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3
Radian Measure
and the Unit Circle
Approach
H
Courtesy Ford Motor Company
ow does an odometer or
speedometer on an automobile
work? The transmission counts how
many times the tires rotate (how many full revolutions take place) per second. A computer then calculates
how far the car has traveled in that second by multiplying the number of revolutions by the tire
circumference. Distance is given by the odometer, and the speedometer takes the distance per second
and converts to miles per hour (or km/h). Realize that the computer chip is programmed to the tire
designed for the vehicle. If a person were to change the tire size (smaller or larger than the original
specifications), then the odometer and speedometer would need to be adjusted.
Suppose you bought a Ford Expedition Eddie Bauer Edition, which comes standard with 17-inch rims
(corresponding to a tire with 25.7-inch diameter), and you decide to later upgrade these tires for 19-inch
rims (corresponding to a tire with 28.2-inch diameter). If the onboard computer is not adjusted, is the
actual speed faster or slower than the speedometer indicator?*
In this case, the speedometer would read 9.6% too low. For example, if your speedometer read 60 mph,
your actual speed would be 65.8 mph. In this chapter, you will see how the angular speed (rotations of
tires per second), radius (of the tires), and linear speed (speed of the automobile) are related.
*Section 3.3, Example 3 and Exercises 53 and 54.
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I N T H I S C H A P T E R , you will learn a second way to measure angles using radians. You will convert between
degrees and radians. You will calculate arc lengths, areas of circular sectors, and angular and linear speeds. Finally, the third
definition of trigonometric functions using the unit circle approach will be given. You will work with the trigonometric
functions in the context of a unit circle.
R A D IAN M EAS U R E AN D TH E
U N IT C I R C LE AP P R OA C H
3.1
3.2
3.3
3.4
Radian Measure
Arc Length
and Area of a
Circular Sector
Linear and
Angular Speeds
Definition 3 of
Trigonometric
Functions: Unit
Circle Approach
• Linear Speed
• Angular Speed
• Relationship
Between Linear and
Angular Speeds
• Trigonometric
Functions and the
Unit Circle (Circular
Functions)
• Properties of Circular
Functions
• The Radian Measure
of an Angle
• Converting Between
Degrees and Radians
LEARNING
■
■
■
■
• Arc Length
• Area of a Circular
Sector
OBJECTIVES
Convert between degrees and radians.
Calculate arc length and the area of a circular sector.
Relate angular and linear speeds.
Draw the unit circle and label the sine and cosine values for special angles
(in both degrees and radians).
129
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SECTION
3.1
RADIAN MEASURE
C O N C E P TUAL O BJ E CTIVE S
S K I LLS O BJ E CTIVE S
■
■
■
Calculate the radian measure of an angle.
Convert between degrees and radians.
Calculate trigonometric function values for angles
given in radians.
■
■
Understand that degrees and radians are both
measures of angles.
Realize that radian measure allows us to write
trigonometric functions as functions of real numbers.
The Radian Measure of an Angle
r
␪ = 1 radian
r
r
In geometry and most everyday applications, angles are measured in degrees. However,
radian measure is another way to measure angles. Using radian measure allows us to write
trigonometric functions as functions not only of angles but also of real numbers in general.
Recall that in Section 1.1 we defined one full rotation as an angle having measure 360°.
Now we think of the angle in the context of a circle. A central angle is an angle that has
its vertex at the center of a circle.
When the intercepted arc’s length is equal to the radius, the measure of the central angle
is 1 radian. From geometry, we know that the ratio of the measures of two angles is equal
to the ratio of the lengths of the arcs subtended by those angles (along the same circle).
s1
r
s1
u1
s2
u2
␪1
r
␪2
r
r
s2
If u1 1 radian, then the length of the subtended arc is equal to the radius, s1 r. This
leads to a general definition of radian measure.
CAUTION
To correctly calculate radians from
s
the formula u r , the radius and
arc length must be expressed in the
same units.
DEFINITION
Radian Measure
If a central angle u in a circle with radius r intercepts
an arc on the circle of length s, then the measure of u,
in radians, is given by
u (in radians) s
r
s
r
␪
r
Note: The formula is valid only if s (arc length) and r
(radius) are expressed in the same units.
Note that both s and r are measured in units of length. When both are given in the same
units, the units cancel, giving the number of radians as a dimensionless (unitless) real
number.
130
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131
One full rotation corresponds to an arc length equal to the circumference 2pr of the
circle with radius r. We see then that one full rotation is equal to 2p radians.
ufull rotation EXAMPLE 1
2pr
2p
r
Finding the Radian Measure of an Angle
What is the measure (in radians) of a central angle u that intercepts an arc of length
4 feet on a circle with radius 10 feet?
Study Tip
Notice in Example 1 that the units,
feet, cancel, therefore leaving u as a
unitless real number, 0.4.
Solution:
s
r
4 ft
0.4 rad
u
10 ft
Write the formula relating radian measure
to arc length and radius.
u
Let s 4 feet and r 10 feet.
■ YOUR TURN
EXAMPLE 2
What is the measure (in radians) of a central angle ␪ that intercepts
an arc of length 3 inches on a circle with radius 50 inches?
Answer: 0.06 rad
■
Classroom Example 3.1.1
Find the measure, in radians,
of the central angle u that
intercepts an arc of length
3 yards on a circle of radius
6 yards.
Finding the Radian Measure of an Angle
What is the measure (in radians) of a central angle u that intercepts an arc of length
6 centimeters on a circle with radius 2 meters?
C O M M O N M I S TA K E
Answer: 12 rad
A common mistake is forgetting to first put the radius and arc length in the same
units.
★ CORRECT
INCORRECT
Write the formula relating radian
measure to arc length and radius.
u (in radians) s
r
Substitute s 6 centimeters and
r 2 meters into the radian expression.
u
6 cm
2m
Convert the radius (2) meters to
centimeters: 2 meters 200 centimeters
6 cm
u
200 cm
CAUTION
Units for arc length and radius must
be the same in order to use
s
u
r
Substitute s 6 centimeters and
r 2 meters into the radian expression.
Classroom Example 3.1.2
Find the measure, in radians,
of the central angle u that
intercepts an arc of length
3 yards on a circle of radius
6 feet.
6 cm
2m
3
u
ERROR (not converting both numerator
and denominator to the same units)
Answer: 32 rad
The units, centimeters, cancel and the
result is a unitless real number.
u 0.03 rad
■ YOUR TURN
What is the measure (in radians) of a central angle u that intercepts
an arc of length 12 millimeters on a circle with radius 4 centimeters?
■
Answer: 0.3 rad
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C H A P T E R 3 Radian Measure and the Unit Circle Approach
Because radians are unitless, the word radians (or rad) is often omitted. If an angle
measure is given simply as a real number, then radians are implied.
W OR DS
M ATH
The measure of u is 4 degrees.
The measure of u is 4 radians.
u 4°
u4
Converting Between Degrees and Radians
To convert between degrees and radians, we must first look for a relationship between
them. We start by considering one full rotation around the circle. An angle corresponding
to one full rotation is said to have measure 360°, and we saw previously that one full
rotation corresponds to u 2p rad.
W OR DS
M ATH
Write the angle measure (in degrees) that
corresponds to one full rotation.
Write the angle measure (in radians) that
corresponds to one full rotation.
Arc length is the circumference of the circle.
s
Substitute s 2pr into u (in radians) .
r
u 360°
s 2pr
u
2pr
2p rad
r
Equate the measures corresponding to one
full rotation.
Divide by 2.
360° 2p rad
180° p rad
Divide by 180° or ␲.
1
p
180°
or 1 p
180°
This leads us to formulas aunit conversations, like
1 hr
b that convert between degrees
60 min
and radians. Let ud represent an angle measure given in degrees and ur represent the
corresponding angle measure given in radians.
C O NVE R TI N G
D E G R E E S TO R AD IAN S
To convert degrees to radians, multiply the degree measure by
ur ud a
C O NVE RTI N G
p
b
180°
p
.
180°
R AD IAN S TO D E G R E E S
To convert radians to degrees, multiply the radian measure by
ud ur a
180°
b
p
180°
.
p
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133
Before we begin converting between degrees and radians, let’s first get a feel for
radians. How many degrees is 1 radian?
W OR DS
M ATH
Multiply 1 radian by
180°
.
p
Approximate p by 3.14.
Use a calculator to evaluate and
round to the nearest degree.
1a
180°
b
p
1a
180°
b
3.14
⬇ 57°
1 rad ⬇ 57°
A radian is much larger than a degree (almost 57 times larger). Let’s compare two
30 rad
angles, one measuring 30 radians and the other measuring 30°. Note that
⬇ 4.77
2p rad/rev
1
revolutions, whereas 30° 12
revolution.
y
y
30º
x
x
30 rad
Converting Degrees to Radians
EXAMPLE 3
Classroom Example 3.1.3
Convert 135° to radians.
