4-1 magnitudes of rotations and measures of arcs

222 Trigonometric Functions

BIG IDEA Magnitudes of rotations are described in revolutions, degrees, and radians.

A rotation is a transformation under which

each point in the plane turns a fi xed

magnitude around a fi xed point called the

center of the rotation. In the fi gure at the

right, A and B are on the same circle

with center Q. Point A has been rotated

counterclockwise about the center Q to

the position of B, its rotation image.

Revolutions and Degrees

There is a way to describe how much A has been rotated to get to B.

Use the measure of the central angle, ∠AQB. Since m∠AQB = 50º, the

magnitude of the rotation is 50º. When rotations are measured in this

way, the rotation of 360º is called one revolution, or a full turn. A rotation

of 180º or 1 _ 2 revolution is called a half turn, and a rotation of 90º or 1 _

4

revolution is called a quarter turn.

1 AO 1C AO

D

1 AO

Full turn360º

Half turn180º

Quarter turn90º

1 revolution counterclockwise

1 _ 2 revolution

counterclockwise 1 _ 4 revolution

counterclockwise

Above, you could also rotate B 50º clockwise to get to A. The clockwise

direction is the negative direction in trigonometry because the four

quadrants are numbered in a counterclockwise order. So the rotation

that maps B onto A is said to be a –50º rotation, or to have magnitude –50º.

You can multiply both sides of the conversion equation

1 revolution = 360º

to fi nd how many degrees there are in any multiple of a revolution.

B

A

Q

50˚50˚

B

A

Q

50˚50˚

Mental Math

A pie chart is constructed to represent the following ice cream preferences: vanilla, 50%; chocolate, 30%; strawberry, 20%. Find the angle measure of each sector of the pie chart.

Lesson

Chapter 4

4-1Magnitudes of Rotations

and Measures of Arcs

Vocabulary

rotation

center of the rotation

rotation image

magnitude of a rotation

revolution, full turn

half turn

quarter turn

radian

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Magnitudes of Rotations and Measures of Arcs 223

Lesson 4-1

B

Q A

B

Q A

B

Q A

1 _ 8 revolution

counterclockwise 2 _ 3 revolution clockwise

1 1 _ 4 revolutions counterclockwise

= 1 _ 8 · 360º = 45º = 2 _

3 (–360º) = –240º = 5 _ 4 · 360º = 450º

In skateboarding, snowboarding, and many other sports, you may see rotations called 360s, 540s, 720s, and so on. In gymnastics and fi gure skating, rotations are measured in revolutions or turns. A 720 means 2 revolutions. In these physical rotations, adding 360º or 1 revolution to a turn creates a different movement. But in mathematical rotations, all that matters is where you begin and where you end. Physically turning 360º means you turned all the way around, but a mathematical rotation of 360º is the same as a mathematical rotation of –360º, and both are the same as if you did nothing at all!

For this reason, the same rotation can have many different magnitudes. For instance, a rotation of 1 _ 8 revolution counterclockwise also has magnitude 1 1 _ 8 revolutions counterclockwise or 7 _ 8 revolution clockwise. In degrees, that rotation has magnitude 45º, 405º, or –315º, respectively. Adding or subtracting 1 revolution (or 360º) to the magnitude of a rotation does not change the rotation.

Radian Measure

In a rotation, points that are farther from the center “move” or “turn” a greater distance along the arc of a circle than points that are closer to the center. For example, at the right, S moves farther to get to E than G moves to get to C. Measuring a rotation in degrees or revolutions does not tell anything about how far a point has moved. To solve this problem, a unit is needed that is related to the length of the arc. That unit is the radian. Radians have been in use for about 125 years and are important in the study of calculus and other advanced mathematics.

Radian measure is based on arc lengths in the unit circle. As you know, if a circle has radius r, its circumference is 2πr. We distinguish between 360º and 2πr because 360º is arc measure and 2πr is arc length.

QY1

Let’s roll Since 2004,

June 21 has been designated

“Go Skateboarding Day.”

Let’s roll Since 2004,

June 21 has been designated

“Go Skateboarding Day.”

� SE is longer than � GC .

