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
hsnb_alg2_pe_0902.indd 469
Angles and Radian Measure
469
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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
hsnb_alg2_pe_0902.indd 471
Angles and Radian Measure
471
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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
2/5/15 1:49 PM
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
hsnb_alg2_pe_0902.indd 473
Angles and Radian Measure
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