3p
Answer:
4
Convert 45° to radians.
Solution:
Multiply 45° by
p
.
180°
Simplify.
(45°)a
p
45°p
b
180°
180°
p
rad
4
p
is the exact value. A calculator can be used to approximate this expression. Scientific
4
p
and graphing calculators have a p button. The decimal approximation of rounded to
4
three decimal places is 0.785.
p
Exact Value:
4
Note:
Approximate Value:
■ YOUR TURN
0.785
Convert 60° to radians.
■
Answer:
p
3
or
1.047
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C H A P T E R 3 Radian Measure and the Unit Circle Approach
Converting Degrees to Radians
EXAMPLE 4
Convert 472° to radians.
Solution:
Classroom Example 3.1.4
a.* Convert 180(2n 1)° to
radians, where n is an
integer.
b. Convert 2000° to radians.
Multiply 472° by
Answer:
100p
a. (2n 1)p b.
9
■
Answer:
23
p or
9
8.029
118
p
45
Use a calculator to approximate.
⬇ 8.238 rad
Convert 460° to radians.
Converting Radians to Degrees
EXAMPLE 5
Convert
2p
to degrees.
3
Solution:
Multiply
Answer: 330°
2p 180°
ⴢ
p
3
2p
180°
by
.
p
3
120°
Simplify.
■
p
b
180°
Simplify (factor out the common 4).
■ YOUR TURN
Classroom Example 3.1.5
11p
Convert
to degrees.
6
472° a
p
.
180°
Answer: 270°
■ YOUR TURN
EXAMPLE 6
Convert
3p
to degrees.
2
Converting Radians to Degrees
Convert 10 radians to degrees.
Solution:
Multiply 10 radians by
180°
.
p
10 ⴢ
Simplify.
2␲
120º =
3
3␲
135º =
4
5␲
150º =
6
180º = ␲
7␲
6
90º =
␲
2
60º =
␲
3
45º =
␲
4
␲
30º =
6
360º = 2␲
11␲
6
7␲
5␲
315º =
225º =
4
4
5␲
4␲
300º =
240º =
3␲
3
3 270º =
2
210º =
330º =
180°
p
1800°
⬇ 573°
p
Since p 180°, we know the following special angles:
p
90°
2
p
60°
3
p
45°
4
p
30°
6
and we can now draw the unit circle with the special angles in both degrees and
radians.
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135
The following table lists sine and cosine values for special angles in both degrees and
radians. Tangent, secant, cosecant, and cotangent values can all be found from sine
and cosine values using quotient and reciprocal identities. The table only lists special
angles in quadrant I and quadrantal angles (0° u 360° or 0 u 2p). Values in
quadrants II, III, and IV can be found using reference angles and knowledge of the
algebraic sign ( or ) of the sine and cosine functions in each quadrant.
VALUE OF
TRIGONOMETRIC FUNCTION
ANGLE, ␪
RADIANS
DEGREES
SIN
␪
COS
0
0°
0
1
p
6
30°
1
2
13
2
p
4
45°
12
2
12
2
p
3
60°
13
2
1
2
p
2
90°
1
0
p
180°
0
1
3p
2
270°
1
0
2p
360°
0
1
EXAMPLE 7
␪
Evaluating Trigonometric Functions
for Angles in Radian Measure
p
Evaluate sin a b exactly.
3
Solution:
Recognize that
p
p
60° or convert to degrees.
3
3
Set a TI/scientific calculator
to radian mode by typing
MODE
䉲
13
2
sin 60° p
Equate sin 60° and sin a b.
3
p
13
sin a b 3
2
p
Evaluate cos a b exactly.
3
If the angle of the trigonometric function to be evaluated has its terminal side in
quadrants II, III, or IV, then we use reference angles and knowledge of the algebraic sign
( or ) in that quadrant. We know how to find reference angles in degrees. Now we will
find reference angles in radians.
䉲
ENTER
. (radian)
Use a TI/scientific calculator to
p
13
b.
check the value of sin a b and a
3
2
Press 2nd ^ p.
p 180°
ⴢ
60°
p
3
Find the value of sin 60°.
■ YOUR TURN
Technology Tip
■
Answer: 12
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C H A P T E R 3 Radian Measure and the Unit Circle Approach
TERMINAL SIDE LIES
IN
...
QI
QII
QIII
QIV
EXAMPLE 8
DEGREES
RADIANS
au
a 180° u
a u 180°
a 360° u
au
apu
aup
a 2p u
Finding Reference Angles in Radians
Find the reference angle for each angle given.
Classroom Example 3.1.8
Find the reference angle for
each angle given.
2p
5p
a.
b.
3
4
Answer:
p
p
a.
b.
3
4
a.
3p
4
b.
11p
6
Solution (a):
y
The terminal side of u lies in quadrant II.
Recall that p radians is 12 of a full revolution,
so 43p is 34 of a half of revolution.
3␲
4
␣
p
The reference angle is made with the
terminal side and the negative x-axis.
x
4p
3p
p
3p
4
4
4
4
Solution (b):
y
The terminal side of u lies in quadrant IV.
Recall that 2p is a complete revolution.
11␲
6
11p
12p
b.
Note that
is not quite 2p aor
6
6
The reference angle is made with the
terminal side and the positive x-axis.
■
Answer:
p
3
■ YOUR TURN
Find the reference angle for
x
␣
2p 5p
.
3
11p
12p
11p
p
6
6
6
6
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3.1 Radian Measure
EXAMPLE 9
Evaluate cos a
Evaluating Trigonometric Functions for Angles
in Radian Measure Using Reference Angles
Technology Tip
5p
b exactly.
4
Use the TI/scientific calculator
5p
to check the value for cos a b and
4
12
.
compare with 2
Solution:
5p
The terminal side of angle
lies in
4
5p
p
p .
quadrant III since
4
4
p
The reference angle is 45°.
4
y
5␲
4
Determine the algebraic sign for the cosine
function in quadrant III.
p
12
cos a b cos 45° 4
2
Answer: cos a
Confirm with a calculator.
0.707 ⬇ 0.707
7p
b exactly.
4
Classroom Example 3.1.9
5p
Evaluate cos a b exactly.
6
Negative ()
Combine the algebraic sign of the cosine
function in quadrant III with the value of the
cosine function of the reference angle.
Evaluate sin a
x
␲
= 45º
4
Find the cosine value for the reference angle.
■ YOUR TURN
137
12
5p
b
4
2
■
Answer: 13
2
12
2
SECTION
3.1
S U M MARY
In this section, a second measure of angles was introduced,
which allows us to write trigonometric functions as functions of
real numbers. A central angle of a circle has radian measure
equal to the ratio of the arc length intercepted by the angle to
s
the radius of the circle, u .
r
Radians and degrees are related by the relation that p 180°.
■
■
To convert from radians to degrees, multiply the
180°
radian measure by
.
p
To convert from degrees to radians, multiply the
p
degree measure by
.
180°
One radian is approximately equal to 57°. Careful attention
must be paid to what mode (degrees or radians) calculators are
set when evaluating trigonometric functions. To evaluate a
trigonometric function for nonacute angles in radians, we use
reference angles (in radians) and knowledge of the algebraic sign
of the trigonometric function.
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C H A P T E R 3 Radian Measure and the Unit Circle Approach
SECTION
3.1
■
EXERCISES
SKILLS
In Exercises 1–10, find the measure (in radians) of a central angle ␪ that intercepts an arc on a circle of radius r with
indicated arc length s.
1. r 10 cm, s 2 cm
2. r 20 cm, s 2 cm
3. r 22 in., s 4 in.
4. r 6 in., s 1 in.
5. r 100 cm, s 20 mm
6. r 1 m, s 2 cm
7. r in., s 8. r 34 cm, s 1
4
1
32
in.
9. r 2.5 cm, s 5 mm
3
14
cm
10. r 1.6 cm, s 0.2 mm
In Exercises 11–24, convert each angle measure from degrees to radians. Leave answers in terms of ␲.
11. 30°
12. 60°
13. 45°
14. 90°
15. 315°
16. 270°
17. 75°
18. 100°
19. 170°
20. 340°
21. 780°
22. 540°
23. 210°
24. 320°
In Exercises 25–38, convert each angle measure from radians to degrees.
25.
p
6
26.
p
4
32.
7p
3
33. 9p
3p
4
27.
34. 6p
28.
7p
6
29.
3p
8
30.
11p
9
35.
19p
20
36.
13p
36
37. 31.
7p
15
5p
12
38. 8p
9
In Exercises 39–44, convert each angle measure from radians to degrees. Round answers to the nearest hundredth of a degree.
39. 4
40. 3
41. 0.85
43. 2.7989
42. 3.27
44.
5.9841
In Exercises 45–50, convert each angle measure from degrees to radians. Round answers to three significant digits.