E

S

CG

O

E

S

CG

O

� SE is longer than � GC .

E

S

CG

O

E

S

CG

O

QY1

A point moves halfway around a circle of radius 5.

a. What is the length of the arc traversed by the point?

b. How many degrees is the rotation?

QY1

A point moves halfway around a circle of radius 5.

a. What is the length of the arc traversed by the point?

b. How many degrees is the rotation?

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Chapter 4

A unit circle (radius = 1) has circumference C = 2π · 1 = 2π, which is approximately 6.28. Consider point A on a unit circle O, as pictured at the right. The magnitude in radians of the rotation with center O that maps A onto P is defi ned as the numerical length of � AP . In this drawing, � AP is 3 _ 8 of a circle, so the length of � AP is 3 _ 8 of the circumference, or 3 _ 8 · 2π = 3π

_ 4 ≈ 2.356. Notice that m∠AOP = 135º. So

135º corresponds to a rotation of 3π

_ 4 radians.

The radian measure of a full turn is 2π because if point A was physically turned through one complete revolution, it would travel over an arc of length 2π. Similarly, the radian measure of a half-turn is π and of a quarter turn is π

_ 2 .

As when rotations are measured in degrees, a clockwise rotation gives rise to a negative radian magnitude. Also, just as adding or subtracting multiples of 360º to the magnitude of a rotation gives rise to the same rotation, so does adding or subtracting multiples of 2π radians. The three circles below show that a rotation of magnitude 7π

_ 6 radians is the

same as a rotation of 7π

_ 6 – 2π = – 5π

_ 6 radians, and is the same as a

rotation of 2π + 7π

_ 6 = 19π

_ 6 radians.

Converting between Radians and Degrees

Some degree measures are easy to convert to radians. For example, 30º is 1 _ 12 of 360º. Use the conversion formula

360º = 1 revolution = 2π radians.

Divide by 12 to obtain

30º = 1 _ 12 revolution = π _ 6 radians.

The table below lists some of the equivalent measures that result from the

basic relationship 2π radians = 360º. Copy and fi ll in the rest of the table.

O 1

P

length of AP = = θ 3π

4�

radian measure = = θ

A

3π

4

O 1

P

length of AP = = θ 3π

4�

radian measure = = θ

A

3π

4

B

1

7π

6

O A

5π

6-

A

B

O

1

19π

6

B

O A1

B

1

7π

6

O A

5π

6-

A

B

O

1

19π

6

B

O A1

30˚π

6 radians30˚π

6 radians

ActivityActivity

Degrees 0º 30º ? ? 90º 120º ? 150º 180º 360º

Radians 0 π

_ 6

π

_ 4 π

_ 3

π

_ 2 ? 3π

_ 4 ? π ?

Revolutions ? ? 1 _ 8 ? ? ? 3 _

8 ? 1 _

2 1

Degrees 0º 30º ? ? 90º 120º ? 150º 180º 360º

Radians 0 π

_ 6

π

_ 4 π

_ 3

π

_ 2 ? 3π

_ 4 ? π ?

Revolutions ? ? 1 _ 8 ? ? ? 3 _

8 ? 1 _

2 1



Lesson 4-1

The values in the Activity are common radian values. You should know them without having to do any paper-and-pencil or calculator work. To convert any rotation measure from one unit to another, use the conversion formula below, since 2π radians = 360º. π radians = 180º

Dividing both sides of the conversion formula by the quantity on one side gives rise to two conversion factors, each equal to 1. 1 = 180º _

π radians or 1 = π radians _ 180º

Example 1a. Convert 1000º to radians exactly.

b. Convert 1000º to radians approximately.

Solution

a. Multiply by the appropriate conversion factor.

1000º = 1000º · π radians _ 180º = 1000 _ 180 π = 50 _ 9 π radiansb. Use a calculator to get a decimal approximation for 50π

_ 9 .

50π

_ 9 ≈ 17.453, so 1000º is about 17.5 radians.

Caution: Computer algebra systems and calculators work in both radians and degrees. Be sure to set the mode to the unit you want.

