A

B

C

D

E

F

O

60˚

3

1

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Name Class Date 10-1

Video Tutor

Finding the Ratio of Arc Length to Radius

The diagram shows three circles centered at point O. The arcs between the sides of the 60° central angle are called intercepted arcs.

AB is on the circle with radius 1 unit,

CD is on the circle with radius 2 units, and

EF is on the circle with radius 3 units.

Each intercepted arc has a different length, but because the arcs are intercepted by the same central angle, each length is the same fraction of the circumference of the circle containing the arc.

A Determine the fraction of each circle’s circumference that the length of each arc represents.

How many degrees are in a circle? •

What fraction of the total number of degrees in a circle is 60°? •

So, what fraction of the circumference of •each circle is the length of each intercepted arc?

B Complete the table below. To find the length of the intercepted arc, use the fraction that you found in part A. Give all answers in terms of π.

REFLECT

1a. What do you notice about the ratios in the fourth column of the table?

E X P L O R E1

Right-Angle TrigonometryConnection: Radian MeasureEssential question: What is radian measure, and how are radians related to degrees?

Radius, r Circumference, C(C = 2πr)

Length of Intercepted Arc, s

Ratio of Arc Length

to Radius, s _ r

1

2

3

CC.9–12.G.C.5

Chapter 10 557 Lesson 1

X

Y

O

rd˚

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1b. When the ratio of the values of one variable y to the corresponding values

of another variable x is a constant k, y is said to be proportional to x, and the

constant k is called the constant of proportionality. Because y

_ x = k, you can

solve for y to get y = kx. In the case of arcs intercepted by a 60° central angle,

is arc length s proportional to radius r? If so, what is the constant of proportionality,

and what equation gives s in terms of r?

1c. Suppose the central angle is 90° instead of 60°. Would arc length s still be proportional to radius r? If so, would the constant of proportionality still be the same? Explain.

Radian Measure In the Explore and its Reflect questions, you should have reached the following conclusions:

1. When a central angle intercepts arcs on circles that have a common center, the ratio of each arc length s to radius r is constant.

2. When the degree measure of the central angle changes, the constant also changes.

These facts allow you to create an alternative way of measuring angles.

Instead of degree measure, you can use radian measure, defined as follows:

If a central angle in a circle of radius r intercepts an arc of length s, then the

angle’s radian measure is θ = s _ r .

Relating Radians to Degrees

Let the degree measure of a central angle in a circle with radius r be d°, as shown. You can derive formulas that relate the angle’s degree measure and its radian measure.

A Find the length s of the intercepted arc XY using the verbal model below. Give the length in terms of π, and simplify.

Circumference of circleLength of arc = · Degrees in arc

Degrees in circle

E X P L O R E2CC.9–12.A.CED.4

Chapter 10 558 Lesson 1

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B Use the result from part A to write the angle’s radian measure θ in terms of d to find a formula for converting degrees to radians.

θ = s _ r = _______ = · d°

C Solve the equation from part B for d° to find a formula for converting radians to degrees.

d° = · θ

REFLECT

2a. What radian measure is equivalent to 360°? Why does this make sense?

Radian measures are usually written in terms of π, using fractions,

such as 2π ___ 3 , rather than mixed numbers.

Converting Between Radians and DegreesE X AM P L E3

A Use the formula θ = π _____ 180° · d° to

convert each degree measure to

radian measure. Simplify the result.

B Use the formula d° = 180° _____ π · θ to

convert each radian measure to

degree measure. Simplify the result.

Degree measure Radian measure

15°

45°

90°

120°

135°

165°

Radian measure Degree measure

π __ 6

π __ 3

5π ___ 12

7π ___ 12

5π ___ 6

π

CC.9–12.N.Q.1

Chapter 10 559 Lesson 1

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REFLECT

3a. Which is greater, 1° or 1 radian? Explain.

3b. A radian is sometimes called a “dimensionless” quantity. Use unit analysis and the definition of radian to explain why this description makes sense.

P R A c t i c E

1. Convert each degree measure to radian measure. Simplify the result.

2. Convert each radian measure to degree measure. Simplify the result.

3. When a central angle of a circle intercepts an arc whose length equals the radius of the circle, what is the angle’s radian measure? Explain.

4. A unit circle has a radius of 1. What is the relationship between the radian measure of a central angle in a unit circle and the length of the arc that it intercepts? Explain.

5. A pizza is cut into 8 equal slices.

a. What is the radian measure of the angle in each slice?

b. If the length along the outer edge of the crust of one slice is about

7 inches, what is the diameter of the pizza to the nearest inch?

(Use the formula θ = s _ r , but note that it gives you the radius of the pizza.)