45. 47°
46. 65°
47. 112°
48. 172°
49. 56.5°
50.
298.7°
In Exercises 51–58, find the reference angle for each of the following angles in terms of both radians and degrees.
51.
2p
3
52.
3p
4
53.
7p
4
54.
5p
4
55.
5p
12
56.
7p
12
57.
4p
3
In Exercises 59–84, find the exact value of the following expressions. Do not use a calculator.
p
59. sin a b
4
p
60. cos a b
6
61. sin a
7p
b
4
62. cos a
2p
b
3
63. sin a
64. cos a
65. sin a
4p
b
3
66. cos a
11p
b
6
3p
b
4
7p
b
6
p
67. sin a b
6
p
68. cos a b
4
69. cos a
71. tan a
72. tan a
11p
b
6
75. tan a
5p
b
6
5p
b
3
5p
b
3
70. sin a
5p
b
6
p
73. tan a b
6
74. tan a
5p
b
6
76. tan a
3p
b
4
77. cot a
3p
b
2
p
78. csc a b
2
79. sec(5p)
80. cot a
3p
b
2
81. sin a
13p
b
4
82. cos a
83. cos a
84. sin a
17p
b
6
8p
b
3
11p
b
3
58.
9p
4
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■
139
A P P L I C AT I O N S
93. Sprinkler. A water sprinkler can reach an arc of 15 feet,
20 feet from the sprinkler as shown. Through how many
radians does the sprinkler rotate?
For Exercises 85 and 86, refer to the following:
Two electronic signals that are not co-phased are called out of
phase. Two signals that cancel each other out are said to be
180° out of phase, or the difference in their phases is 180°.
85. Electronic Signals. How many radians out of phase are
two signals whose phase difference is 270°?
86. Electronic Signals. How many radians out of phase are
two signals whose phase difference is 110°?
87. Construction. In China, you find circular clan homes
called tulou. Some tulou are three or four stories high and
exceed 70 meters in diameter. If a wedge or section on the
third floor of such a building has a central angle measuring
36°, how many radians is this?
15 ft
20 ft
Kin Cheung/Reuters/Landov
94. Sprinkler. A sprinkler is set to reach an arc of 35 feet,
15 feet from the sprinkler. Through how many radians does
the sprinkler rotate?
95. Engine. If a car engine is said to be running at 1500
RPMs (revolutions per minute), through how many radians
is the engine turning every second?
96. Engine. If a car engine is said to rotate 15,000° per second,
through how many radians does the engine turn each second?
For Exercises 97 and 98, refer to the following:
A traction splint is commonly used to treat complete long bone
fractures of the leg. The angle between the leg and torso is an
oblique angle u. The reference angle a is the acute angle
between the leg in traction and the bed.
88. Construction. In China, you find circular clan homes called
tulou. Some tulou are three or four stories high and exceed
70 meters in diameter. If a wedge or section on the third
floor of such a building has a central angle measuring 72°,
how many radians is this?
89. Clock. How many radians does the second hand of a clock
turn in 2 12 minutes?
90. Clock. How many radians does the second hand of a clock
turn in 3 minutes and 15 seconds?
91. London Eye. The London Eye has 32 capsules (each
capable of holding 25 passengers with an unobstructed
view of London). What is the radian measure of the angle
made between the center of the wheel and the spokes
aligning with each capsule?
92. Space Needle. The space needle in Seattle has a restaurant
that offers views of Mount Rainier and Puget Sound. The
restaurant completes one full rotation in approximately
45 minutes. How many radians will the restaurant have
rotated in 25 minutes?
␣
␪
3p
, find the measure of the
4
reference angle in both radians and degrees.
97. Health/Medicine. If u ⫽
2p
, find the measure of the
3
reference angle in both radians and degrees.
98. Health/Medicine. If u ⫽
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For Exercises 99–102, refer to the following:
A water molecule is composed of one oxygen atom and two
hydrogen atoms and exhibits a bent shape with the oxygen
atom at the center.
Net negative charge
␦–
O
H
␦+
105º
Attraction of bonding
electrons to the oxygen
creates local negative
and positive particle
charges
100. Chemistry. The angle between the S-O bonds in sulfur
dioxide (SO2) is approximately 120. Find the angle
between the S-O bonds of sulfur dioxide in radians.
101. Chemistry/Environment. Nitrogen dioxide (NO2) is a
toxic gas and prominent air pollutant. The angle between
the N-O bond is 134.3. Find the angle between the N-O
bonds in radians.
119.7 pm
N
O
O
134.3º
H
␦+
Net positive charge
99. Chemistry. The angle between the O-H bonds in a water
molecule is approximately 105. Find the angle between
the O-H bonds of a water molecule in radians.
102. Chemistry/Environment. Methane (CH4) is a chemical
compound and potent greenhouse gas. The angle between
the C-H bonds is 109.5°. Find the angle between the
C-H bonds in radians.
H
108.70 pm
C
H
■
H
109.5º H
C AT C H T H E M I S TA K E
In Exercises 103–106, explain the mistake that is made.
103. What is the measure (in radians) of a central angle u that
intercepts an arc of length 6 centimeters on a circle with
radius 2 meters?
p
105. Evaluate 6 tan(45) 5 sec a b.
3
Solution:
Evaluate tan(45)
Solution:
p
and sec a b.
3
Write the formula for radians.
6
2
Substitute s 6, r 2.
u
Write the angle in terms of radians.
u 3 rad
Substitute the
values of the
trigonometric
functions.
This is incorrect. What mistake was made?
104. What is the measure (in radians) of a central angle u that
intercepts an arc of length 2 inches on a circle with radius
1 foot?
Solution:
Write the formula for radians.
s
u
r
Substitute s 2, r 1.
u
Write the angle in terms of radians.
u 2 rad
This is incorrect. What mistake was made?
2
1
Simplify.
tan(45) 1
p
sec a b 2
3
p
6 tan(45) 5 sec a b 6(1) 5(2)
3
p
6 tan(45) 5 sec a b 16
3
This is incorrect. What mistake was made?
106. Approximate with a calculator cos(42) tan(65) sin(12).
Round to three decimal places.
Solution:
Evaluate the trigonometric functions individually.
cos(42) ⬇ 0.743
tan(65) ⬇ 2.145
sin(12) ⬇ 0.208
Substitute the values into the expression.
cos(42) tan(65) sin(12) ⬇ 0.743 2.145 0.208
Simplify.
cos(42) tan(65) sin(12) ⬇ 2.680
This is incorrect. What mistake was made?
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■
141
CONCEPTUAL
In Exercises 107–110, determine whether each statement is true or false.
107. An angle with measure 4 radians is a quadrant II angle.
108. Angles expressed exactly in radian measure are always
given in terms of p.
109. For an angle with positive measure, it is possible for the
numerical values of the degree and radian measures to
be equal.
■
111. Find the sum of complementary angles in radian measure.
112. How many complete revolutions does an angle with
measure 92 radians make?
CHALLENGE
113. The distance between Atlanta, Georgia, and Boston,
Massachusetts, is approximately 900 miles along the
curved surface of the Earth. The radius of the Earth is
approximately 4000 miles. What is the central angle with
vertex at the center of the Earth and sides of the angles
intersecting the surface of the Earth in Atlanta and Boston?
114. The radius of the Earth is approximately 6400 kilometers.
If a central angle, with vertex at the center of the Earth,
intersects the surface of the Earth in London (UK) and
Rome (Italy) with a central angle of 0.22 radians, what
is the distance along the Earth’s surface between London
and Rome? Round to the nearest hundred kilometers.
■
110. The sum of the angles with radian measure in a triangle
is p.
115. At 8:20, what is the radian measure of the smaller angle
between the hour hand and minute hand?
116. At 9:05, what is the radian measure of the larger angle
between the hour hand and minute hand?
117. Find the exact value for
p
p
5 cos a3x b 2 sin(2x) 5 for x .
2
3
118. Find the exact value for
p
x
2 cos a3x b 2 sin a b 5 for x p.
3
6
TECH NOLOGY
119. With a calculator set in radian mode, find sin 42. With a
180°
calculator set in degree mode, find sin a42
b. Why do
p
your results make sense?
SECTION
3.2
120. With a calculator set in radian mode, find cos 5. With a
180°
b. Why do
calculator set in degree mode, find cos a5
p
your results make sense?
AR C LE N GTH AN D AR EA O F
A C I R C U L A R S E CTO R
S K I LLS O BJ E CTIVE S
C O N C E P TUAL O BJ E CTIVE
■
■
■
■
Calculate the length of an arc along a circle.
Find the area of a circular sector.
Solve application problems involving circular arc
lengths and sectors.
Understand that to use the arc length formula, the
angle measure must be in radians.