How large in degrees is an angle or rotation of magnitude 1 radian? Think: If point B on a unit circle is rotated 1 unit around the center to Q, as shown at the right, what is m∠BOQ? The circumference of a unit circle is 2π ≈ 6.28. So there are about 6.28 radians in one revolution, and one radian is slightly less than 1 _ 6 revolution. But 1 _ 6 revolution is equivalent to 60º. So one radian should be slightly less than 60º.

Example 2Convert 1 radian to degrees.

Solution Use the conversion factor 180º _

π radians .

So 1 radian = 1 radian • 180º _ π radians = 180º _ π ≈ 57.3º.

Why Are Radians Used?

You may have studied angles and rotations for years and never used radians. You may be wondering why radians are used and if they are ever needed. One advantage of radians over degrees is that certain formulas are simpler when written with radians.

Q

BO 1

length of BQ = 1

radian measure = 1

�

Q

BO 1

length of BQ = 1

radian measure = 1

�

1

1

1

1

1

1 ≈0.28

1

1

1

1

1

1

1 ≈0.28

1



Chapter 4

Example 3Find the length of an arc of a 50º central angle in a circle of radius 6 feet.

Solution The 50º arc is 50 _ 360 of the circumference of the circle. The circumference has length 2πr, or ? ft. So, the length of the arc is 50 _ 360 • ? , which simplifi es to ? ft exactly, or

? ft, to the nearest hundredth.

Example 3 is easily generalized. Notice how much simpler the formula

is if the central angle is measured in radians.

Circle Arc Length Formula

If s is the length of the arc of a central angle of θ radians in a circle of radius r, then s = rθ.

Proof The central angle is θ

_ 2π of a revolution. So the length s of the arc is θ

_ 2π

of

the circumference. The circumference of the circle is 2πr.

Thus, s = θ

_ 2π · 2πr = rθ.

QY2

Example 4A swing hangs from chains that are 8 ft long. How far does the seat of the

swing travel if it moves through an angle of 1.25 radians?

Solution 1 Since the angle is 1.25 radians, the length of the intercepted arc on the unit circle is 1.25. The arc length on the 8-foot circle is 8 times the length of the arc on the unit circle. The distance traveled is 8 • 1.25 = 10 feet.

Solution 2 Use the Circle Arc Length Formula.

s = rθ. The swing travels 8 · 1.25, or 10 feet.

Radians are so commonly used in mathematics that when no unit is

given in a problem that could be in degrees or radians, it is understood

that the measure is in radians.

Questions

COVERING THE IDEAS

1. Convert 9 _ 10

revolution to degrees.

2. Convert –805º to revolutions, rounding to the nearest tenth.

GUIDEDGUIDED

6 ft50˚

6 ft50˚

6 ft50˚

6 ft50˚

s

rθ

s

rθ

s

rθ

s

rθ

QY2

Find the length of a 5 _ 18 π radian arc in a circle of radius 6 feet.

QY2

Find the length of a 5 _ 18 π radian arc in a circle of radius 6 feet.

r = 1r = 8

length = 1.25 feet

length = 8 · 1.25 feet(not to scale)

r = 1r = 8

length = 1.25 feet

length = 8 · 1.25 feet(not to scale)



Lesson 4-1

3. At the right is a graph of a circle with radius 1.

a. Give the length of � AC . b. What is the smallest positive magnitude in radians of the

rotation with center O that maps A to C?

In 4 and 5, draw a circle with radius 1.

a. On this circle, heavily shade an arc with the given length.

b. Give the degree measure of the central angle of this arc.

4. 2π

_ 3 5. 1

In 6 and 7, draw a unit circle and an arc with the given radian measure.

6. 3π

_ 2 7. 2

8. Convert the measure to degrees. Round to the nearest thousandth. a. –0.2 radians b. –0.2π radians 9. If a skateboarder does a “540,” what is the magnitude of the rotation a. in revolutions? b. in radians?In 10 and 11, convert to radians exactly without using a calculator.

10. 225º 11. –80º

12. a. Draw an angle representing a rotation with measure 7π

_ 12 radians. b. Give two other radian measures of the same rotation.

13. Use the circle at the right. If m∠ABC = 5π

_ 6 radians and the radius

of the circle is 33, compute the length of � AC .