Degree measure 18° 24° 72° 84° 108° 126° 132°

Radian measure

Radian measure π ___

15 π __

5 4π ___

15 3π ___

10 8π ___

15 13π ____

15 9π ___

10

Degree measure

Chapter 10 560 Lesson 1

295

Name ________________________________________ Date __________________ Class __________________

Original content Copyright © by Holt McDougal. Additions and changes to the original content are the responsibility of the instructor.

1 Holt McDougal Algebra 1

Practice The Unit Circle

Convert each measure from degrees to radians or from radians to degrees.

1. π512

2. 215° 3. π−

2918

________________________ _________________________ ________________________

4. −180° 5. π53

6. π−

76

________________________ _________________________ ________________________

7. 400° 8. π310

9. 35°

________________________ _________________________ ________________________

Use the unit circle to find the exact value of each trigonometric function.

10. π2cos3

11. π5tan4

12. π5tan6

________________________ _________________________ ________________________

13. sin 315° 14. cos 225° 15. tan 60°

________________________ _________________________ ________________________

Use a reference angle to find the exact value of the sine, cosine, and tangent of each angle. 16. 150° 17. −225° 18. −300°

________________________ _________________________ ________________________

19. π116

20. π−

23

21. π54

________________________ _________________________ ________________________

Solve. 22. San Antonio, Texas, is located about 30° north of the equator.

If Earth’s radius is about 3959 miles, approximately how many miles is San Antonio from the equator?

___________________

Chapter

2

A2_MGAESE902135_M00L00PP.indd 295 6/16/12 12:01:40 AM

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10-1Name Class Date

Additional Practice

Chapter 10 561 Lesson 1

296

Name ________________________________________ Date __________________ Class __________________

Original content Copyright © by Holt McDougal. Additions and changes to the original content are the responsibility of the instructor.

1 Holt McDougal Algebra 1

Practice Right-Angle Trigonometry

Find the value of the sine, cosine, and tangent functions for θ. 1. 2. 3.

________________________ ________________________ ____________________

Use a trigonometric function to find the value of x. 4. 5. 6.

________________________ ________________________ ____________________

Find the values of the six trigonometric functions for θ. 7. 8. 9.

_______________________________ _______________________________ ________________________

_______________________________ _______________________________ ________________________

Solve. 10. A water slide is 26 feet high. The angle between the slide

and the water is 33.5°. What is the length of the slide? ___________________ 11. A surveyor stands 150 feet from the base of a viaduct and

measures the angle of elevation to be 46.2°. His eye level is 6 feet above the ground. What is the height of the viaduct to the nearest foot? ___________________

12. The pilot of a helicopter measures the angle of depression to a landing spot to be 18.8°. If the pilot’s altitude is 1640 meters, what is the horizontal distance to the landing spot to the nearest meter? ___________________

Problem Solving The Unit Circle

Gabe is spending two weeks on an archaeological dig. He finds a fragment of a circular plate that his leader thinks may be valuable. The arc length of the fragment

is about 16

the circumference of the original complete

plate and measures 1.65 inches. 1. A similar plate found earlier has a diameter of 3.14 inches.

Could Gabe’s fragment match this plate? a. Write an expression for the radius, r, of the earlier plate. ____________________ b. What is the measure, in radians, of a central angle, θ,

that intercepts an arc that is 16

the length of the

circumference of a circle? ____________________ c. Write an expression for the arc length, S, intercepted

by this central angle. ____________________

d. How long would the arc length of a fragment be if it were 16

the circumference of the plate? ____________________ e. Could Gabe’s plate be a matching plate? Explain.

_________________________________________________________________________________________

_________________________________________________________________________________________

2. Toby finds another fragment of arc length 2.48 inches. What fraction of the outer edge of Gabe’s plate would it be if this fragment were part of Gabe’s plate? ____________________

The diameter of a merry-go-round at the playground is 12 feet. Elijah stands on the edge and his sister pushes him around. Choose the letter for the best answer 3. How far does Elijah travel if he moves

through an angle of π5 radians?4

A 12.0 ft C 23.6 ft

B 15.1 ft D 47.1 ft

4. Through what angle does Elijah move if he travels a distance of 80 feet around the circumference?

F π40 radians3

H 40 radians3

G 80 radians3

J 20 radians3

Virgil sets his boat on a 1000-yard course keeping a constant distance from a rocky outcrop. Choose the letter for the best answer. 5. If Virgil keeps a distance of 200 yards,

through what angle does he travel? A 5π radians C 10 radians B 5 radians D 10π radians

6. If Virgil keeps a distance of 500 yards, what fraction of the circumference of a circle does he cover?

F π1 H

π3

4

G π3

J π34

Chapter

2

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Problem Solving

Chapter 10 562 Lesson 1