In Section 3.1, radian measure was defined in terms of the ratio of a circular arc of length
s and length of the circle’s radius r.
s
u (in radians) r
In this section (3.2) and the next (3.3), we look at applications of radian measure that
involve calculating arc lengths and areas of circular sectors and calculating angular and
linear speeds.
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Arc Length
From geometry we know the length of an arc of a circle is proportional to its central angle.
In Section 3.1, we learned that for the special case when the arc length is equal to the
circumference of the circle, the angle measure in radians corresponding to one full rotation is
2p. Let us now assume that we are given the central angle and we want to find the arc length.
W OR DS
M ATH
s
r
s
rⴢu ⴢr
r
u
Write the definition of radian measure.
Multiply both sides of the equation by r.
ru s
Simplify.
The formula s r u is true only when u is in radians. To develop a formula when u is in
p
degrees, we multiply u by
to convert the angle measure to radians.
180°
DEFINITION
Study Tip
If a central angle u in a circle with radius r intercepts an arc on the circle of length
s, then the arc length s is given by
To use the relationship
s r ur
s ru
s rud a
the angle u must be in radians.
EXAMPLE 1
Classroom Example 3.2.1
Find the arc length of a sector
determined by central angle
11p
on a circle with radius
6
24 meters.
Arc Length
ur is in radians.
p
b ud is in degrees.
180°
Finding Arc Length When the Angle
Has Radian Measure
In a circle with radius 10 centimeters, an arc is intercepted by a central angle with
7p
. Find the arc length.
measure
4
Solution:
Write the formula for arc length when
the angle has radian measure.
s r ur
Answer: 44p m
Substitute r 10 centimeters and ur 7p
.
4
s
Simplify.
■
Answer: 5p in.
■ YOUR TURN
s (10 cm)a
7p
b
4
35p
cm
2
In a circle with radius 15 inches, an arc is intercepted by a central
p
. Find the arc length.
3
angle with measure
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EXAMPLE 2
Finding Arc Length When the Angle
Has Degree Measure
In a circle with radius 7.5 centimeters, an arc is intercepted by a central angle with
measure 76°. Find the arc length. Approximate the arc length to the nearest centimeter.
Solution:
Write the formula for arc length when
the angle has degree measure.
s r ud a
Substitute r 7.5 centimeters and ud 76°.
s (7.5 cm)(76°)a
Evaluate the result with a calculator.
s ⬇ 9.948 cm
Round to the nearest centimeter.
s ⬇ 10 cm
■ YOUR TURN
EXAMPLE 3
p
b
180°
p
b
180°
In a circle with radius 20 meters, an arc is intercepted by a central
angle with measure 113°. Find the arc length. Approximate the arc
length to the nearest meter.
■
Answer: 39 m
■
Answer: 7121 km
Path of International Space Station
The International Space Station (ISS) is in an
approximate circular orbit 400 kilometers above
the surface of the Earth. If the ground station
tracks the space station when it is within a 45°
central angle of this circular orbit about the center
of the Earth above the tracking antenna, how
many kilometers does the ISS cover while it is
being tracked by the ground station? Assume
that the radius of the Earth is 6400 kilometers.
Round to the nearest kilometer.
ISS
400 km
45º
6400 km
Solution:
p
b
180°
Write the formula for arc length when the
angle has degree measure.
s r ud a
Recognize that the radius of the orbit is
r 6400 400 6800 kilometers and
that ud 45°.
s (6800 km)(45°)a
Evaluate with a calculator.
s ⬇ 5340.708 km
Round to the nearest kilometer.
s ⬇ 5341 km
p
b
180°
The ISS travels approximately 5341 kilometers during the ground station tracking.
■ YOUR TURN
If the ground station in Example 3 could track the ISS within a 60°
central angle of its circular orbit about the center of the Earth, how
far would the ISS travel during the ground station tracking?
143
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C H A P T E R 3 Radian Measure and the Unit Circle Approach
EXAMPLE 4
Classroom Example 3.2.4
Consider two gears working
together such that the
smaller gear has a radius of
10 centimeters, while the
larger gear has a radius
measuring 25 centimeters.
Through how many degrees
does the small gear rotate
when the large gear makes
one complete rotation?
Gears
Gears are inside many devices like automobiles and power meters. When the smaller gear
drives the larger gear, then typically the driving gear is rotated faster than a larger gear would
be if it were the drive gear. In general, smaller ratios of radius of the driving gear to the
driven gear are called for when machines are expected to yield more power. The smaller
gear has a radius of 3 centimeters, and the larger gear has a radius of 6.4 centimeters. If
the smaller gear rotates 170°, how many degrees has the larger gear rotated? Round the
answer to the nearest degree.
3 cm
6.4 cm
Answer: 900°
Solution:
Technology Tip
Recognize that the small gear arc length the large gear arc length.
When solving for ud, be sure
to use a pair of parentheses for the
product in the denominator.
Smaller Gear
180° 17p cm
ⴢ
p
6(6.4 cm)
180° ⴢ 17
6(6.4)
ud Write the formula for arc length when
the angle has degree measure.
s r ud a
p
b
180°
Substitute the values for the smaller gear:
r 3 centimeters and ud 170°.
ssmaller (3 cm)(170°)a
Simplify.
ssmaller a
17p
b cm
6
Remember that the larger gear’s arc length
is equal to the smaller gear’s arc length.
sa
17p
b cm
6
Write the formula for arc length when
the angle has degree measure.
s r ud a
p
b
180°
Larger Gear
p
b
180°
Study Tip
Notice that when calculating ud
in Example 4, the centimeter units
cancel but its degree measure
remains.
Substitute s a
17p
b centimeter and
6
r 6.4 centimeters.
a
p
17p
cmb (6.4 cm)ud a
b
6
180°
Simplify.
180° 17p cm
ⴢ
p
6(6.4 cm)
ud ⬇ 79.6875°
Round to the nearest degree.
ud 80°
Solve for ud.
ud The larger gear rotates approximately 80°.
Area of a Circular Sector
A restaurant lists a piece of French silk pie as having 400 calories. How does the chef
arrive at that number? She calculates the calories of all the ingredients that went into
making the entire pie and then divides by the number of slices the pie yields. For
example, if an entire pie has 3200 calories and it is sliced into 8 equal pieces, then each
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3.2 Arc Length and Area of a Circular Sector
piece has 400 calories. Although that example involves volume, the idea is the same
with areas of sectors of circles. Circular sectors can be thought of as “pieces of a pie.”
Recall that arc lengths of a circle are proportional to the central angle (in radians) and
the radius. Similarly, a circular sector is a portion of the entire circle. Let A represent the
area of the sector of the circle and ur represent the central angle (in radians) that forms
the sector. Then, let us consider the entire circle whose area is pr 2 and the angle that
represents one full rotation has measure 2p (radians).
W OR DS
s
r
A
pr 2
Write the ratio of the central angle ␪r to the
measure of one full rotation.
ur
2p
The ratios must be equal (proportionality of
sector to circle).
A
Multiply both sides of the equation by pr 2.
pr
pr 2 ⴢ
2
ur
2p
ur
ⴢ pr 2
2p
1
A r 2 ur
2
A
pr 2
Simplify.
Study Tip
Area of a Circular Sector
The area of a sector of a circle with radius r and central angle u is given by
EXAMPLE 5
␪
⟨
M ATH
Write the ratio of the area of the sector to the
area of the entire circle.
DEFINITION
r
145
1
A r 2ur
2
ur is in radians.
1
p
A r 2ud a
b
2
180°
ud is in degrees.
A 12 r 2 u
the angle ␪ must be in radians.
Finding the Area of a Circular Sector When
the Angle Has Radian Measure
Find the area of the sector associated with a single slice of pizza if the entire pizza has a
14-inch diameter and the pizza is cut into 8 equal pieces.
Solution:
The radius is half the diameter.
To use the relationship
Classroom Example 3.2.5
Find the area of the sector
with diameter 16 feet and
7p
central angle
.
8
Answer: 28 p ft2
14
7 in.
r
2
2p
p
8
4
Find the angle of each slice if the pizza is cut
into 8 pieces ( 18 of the complete 2p revolution).
ur Write the formula for circular sector area
in radians.
1
A r 2ur
2
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C H A P T E R 3 Radian Measure and the Unit Circle Approach
Answer: 8p in.2 ⬇ 25 in.2
p
Substitute r 7 inches and ur into
4
the area equation.
A
1
p
(7 in.) 2 a b
2
4
Simplify.
A
49p 2
in.
8
Approximate the area with a calculator.
A ⬇ 19 in.2
■ YOUR TURN
EXAMPLE 6
Classroom Example 3.2.6
Find the exact area of the
sector with diameter 1.4 inches
and central angle 225°.
49p 2
Answer:
in.
160
Find the area of a slice of pizza (cut into 8 equal pieces) if the entire
pizza has a 16-inch diameter.