In 14 and 15, use a circle with diameter 8 cm.

14. Find the length of the arc intercepted by an angle of 3π

_ 4 .

15. Find the length of the arc intercepted by an angle of 63º.

16. Suppose the blades of a wind turbine are 16' long. What is the distance traveled by a point on its tip as the blade rotates 3 _ 5 of a revolution?

APPLYING THE MATHEMATICS

17. On the clock tower on the Houses of Parliament in London, England, the minute hand is about 14 feet long. How many feet does the end of the minute hand move in 5 minutes?

18. An angle whose measure is π _ 2 is about ? times as large as an

angle whose measure is π _ 2 º.

19. Musicians use a metronome to produce a steady beat as they practice. Many mechanical metronomes have a swinging arm with a weight to control the tempo. Suppose that a metronome arm is 4 inches long, moves through an angle of π

_ 3 , and beats at a rate of 160 beats per minute. How far does the tip of the arm travel in

a. 1 beat? b. 1 hour?

Ax

y

1C O Ax

y

1C O

A

B

C

33

5π

6

A

B

C

33

5π

6



Chapter 4

20. The planet Jupiter rotates on its axis at a rate of approximately

0.6334 radians per hour. What is the approximate length of the

Jovian day (the time it takes Jupiter to make a complete revolution)?

21. Suppose you can ride a bike with 22" wheels (in diameter) so that

the wheels rotate 150 revolutions per minute.

a. Find the number of inches traveled during each revolution.

b. How many inches are traveled each minute?

c. Use your answer from Part b to fi nd the speed, in miles per

hour, that you are traveling.

22. Recall that when greater precision is desired, a degree is split

into 60 minutes (abbreviated '). The diagram at the right shows

a cross section of Earth. G represents Grand Rapids, MI and

M represents Montgomery, AL. Assume that Grand Rapids is

directly north of Montgomery. If the radius of Earth is about

3960 miles, estimate the air distance from Grand Rapids

to Montgomery.

REVIEW

23. Use the Graph-Translation Theorem to fi nd the equation of the

image of y = x2 under T : (x, y) → (x + 3, y - 2). (Lesson 3-2)

24. You discovered a new element, Yournameium, which has a half-

life of 15 hours. Suppose the initial amount is A0. How much will

remain after each number of hours? (Lesson 2-4)

a. 45 b. 8 c. 42 d. t

25. The New York Times held a contest pitting professionals’ stock

choices with stocks chosen by throwing darts at a dartboard every

six months. The values in the table and box plots below represent

the points gained or lost by the stocks. (Lessons 1-7, 1-6, 1-4, 1-2)

a. Find the mean, median, and standard deviation for each data set.

b. Compare and contrast the distributions using the box plots.

26. Given the point P on the circle with center (0, 0) at the right, fi ll in

the coordinates of the remaining refl ection images in the various

quadrants. (Previous Course)

EXPLORATION

27. Derive a formula for the area of a sector in terms of the radius r of

the circle and the length x of its boundary arc x in radians.

Diagram not to scale

G

M

42˚58’

32˚22’Centerof Earth

Diagram not to scale

G

M

42˚58’

32˚22’Centerof Earth

0 10 20 30 40 50-10-20-30 60

Pros

Darts

Change in Stock Value

0 10 20 30 40 50-10-20-30 60

Pros

Darts

Change in Stock Value

P = (-.15, .99) Q = ( ? , ? )

S = ( ? , ? ) R = ( ? , ? )

P = (-.15, .99) Q = ( ? , ? )

S = ( ? , ? ) R = ( ? , ? )

QY ANSWERS

1. a. 5π b. 180º

2. 5π

_ 3 ≈ 5.24 ft

QY ANSWERS

1. a. 5π b. 180º

2. 5π

_ 3 ≈ 5.24 ft

Period # Pros Darts

1 51.2 11.72 25.2 1.13 –3.3 –3.14 7.7 –1.45 –21.0 7.76 –13.0 15.47 –2.5 3.68 –19.6 5.79 6.3 –5.7

10 –5.1 6.911 14.1 1.8


4-1 magnitudes of rotations and measures of arcs

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