Angles and Radian Measure

9.2

Angles and Radian Measure

Essential Question

How can you find the measure of an angle

in radians?

Let the vertex of an angle be at the origin, with one side of the angle on the positive

x-axis. The radian measure of the angle is a measure of the intercepted arc length on

a circle of radius 1. To convert between degree and radian measure, use the fact that

¦Ð radians

180¡ã

¡ª = 1.

Writing Radian Measures of Angles

Work with a partner. Write the radian measure of each angle with the given

degree measure. Explain your reasoning.

a.

b.

y

y

90¡ã

radian

measure

degree

measure

60¡ã

120¡ã

135¡ã

45¡ã

¦Ð

30¡ã

150¡ã

0¡ã

360¡ã x

180¡ã

x

210¡ã

225¡ã

315¡ã

330¡ã

240¡ã

270¡ã

300¡ã

Writing Degree Measures of Angles

Work with a partner. Write the degree measure of each angle with the given

radian measure. Explain your reasoning.

y

degree

measure

radian

measure

7¦Ð

9

5¦Ð

9

4¦Ð

9

2¦Ð

9

x

11¦Ð

9

13¦Ð 14¦Ð

9

9

REASONING

ABSTRACTLY

To be proficient in math,

you need to make sense

of quantities and their

relationships in problem

situations.

16¦Ð

9

Communicate Your Answer

3. How can you find the measure of an angle

in radians?

y

4. The figure shows an angle whose measure is

30 radians. What is the measure of the angle in

degrees? How many times greater is 30 radians

than 30 degrees? Justify your answers.

x

30 radians

Section 9.2

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Angles and Radian Measure

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9.2 Lesson

What You Will Learn

Draw angles in standard position.

Find coterminal angles.

Core Vocabul

Vocabulary

larry

initial side, p. 470

terminal side, p. 470

standard position, p. 470

coterminal, p. 471

radian, p. 471

sector, p. 472

central angle, p. 472

Previous

radius of a circle

circumference of a circle

Use radian measure.

Drawing Angles in Standard Position

In this lesson, you will expand your study of angles to include angles with measures

that can be any real numbers.

Core Concept

Angles in Standard Position

90¡ã y

terminal

side

In a coordinate plane, an angle can be formed

by fixing one ray, called the initial side, and

rotating the other ray, called the terminal side,

about the vertex.

0¡ã

An angle is in standard position when its vertex

is at the origin and its initial side lies on the

positive x-axis.

x

180¡ã vertex initial

360¡ã

side

270¡ã

The measure of an angle is positive when the rotation of its terminal side is

counterclockwise and negative when the rotation is clockwise. The terminal side

of an angle can rotate more than 360¡ã.

Drawing Angles in Standard Position

Draw an angle with the given measure in standard position.

a. 240¡ã

b. 500¡ã

c. ?50¡ã

b. Because 500¡ã is 140¡ã

more than 360¡ã, the

terminal side makes

one complete rotation

360¡ã counterclockwise

plus 140¡ã more.

c. Because ?50¡ã is

negative, the terminal

side is 50¡ã clockwise

from the positive

x-axis.

SOLUTION

a. Because 240¡ã is 60¡ã

more than 180¡ã, the

terminal side is 60¡ã

counterclockwise past

the negative x-axis.

y

y

240¡ã

y

140¡ã

x

500¡ã

60¡ã

Monitoring Progress

x

x

?50¡ã

Help in English and Spanish at

Draw an angle with the given measure in standard position.

1. 65¡ã

470

Chapter 9

hsnb_alg2_pe_0902.indd 470

2. 300¡ã

3. ?120¡ã

4. ?450¡ã

Trigonometric Ratios and Functions

2/5/15 1:48 PM

Finding Coterminal Angles

STUDY TIP

If two angles differ by a

multiple of 360¡ã, then the

angles are coterminal.

In Example 1(b), the angles 500¡ã and 140¡ã are coterminal because their terminal

sides coincide. An angle coterminal with a given angle can be found by adding or

subtracting multiples of 360¡ã.

Finding Coterminal Angles

Find one positive angle and one negative angle that are coterminal with (a) ?45¡ã

and (b) 395¡ã.

SOLUTION

There are many such angles, depending on what multiple of 360¡ã is added or

subtracted.

a. ?45¡ã + 360¡ã = 315¡ã

?45¡ã ? 360¡ã = ?405¡ã

b. 395¡ã ? 360¡ã = 35¡ã

395¡ã ? 2(360¡ã) = ?325¡ã

y

y

?325¡ã

35¡ã

315¡ã

?45¡ã

x

395¡ã

?405¡ã

Monitoring Progress

x

Help in English and Spanish at

Find one positive angle and one negative angle that are coterminal with the

given angle.

5. 80¡ã

STUDY TIP

Notice that 1 radian

is approximately equal

to 57.3¡ã.