Finding the Area of a Circular Sector When
the Angle Has Degree Measure
Sprinkler heads come in all different sizes depending on the angle of rotation desired. If a
sprinkler head rotates 90° and has enough pressure to keep a constant 25-foot spray, what
is the area of the sector of the lawn that gets watered? Round to the nearest square foot.
Solution:
1
p
A r 2ud a
b
2
180°
1
p
A (25 ft) 2 (90°)a
b
2
180°
Write the formula for circular sector
area in degrees.
Substitute r 25 feet and ␪d 90
into the area equation.
■
Answer: 450p ft2 ⬇ 1414 ft2
625p 2
b ft ⬇ 490.87 ft2
4
Simplify.
Aa
Round to the nearest square foot.
A ⬇ 491 ft2
■ YOUR TURN
If a sprinkler head rotates 180° and has enough pressure to keep a
constant 30-foot spray, what is the area of the sector of the lawn it
can water? Round to the nearest square foot.
SECTION
3.2
S U M MARY
SMH
In this section, we used the proportionality concept (both the arc
length and area of a sector are proportional to the central angle
of a circle). The definition of radian measure was used to
develop formulas for the arc length of a circle when the
central angle is given in either radians or degrees.
s r ur
s r ud a
p
b
180°
The formula for the area of a sector of a circle was also
developed for the cases in which the central angle is given in
either radians or degrees.
ur is in radians.
1
A r 2ur
2
ur is in radians.
ud is in degrees.
1
p
A r 2ud a
b
2
180°
ud is in degrees.
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3.2 Arc Length and Area of a Circular Sector
147
SECTION
3.2
■
EXERCISES
SKILLS
In Exercises 1–12, find the exact length of each arc made by the indicated central angle and radius of each circle.
1. u 3, r 4 mm
5. u 2p
, r 3.5 m
7
9. u 8°, r 1500 km
2. u 4, r 5 cm
6. u 3. u p
, r 10 in.
4
10. u 3°, r 1800 km
p
, r 8 ft
12
4. u p
, r 6 yd
8
7. u 22°, r 18 ␮m
8. u 14°, r 15 ␮m
11. u 48°, r 24 cm
12. u 30°, r 120 cm
In Exercises 13–24, find the exact length of each radius given the arc length and central angle of each circle.
13. s 17. s 21. s 5p
p
ft, u 2
10
12p
4p
yd, u 5
5
8p
mi, u 40°
3
14. s 5p
p
m, u 6
12
18. s 4p in., u 22. s 3p
2
p
␮m, u 30°
4
15. s 24 p
3p
in., u 5
5
16. s 5p
p
km, u 9
180
19. s 4p
yd, u 20°
9
20. s 11p
cm, u 15°
6
23. s 2p
km, u 45o
11
24. s 3p
ft, u 35o
16
In Exercises 25–36, use a calculator to approximate the length of each arc made by the indicated central angle and radius
of each circle. Round answers to two significant digits.
25. u 3.3, r 0.4 mm
26. u 2.4, r 5.5 cm
29. u 4.95, r 30 mi
30. u 33. u 29°, r 2500 km
34. u 11°, r 2200 km
7p
, r 17 mm
8
27. u p
, r 8 yd
15
28. u p
, r 6 ft
10
31. u 79.5°, r 1.55 ␮m
32. u 19.7°, r 0.63 ␮m
35. u 57°, r 22 ft
36. u 127°, r 58 in.
In Exercises 37–48, find the area of the circular sector given the indicated radius and central angle. Round answers to three
significant digits.
5p
p
p
3p
, r 2.2 km
, r 13 mi
37. u , r 7 ft
38. u , r 3 in.
39. u 40. u 6
5
8
6
2p
3p
, r 10 cm
, r 33 m
41. u 42. u 43. u 56°, r 4.2 cm
44. u 27°, r 2.5 mm
11
3
45. u 1.2°, r 1.5 ft
46. u 14°, r 3.0 ft
47. u 22.8o, r 2.6 mi
48. u 60°, r 15 km
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C H A P T E R 3 Radian Measure and the Unit Circle Approach
A P P L I C AT I O N S
David Ball/Index Stock/Photolibrary
49. Low Earth Orbit Satellites. A low Earth orbit (LEO)
satellite is in an approximate circular orbit 300 kilometers
above the surface of the Earth. If the ground station tracks
the satellite when it is within a 45° cone above the tracking
antenna (directly overhead), how many kilometers does
the satellite cover during the ground station track? Assume
the radius of the Earth is 6400 kilometers. Round your
answer to the nearest kilometer.
50. Low Earth Orbit Satellites. A low Earth orbit (LEO)
satellite is in an approximate circular orbit 250 kilometers
above the surface of the Earth. If the ground station tracks
the satellite when it is within a 30° cone above the tracking
antenna (directly overhead), how many kilometers does the
satellite cover during the ground station track? Assume the
radius of the Earth is 6400 kilometers. Round your answer
to the nearest kilometer.
51. Big Ben. The famous clock tower in London has a
minute hand that is 14 feet long. How far does the tip of
the minute hand of Big Ben travel in 25 minutes? Round
your answer to the nearest foot.
Getty Images, Inc.
52. Big Ben. The famous clock tower in London has a minute
hand that is 14 feet long. How far does the tip of the
minute hand of Big Ben travel in 35 minutes? Round your
answer to two decimal places.
53. London Eye. The London Eye is a wheel that has
32 capsules and a diameter of 400 feet. What is the
distance someone has traveled once they reach the highest
point for the first time?
54. London Eye. Assuming the wheel stops at each capsule in
Exercise 53, what is the distance someone has traveled from
the point he or she first gets in the capsule to the point at
which the Eye stops for the sixth time during the ride?
55. Gears. The smaller gear shown below has a radius of
5 centimeters, and the larger gear has a radius of 12.1
centimeters. If the smaller gear rotates 120°, how many
degrees has the larger gear rotated? Round the answer to
the nearest degree.
56. Gears. The smaller gear has a radius of 3 inches, and
the larger gear has a radius of 15 inches (see the figure
above). If the smaller gear rotates 420°, how many degrees
has the larger gear rotated? Round the answer to the
nearest degree.
57. Bicycle Low Gear. If a bicycle has 26-inch diameter
wheels, the front chain drive has a radius of 2.2 inches,
and the back drive has a radius of 3 inches, how far does
the bicycle travel for every one rotation of the cranks
(pedals)?
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3.2 Arc Length and Area of a Circular Sector
58. Bicycle High Gear. If a bicycle has 26-inch diameter
wheels, the front chain drive has a radius of 4 inches, and
the back drive has a radius of 1 inch, how far does the
bicycle travel for every one rotation of the cranks (pedals)?
Getty Images, Inc.
59. Odometer. A Ford Expedition Eddie Bauer Edition comes
standard with 17-inch rims (which corresponds to a tire
with 25.7-inch diameter). Suppose you decide to later
upgrade these tires for 19-inch rims (corresponding to a tire
with 28.2-inch diameter). If you do not get your onboard
computer reset for the new tires, the odometer will not be
accurate. After your new tires have actually driven 1000
miles, how many miles will the odometer report the
Expedition has been driven? Round to the nearest mile.
60. Odometer. For the same Ford Expedition Eddie Bauer
Edition in Exercise 59, after you have driven 50,000 miles,
how many miles will the odometer report the Expedition has
been driven if the computer is not reset to account for the
new oversized tires? Round to the nearest mile.
149
67. Bicycle Wheel. A bicycle wheel 26 inches in diameter
travels 20 inches in 0.10 seconds. What is the speed of
the wheel in revolutions per second?
68. Bicycle Wheel. A bicycle wheel 26 inches in diameter
travels at four revolutions per second. Through how many
radians does the wheel turn in 0.5 seconds?
For Exercises 69 and 70, refer to the following:
Sniffers outside a chemical munitions disposal site monitor the
atmosphere surrounding the site to detect any toxic gases. In the
event that there is an accidental release of toxic fumes, the data
provided by the sniffers make it possible to determine both the
distance d that the fumes reach as well as the angle of spread u
that sweep out a circular sector.
69. Environment. If the maximum angle of spread is 105° and
the maximum distance at which the toxic fumes were
detected was 9 miles from the site, find the area of the
circular sector affected by the accidental release.
70. Environment. To protect the public from the fumes,
officials must secure the perimeter of this area. Find the
perimeter of the circular sector in Exercise 69.
For Exercises 71 and 72, refer to the following:
The structure of human DNA is a linear double helix formed
of nucleotide base pairs (two nucleotides) that are stacked with
spacing of 3.4 angstroms (3.4 1012 m), and each base pair is
rotated 36 with respect to an adjacent pair and has 10 base
pairs per helical turn. The DNA of a virus or a bacterium,
however, is a circular double helix (see the figure below) with
the structure varying among species.
