterns you see in the table, how could you find the arc length if given the radian measure of the angle and the radius of the circle?
2. What is the relationship between arc length, radius, and radian measure?

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

In Try This 4 you may have identified the relationship between arc length, radius, and central angle:

arc length = central angle in radians × radius of the circle

Using variables, this expression can be written as a = θr, where

• a is the arc length
• θ is the central angle in radians
• r is the radius (arc length and radius are measured in the same units)

Connect

Lesson 2 Assignment

Lesson 2 Summary

In this lesson you looked at objects rotating over and over again. The time-lapse image shows stars rotating around and around, as seen from the North Pole.

You learned that coterminal angles have the same terminal arm in standard position. There are an infinite number of coterminal angles for any given angle θ. Coterminal angles can be expressed in general form as θ + (360°)n, n ∈ I or θ + 2πn, n ∈ I.

Arc length is related to the radius of the circle and the central angle in radians. You learned that you could use either proportion of the circumference or the formula a = θr to determine arc length.

You went around and around in circles in this lesson. In Lesson 3 you will continue to study some of the properties of circles and determine how this information can help you with trigonometry.