180¡ã = ¦Ð radians

180¡ã

¦Ð

¡ª = 1 radian

57.3¡ã ¡Ö 1 radian

6. 230¡ã

8. ?135¡ã

7. 740¡ã

Using Radian Measure

Angles can also be measured in radians. To define

a radian, consider a circle with radius r centered at

the origin, as shown. One radian is the measure of

an angle in standard position whose terminal side

intercepts an arc of length r.

y

r

1 radian

Because the circumference of a circle is 2¦Ðr, there

are 2¦Ð radians in a full circle. So, degree measure

and radian measure are related by the equation

360¡ã = 2¦Ð radians, or 180¡ã = ¦Ð radians.

r

x

Core Concept

Converting Between Degrees and Radians

Degrees to radians

Radians to degrees

Multiply degree measure by

Multiply radian measure by

¦Ð radians

180¡ã

180¡ã

¦Ð radians

¡ª.

¡ª.

Section 9.2

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Convert Between Degrees and Radians

Convert the degree measure to radians or the radian measure to degrees.

¦Ð

b. ?¡ª

12

a. 120¡ã

READING

The unit ¡°radians¡± is often

omitted. For instance, the

¦Ð

measure ?¡ª radians may

12

¦Ð

be written simply as ?¡ª.

12

SOLUTION

¦Ð radians

a. 120¡ã = 120 degrees ¡ª

180 degrees

(

¦Ð

¦Ð

180¡ã

b. ?¡ª = ?¡ª radians ¡ª

12

12

¦Ð radians

)

)(

(

2¦Ð

=¡ª

3

)

= ?15¡ã

Concept Summary

Degree and Radian Measures of Special Angles

The diagram shows equivalent degree and

radian measures for special angles from

0¡ã to 360¡ã (0 radians to 2¦Ð radians).

You may find it helpful to memorize the

equivalent degree and radian measures of

special angles in the first quadrant and for

¦Ð

90¡ã = ¡ª radians. All other special angles

2

shown are multiples of these angles.

Monitoring Progress

5¦Ð

6

¦Ð

7¦Ð

6

y

¦Ð

2

radian

¦Ð measure

3 ¦Ð

4

90¡ã

¦Ð

120¡ã

60¡ã

6

135¡ã

45¡ã

30¡ã

150¡ã degree

2¦Ð

3¦Ð 3

4

180¡ã

measure

0¡ã

360¡ã

0 x

2¦Ð

210¡ã

330¡ã

225¡ã

315¡ã

11¦Ð

240¡ã

300¡ã

6

270¡ã

5¦Ð

7¦Ð

4 4¦Ð

4

5¦Ð

3¦Ð

3

3

2

Help in English and Spanish at

Convert the degree measure to radians or the radian measure to degrees.

5¦Ð

9. 135¡ã

10. ?40¡ã

11. ¡ª

12. ?6.28

4

A sector is a region of a circle that is bounded by two radii and an arc of the circle.

The central angle ¦È of a sector is the angle formed by the two radii. There are simple

formulas for the arc length and area of a sector when the central angle is measured

in radians.

Core Concept

Arc Length and Area of a Sector

The arc length s and area A of a sector with

radius r and central angle ¦È (measured in

radians) are as follows.

sector

r

Arc length: s = r¦È

Area: A = ¡ª12 r 2¦È

472

Chapter 9

hsnb_alg2_pe_0902.indd 472

central

angle ¦È

arc

length

s

Trigonometric Ratios and Functions

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Modeling with Mathematics

A softball field forms a sector with the dimensions shown. Find the length of the

outfield fence and the area of the field.

SOLUTION

1. Understand the Problem You are given the

dimensions of a softball field. You are asked

to find the length of the outfield fence and the

area of the field.

outfield

fence

200 ft

2. Make a Plan Find the measure of the central

angle in radians. Then use the arc length and

area of a sector formulas.

90¡ã

3. Solve the Problem

200 ft

Step 1 Convert the measure of the central angle to radians.

¦Ð radians

90¡ã = 90 degrees ¡ª

180 degrees

(

COMMON ERROR

You must write the

measure of an angle

in radians when using

these formulas for the

arc length and area of

a sector.

)

¦Ð

= ¡ª radians

2

Step 2 Find the arc length and the area of the sector.

1

Area: A = ¡ªr 2¦È

2

Arc length: s = r ¦È

¦Ð

= 200 ¡ª

2

¦Ð

1

= ¡ª (200)2 ¡ª

2

2

= 100¦Ð

= 10,000¦Ð

¡Ö 314

¡Ö 31,416

( )

ANOTHER WAY

Because the central

angle is 90¡ã, the sector

represents ¡ª14 of a circle

with a radius of 200 feet.

So,

s = ¡ª14

The length of the outfield fence is about 314 feet. The area of the field

is about 31,416 square feet.

4. Look Back To check the area of the field,

consider the square formed using the two

200-foot sides.

? 2¦Ðr = ¡ª ? 2¦Ð (200)

1

4

= 100¦Ð

By drawing the diagonal, you can see that

the area of the field is less than the area of the

square but greater than one-half of the area of

the square.

and

A = ¡ª41

? ¦Ðr

2

= ¡ª14

? ¦Ð (200)

= 10,000¦Ð.

( )

2

1

¡ª2

? (area of square)

1

2

200 ft

area of square

?

90¡ã

200 ft

?

¡ª (200)2 < 31,416 < 2002

20,000 < 31,416 < 40,000

Monitoring Progress

?

Help in English and Spanish at

13. WHAT IF? In Example 4, the outfield fence is 220 feet from home plate. Estimate

the length of the outfield fence and the area of the field.

Section 9.2

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