61. Sprinkler Coverage. A sprinkler has a 20-foot spray and
covers an angle of 45°. What is the area that the sprinkler
waters?
62. Sprinkler Coverage. A sprinkler has a 22-foot spray and
covers an angle of 60°. What is the area that the sprinkler
waters?
Twists
63. Windshield Wiper. A windshield wiper that is 12 inches
long (blade and arm) rotates 70°. If the rubber part is
8 inches long, what is the area cleared by the wiper?
Round to the nearest square inch.
64. Windshield Wiper. A windshield wiper that is 11 inches
long (blade and arm) rotates 65°. If the rubber part is
7 inches long, what is the area cleared by the wiper?
Round to the nearest square inch.
65. Bicycle Wheel. A bicycle wheel 26 inches in diameter
travels 45° in 0.05 seconds. Through how many revolutions
does the wheel turn in 30 seconds?
66. Bicycle Wheel. A bicycle wheel 26 inches in diameter
2p
travels
in 0.075 seconds. Through how many
3
revolutions does the wheel turn in 30 seconds?
(Source: http://www.biophysics.org/Portals/1/
PDFs/Education/Vologodskii.pdf.)
71. Biology. If the circular DNA of a virus has 10 twists (or
turns) per circle and an inner diameter of 4.5 nanometers,
find the arc length between consecutive twists of the DNA.
72. Biology. If the circular DNA of a virus has 40 twists (or
turns) per circle and an inner diameter of 2.0 nanometers,
find the arc length between consecutive twists of the DNA.
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C H A P T E R 3 Radian Measure and the Unit Circle Approach
C AT C H T H E M I S TA K E
In Exercises 73 and 74, explain the mistake that is made.
73. A circle with radius 5 centimeters has an arc that is made
from a central angle with measure 65°. Approximate the
arc length to the nearest millimeter.
Solution:
Solution:
Write the formula for arc length.
s ru
Substitute r 5 centimeters and
u 65° into the formula.
Write the formula for area
of a circular sector.
1
A r 2ur
2
s (5 cm)(65)
A
Simplify.
s 325 cm
Substitute r 2.2 centimeters and
u 25° into the formula.
Simplify.
A 60.5 cm2
This is incorrect. What mistake was made?
■
74. For a circle with radius r 2.2 centimeters, find the area
of the circular sector with central angle measuring u 25°.
Round the answer to three significant digits.
1
(2.2 cm) 2 (25°)
2
This is incorrect. What mistake was made?
CONCEPTUAL
In Exercises 75–78, determine whether each statement is true or false.
75. The length of an arc with central angle 45° in a unit
circle is 45.
p
76. The length of an arc with central angle in a unit circle
3
p
is .
3
77. If the radius of a circle doubles, then the arc length
(associated with a fixed central angle) doubles.
79. If a smaller gear has radius r1 and a larger gear has radius
r2 and the smaller gear rotates u°1 what is the degree
measure of the angle the larger gear rotates?
80. If a circle with radius r1 has an arc length s1 associated
with a particular central angle, write the formula for the
area of the sector of the circle formed by that central
angle, in terms of the radius and arc length.
78. If the radius of a circle doubles, then the area of the sector
(associated with a fixed central angle) doubles.
■
CHALLENGE
For Exercises 81–84, refer to the following:
You may think that a baseball field is a circular sector but it is not.
If it were, the distances from home plate to left field, center field,
and right field would all be the same (the radius). Where the
infield dirt meets the outfield grass and along the fence in the
outfield are arc lengths associated with a circle of radius 95 feet
and with a vertex located at the pitcher’s mound (not home plate).
Infield / Outfield Grass Line:
95-ft radius from front of pitching rubber
Second
base
ft n
90 wee
t es
be bas
Infield
Pitching mound
Third base
13-ft radius
First base
13-ft radius
Home plate
13-ft radius
in
e
ul
l
Fo
e
in
l
ul
Fo
9-ft radius
81. What is the area enclosed in the circular sector with radius
95 feet and central angle 150°? Round to the nearest
hundred square feet.
82. Approximate the area of the infield by adding the area in
blue to the result in Exercise 81. Neglect the area near first
and third bases and the foul line. Round to the nearest
hundred square feet.
83. If a batter wants to bunt a ball so that it is fair (in front of
home plate and between the foul lines) but keep it in the
dirt (in the sector in front of home plate), within how small
of an area is the batter trying to keep his bunt? Round to
the nearest square foot.
84. Most bunts would fall within the blue triangle in the
diagram on the left. Assume the catcher only fields bunts
that fall in the sector described in Exercise 83 and the
pitcher only fields bunts that fall on the pitcher’s mound.
Approximately how much area do the first baseman and
third baseman each need to cover? Round to the nearest
square foot.
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SECTION
3.3
LINEAR AND ANGULAR SPEEDS
C O N C E P TUAL O BJ E CTIVE
S K I LLS O BJ E CTIVE S
■
■
■
Calculate linear speed.
Calculate angular speed.
Solve application problems involving angular and
linear speeds.
■
Relate angular speed to linear speed.
In the chapter opener about a Ford Expedition with standard 17-inch rims, we learned that
the onboard computer that determines distance (odometer reading) and speed (speedometer)
combines the number of tire rotations and the size of the tire. Because the onboard
computer is set for 17-inch rims (which corresponds to a tire with 25.7-inch diameter),
if the owner decided to upgrade to 19-inch rims (corresponding to a tire with 28.2-inch
diameter), the computer would have to be updated with this new information. If the
computer is not updated with the new tire size, both the odometer and speedometer
readings will be incorrect.
You will see in this section that the angular speed (rotations of tires per second), radius (of
the tires), and linear speed (speed of the automobile) are related. In the context of a circle, we
will first define linear speed, then angular speed, and then relate them using the radius.
Linear Speed
It is important to note that although velocity and speed are often used as synonyms, speed
is how fast you are traveling, whereas velocity is the speed in which you are
traveling and the direction you are traveling. In physics the difference between speed and
velocity is that velocity has direction and is written as a vector (Chapter 7), and speed is
the magnitude of the velocity vector, which results in a real number. In this chapter, speed
will be used.
Recall the relationship between distance, rate, and time: d rt. Rate is speed, and in
words this formula can be rewritten as
distance speed ⴢ time
or
speed distance
time
It is important to note that we assume speed is constant. If we think of a car driving around
a circular track, the distance it travels is the arc length s, and if we let v represent speed
and t represent time, we have the formula for speed around a circle (linear speed):
v
DEFINITION
s
t
s
Linear Speed
If a point P moves along the circumference of a circle at a constant speed, then the
linear speed v is given by
s
v t
where s is the arc length and t is the time.
151
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C H A P T E R 3 Radian Measure and the Unit Circle Approach
EXAMPLE 1
Classroom Example 3.3.1*
A car travels at a constant
speed around a circular track
with circumference equal
to 1.5 miles. How many
laps would the car need to
complete in 20 minutes in
order to average a linear
speed of 75 miles per hour?
Answer: 1623 laps
■
Answer: 105 mph
Linear Speed
A car travels at a constant speed around a circular track with circumference equal to
2 miles. If the car records a time of 15 minutes for 9 laps, what is the linear speed of the
car in miles per hour?
Solution:
Calculate the distance traveled
around the circular track.
s (9 laps)a
Substitute t 15 minutes and
s
s 18 miles into v .
t
v
18 mi
15 min
Convert the linear speed from miles
per minute to miles per hour.
va
Simplify.
v 72 mph
■ YOUR TURN
2 mi
b 18 mi
lap
18 mi
60 min
ba
b
15 min
1 hr
A car travels at a constant speed around a circular track with
circumference equal to 3 miles. If the car records a time of 12 minutes
for 7 laps, what is the linear speed of the car in miles per hour?
Angular Speed
To calculate linear speed, we find how fast a position along the circumference of a circle is
changing. To calculate angular speed, we find how fast the central angle is changing.
Study Tip
The units of angular speed will be in
radians per unit time (e.g., radians
per minute).
DEFINITION
Angular Speed
If a point P moves along the circumference of a circle at a constant speed, then the
central angle ␪ that is formed with the terminal side passing through point P also
changes over some time t at a constant speed. The angular speed ␻ (omega) is
given by
u
v t
EXAMPLE 2
Classroom Example 3.3.2
A lighthouse in the middle of
a channel rotates its light in a
circular motion with constant
speed. If the beacon of light
completes three rotations every
12 seconds, find its angular
speed in radians per minute.
Answer: 30p rad/min
where ␪ is given in radians
Angular Speed
A lighthouse in the middle of a channel rotates its light in a
circular motion with constant speed. If the beacon of light
completes one rotation every 10 seconds, what is the angular
speed of the beacon in radians per minute?
s
Solution:
Calculate the angle measure in radians
associated with one rotation.
Substitute u 2p and t 10 seconds
u
into v .
t
u 2p
v
2p (rad)
10 sec
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3.3 Linear and Angular Speeds
2p (rad) 60 sec
ⴢ
10 sec
1 min
Convert the angular speed from radians per second
to radians per minute.
v
Simplify.
v 12p rad /min
■ YOUR TURN
153
If the lighthouse in Example 2 is adjusted so that the beacon rotates
one time every 40 seconds, what is the angular speed of the beacon in
radians per minute?
■
Answer: v 3p rad/min
Relationship Between Linear
and Angular Speeds
In the chapter opener, we discussed the Ford Expedition with 17-inch standard rims that
would have odometer and speedometer errors if the owner decided to upgrade to 19-inch
rims without updating the onboard computer. That is because angular speed (rotations of
tires per second), radius (of the tires), and linear speed (speed of the automobile) are
related. To see how, let us start with the definition of arc length (Section 3.2), which comes
from the definition of radian measure (Section 3.1).
W OR DS
M ATH
Write the definition of radian measure.
u
s
r
s ru
s
ru
t
t
s
u
r
t
t
s
␪
v ⴝ and ␻ ⴝ
t
t
Write the definition of arc length (u in radians).
Divide both sides by t.
Rewrite the right side of the equation.
Recall the definitions of linear and angular speeds.
s
␪
u
s
Substitute v ⴝ and ␻ ⴝ into r .
t
t
t
t
R E LATI N G
v rv
LI N E AR AN D AN G U LAR S P E E D S
y
If a point P moves at a constant speed along the
circumference of a circle with radius r, then the
linear speed v and the angular speed v are
related by
v rv
or
v
v
r
Note: This relationship is true only when u is
given in radians.
P
Study Tip
s
r
␪
x
This relationship between linear
speed and angular speed assumes the
angle is given in radians.
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C H A P T E R 3 Radian Measure and the Unit Circle Approach
We now will investigate the Ford Expedition scenario with upgraded tires. Notice that
tires of two different radii with the same angular speed have different linear speeds since
v rv. The larger tire (larger r) has the faster linear speed.
EXAMPLE 3
14.1 in.
Relating Linear and Angular Speeds
A Ford F-150 comes standard with tires that have a diameter of 25.7 inches. If the owner
decided to upgrade to tires with a diameter of 28.2 inches without having the onboard
computer updated, how fast will the truck actually be traveling when the speedometer
reads 75 miles per hour?
Solution:
12.85 in.
The computer in the F-150 “thinks” the tires are 25.7 inches in diameter and knows the
angular speed. Use the programmed tire diameter and speedometer reading to calculate the
angular speed. Then use that angular speed and the upgraded tire diameter to get the actual
speed (linear speed).
S TEP 1
Calculate the angular speed of the tires.
Write the formula for the angular speed.
v
v
r
Substitute v 75 miles per hour and
25.7
12.85 inches into the formula.
r
2
v
75 mi/hr
12.85 in.
1 mile 5280 feet 63,360 inches.
v
75(63,360) in./hr
12.85 in.
Simplify.
v ⬇ 369,805
Study Tip
We could have solved Example 3 the
following way:
75 mph
x
25.7 in.
28.2 in.
28.2 in.
x
75 mph
25.7 in.
⬇ 82.296 mph
S TEP 2
rad
hr
Calculate the actual linear speed of the truck.
Write the linear speed formula.
28.2
14.1 inches
2
and v ⬇ 369,805 radians per hour.
Substitute r v rv
v (14.1 in.)a369,805
rad
b
hr
Simplify.
v ⬇ 5,214,251
in.
hr
1 mile 5280 feet 63,360 inches.
v ⬇ 5,214,251
1 mi
in.
ⴢ
hr 63,360 in.
v ⬇ 82.296
mi
hr
Although the speedometer indicates a speed of 75 miles per hour, the actual speed is
approximately 82 miles per hour .
■
Answer: Approximately 62 mph
■ YOUR TURN
Suppose the owner of the F-150 in Example 3 decides to downsize the
tires from their original 25.7-inch diameter to a 24.4-inch diameter. If
the speedometer indicates a speed of 65 miles per hour, what is the
actual speed of the truck?
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3.3 Linear and Angular Speeds
155
SECTION
3.3
S U M MARY
In this section, circular motion was defined in terms of linear
speed (speed along the circumference of a circle) v and
angular speed (speed of angle rotation) v.
s
Linear speed: v t
u
Angular speed: v t , where u is given in radians.
Linear and angular speeds associated with circular motion are
related through the radius r of the circle.
v rv
or
v
vr
It is important to note that these formulas hold true only when
angular speed is given in radians per unit of time.
SECTION
3.3
■
EXERCISES
SKILLS
In Exercises 1–10, find the linear speed of a point that moves with constant speed in a circular motion if the point travels
along the circle of arc length s in time t. Label your answer with correct units.
1. s 2 m, t 5 sec
2. s 12 ft, t 3 min
3. s 68,000 km, t 250 hr
4. s 7,524 mi, t 12 days
5. s 1.75 nm (nanometers), t 0.25 ms (milliseconds)
6. s 3.6 ␮m (microns), t 9 ns (nanoseconds)
7. s 8. s 25 cm, t 8 hr
1
16
in., t 4 min
3
9. s 10
m, t 5.2 sec
10. s 12.2 mm, t 3.4 min
In Exercises 11–20, find the distance traveled (arc length) of a point that moves with constant speed v along a circle in time t.
11. v 2.8 m/sec, t 3.5 sec
12. v 6.2 km/hr, t 4.5 hr
13. v 4.5 mi/hr, t 20 min
14. v 5.6 ft/sec, t 2 min
15. v 60 mi/hr, t 15 min
16. v 72 km/hr, t 10 min
17. v 750 km/min, t 4 days
18. v 120 ft/sec, t 27 min
19. v 23 ft/s, t 3 min
20. v 46 km/hr, t 20 min
In Exercises 21–32, find the angular speed associated with rotating a central angle ␪ in time t.
21. u 25p, t 10 sec
25. u 7p
, t 12 hr
2
29. u 780°, t 3 min
22. u 3p
1
, t sec
4
6
p
1
,t
min
2
10
23. u 100 p, t 5 min
24. u 26. u 18.3, t 30.45 hr
27. u 200°, t 5 sec
28. u 60°, t 0.2 sec
30. u 420°, t 6 min
31. u 900°, t 3.5 sec
32. u 350°, t 5.6 sec
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C H A P T E R 3 Radian Measure and the Unit Circle Approach
In Exercises 33–42, find the linear speed of a point traveling at a constant speed along the circumference of a circle with
radius r and angular speed ␻.
33. v 2p rad
, r 9 in.
3 sec
34. v 3p rad
, r 8 cm
4 sec
35. v p rad
, r 5 mm
20 sec
36. v 5p rad
, r 24 ft
16 sec
37. v 4p rad
, r 2.5 in.
15 sec
38. v 8p rad
, r 4.5 cm
15 sec
39. v 7
16p rad
, r yd
3 sec
3
40. v p rad
, r 10.2 in.
8 min
41. v 10p
rad
, r 40 cm
sec
42. v 27.3
rad
, r 22.6 mm
sec
In Exercises 43–52, find the distance a point travels along a circle s, over a time t, given the angular speed ␻, and radius of
the circle r. Round to three significant digits.
43. r 5 cm, v p rad
, t 10 sec
6 sec
44. r 2 mm, v 6p
p rad
, t 10 min
15 sec
46. r 3.2 ft, v 47. r 12 m, v 3p rad
, t 100 sec
2 sec
48. r 6.5 cm, v 49. r 30 cm, v p rad
, t 25 sec
10 sec
45. r 5.2 in., v 50. r 5 cm, v rad
, t 11 sec
sec
p rad
, t 3 min
4 sec
2p rad
, t 50.5 min
15 sec
5p rad
, t 9 min
3 sec
51. r 15 in., v 5 rotations per second, t 15 min (express distance in miles*)
52. r 17 in., v 6 rotations per second, t 10 min (express distance in miles*)
*1 mi 5280 ft
■
A P P L I C AT I O N S
53. Tires. A car owner decides to upgrade from tires with
a diameter of 24.3 inches to tires with a diameter of
26.1 inches. If she doesn’t update the onboard computer,
how fast will she actually be traveling when the
speedometer reads 65 mph?
54. Tires. A car owner decides to upgrade from tires with
a diameter of 24.8 inches to tires with a diameter of
27.0 inches. If she doesn’t update the onboard computer,
how fast will she actually be traveling when the
speedometer reads 70 mph?
55. Planets. The Earth rotates every 24 hours (actually
23 hours, 56 minutes, and 4 seconds) and has a diameter of
7926 miles. If you’re standing on the equator, how fast are
you traveling in miles per hour (how fast is the Earth
spinning)? Compute this using 24 hours and then with 23
hours, 56 minutes, 4 seconds as time of rotation.
56. Planets. The planet Jupiter rotates every 9.9 hours and has
a diameter of 88,846 miles. If you’re standing on its
equator, how fast are you traveling in miles per hour?
57. Carousel. A boy wants to jump onto a moving carousel
that is spinning at the rate of five revolutions per minute.
If the carousel is 60 feet in diameter, how fast must the
boy run, in feet per second, to match the speed of the
carousel and jump on?
58. Carousel. A boy wants to jump onto a playground
carousel that is spinning at the rate of 30 revolutions
per minute. If the carousel is 6 feet in diameter, how fast
must the boy run, in feet per second, to match the speed
of the carousel and jump on?
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3.3 Linear and Angular Speeds
157
Courtesy NASA
59. Music. Some people still have their phonograph
collections and play the records on turntables. A
phonograph record is a vinyl disc that rotates on the
turntable. If a 12-inch-diameter record rotates at 33 13
revolutions per minute, what is the angular speed in
radians per minute?
Niall McDiarmid/Alamy
65. NASA. If two humans are on opposite (red and blue) ends
of the centrifuge and their linear speed is 200 miles per
hour, how fast is the arm rotating?
66. NASA. If two humans are on opposite (red and blue) ends
of the centrifuge and they rotate one full rotation every
second, what is their linear speed in feet per second?
For Exercises 67 and 68, refer to the following:
To achieve similar weightlessness as that on NASA’s
centrifuge, ride the Gravitron at a carnival or fair. The
Gravitron has a diameter of 14 meters, and in the first 20
seconds it achieves zero gravity and the floor drops.
Patrick Reddy/America 24-7/Getty Images, Inc.
60. Music. Some people still have their phonograph collections
and play the records on turntables. A phonograph record is
a vinyl disc that rotates on the turntable. If a 12-inch-diameter
record rotates at 33 13 revolutions per minute, what is the
linear speed of a point on the outer edge in inches per
minute?
61. Bicycle. How fast is a bicyclist traveling in miles per hour
if his tires are 27 inches in diameter and his angular speed
is 5p radians per second?
62. Bicycle. How fast is a bicyclist traveling in miles per hour
if his tires are 22 inches in diameter and his angular speed
is 5p radians per second?
63. Electric Motor. If a 2-inch-diameter pulley that’s being
driven by an electric motor and running at 1600 revolutions
per minute is connected by a belt to a 5-inch-diameter
pulley to drive a saw, what is the speed of the saw in
revolutions per minute?
64. Electric Motor. If a 2.5-inch-diameter pulley that’s
being driven by an electric motor and running at
1800 revolutions per minute is connected by a belt to a
4-inch-diameter pulley to drive a saw, what is the speed
of the saw in revolutions per minute?
67. Gravitron. If the Gravitron rotates 24 times per minute,
find the linear speed of the people riding it in meters per
second.
For Exercises 65 and 66, refer to the following:
69. Clock. What is the linear speed of a point on the end of a
10-centimeter second hand given in meters per second?
NASA explores artificial gravity as a way to counter the
physiologic effects of extended weightlessness for future space
exploration. NASA’s centrifuge has a 58-foot-diameter arm.
68. Gravitron. If the Gravitron rotates 30 times per minute,
find the linear speed of the people riding it in kilometers
per hour.
70. Clock. What is the angular speed of a point on the end of
a 10-centimeter second hand given in radians per second?
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C H A P T E R 3 Radian Measure and the Unit Circle Approach
C AT C H T H E M I S TA K E
In Exercises 71 and 72, explain the mistake that is made.
71. If the radius of a set of tires on a car is 15 inches and the
tires rotate 180° per second, how fast is the car traveling
(linear speed) in miles per hour?
Solution:
Solution:
Write the formula for
linear speed.
v rv
Write the formula for
linear speed.
v rv
Let r 15 inches and
v 180° per second.
v (15 in.)(180°/sec)
Let r 10 inches and
v 180° per second.
v (10 in.)(180°/sec)
Simplify.
v 2700 in./sec
Simplify.
v 1800 in./sec
Let 1 mile 5280 feet
63,360 inches and
1 hour 3600 seconds.
va
Let 1 mile 5280 feet
63,360 inches and
1 hour 3600 seconds.
va
Simplify.
v ⬇ 153.4 mph
Simplify.
v ⬇ 102.3 mph
2700 ⴢ 3600
b mph
63,360
This is incorrect. The correct answer is approximately
2.7 miles per hour. What mistake was made?
■
72. If a bicycle has tires with radius 10 inches and the tires
rotate 90° per 12 second, how fast is the bicycle traveling
(linear speed) in miles per hour?
1800 ⴢ 3600
b mph
63,360
This is incorrect. The correct answer is approximately
1.8 miles per hour. What mistake was made?
CONCEPTUAL
In Exercises 73 and 74, determine whether each statement is
true or false.
73. Angular and linear speed are inversely proportional.
74. Angular and linear speed are directly proportional.
75. In the chapter opener about the Ford Expedition, if the
standard tires have radius r1 and the upgraded tires have
radius r2, assuming the owner does not get the onboard
computer adjusted, find the actual speed the Ford is
traveling, v2, in terms of the indicated speed on the
speedometer, v1.
76. For the Ford in Exercise 75, find the actual mileage the
Ford has traveled, s2, in terms of the indicated mileage on
the odometer, s1.
In Exercises 77 and 78, use the diagram below:
The large gear has a radius of 6 centimeters, the medium gear
has a radius of 3 centimeters, and the small gear has a radius
of 1 centimeter.
1 cm
3 cm
6 cm
77. If the small gear rotates 1 revolution per second,
what is the linear speed of a point traveling along the
circumference of the large gear?
78. If the small gear rotates 1.5 revolutions per second,
what is the linear speed of a point traveling along the
circumference of the large gear?
■
CHALLENGE
79. A boy swings a red ball attached to a 10-foot string around
his head as fast as he can. He then picks up a blue ball
attached to a 5-foot string and swings it at the same
angular speed. How does the linear velocity of the blue ball
compare to that of the red ball.
80. One of the cars on a Ferris wheel, 100 feet in diameter,
goes all of the way around in 35 seconds. What is the linear
speed of a point halfway between the car and the hub?
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SECTION
3.4
Page 159
DEFINITION 3 OF TRIGONOMETRIC
F U N CTI O N S: U N IT C I R C LE AP P R OAC H
C O N C E P TUAL O BJ E CTIVE S
S K I LLS O BJ E CTIVE S
■
■
■
Draw the unit circle illustrating the special angles
and label the sine and cosine values.
Determine the domain and range of trigonometric
(circular) functions.
Classify circular functions as even or odd.
■
■
■
Understand that trigonometric functions using the
unit circle approach are consistent with both of the
previous definitions (right triangle trigonometry and
trigonometric functions of nonacute angles in the
Cartesian plane).
Relate x-coordinates and y-coordinates of points on
the unit circle to the values of the cosine and sine
functions.
Visualize periodic properties of trigonometric
(circular) functions.
Recall that the first definition of trigonometric functions we developed was in terms of
ratios of sides of right triangles (Section 1.3). Then, in Section 2.2, we superimposed right
triangles on the Cartesian plane, which led to a second definition of trigonometric functions
(for any angle) in terms of ratios of x- and y-coordinates of a point and the distance from
the origin to that point. In this section, we inscribe right triangles into the unit circle
in the Cartesian plane, which will yield a third definition of trigonometric functions. It is
important to note that all three definitions are consistent with one another.
Trigonometric Functions and the
Unit Circle (Circular Functions)
y
(0, 1)
Recall that the equation for the unit circle (radius of 1 centered at the origin) is given by
x2 y2 1. We will use the term circular function later in this section, but it is important
to note that a circle is not a function (it does not pass the vertical line test).
If we form a central angle u in the unit circle such that the terminal side lies in
quadrant I, we can use the previous two definitions of the sine and cosine functions when
r 1 (i.e., on the unit circle) and noting that we can form a right triangle with legs of
lengths x and y and hypotenuse r 1.
TRIGONOMETRIC
FUNCTION
RIGHT TRIANGLE
TRIGONOMETRY
CARTESIAN
PLANE
sin u
opposite
y
y
hypotenuse
1
y
y
y
r
1
cos u
adjacent
x
x
hypotenuse
1
x
x
x
r
1
r=1
(–1, 0)
␪
(x, y)
y
x
x
(1, 0)
(0, –1)
159
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C H A P T E R 3 Radian Measure and the Unit Circle Approach
Notice that any point (x, y) on the unit circle can be written as (cos u, sin u) , where u
is the measure of a trigonometric angle defined in Chapter 2. If we recall the unit circle
coordinate values for special angles (Section 2.1), we can now summarize the exact values
for the sine and cosine functions in the illustration below.
(x, y) = (cos ␪, sin ␪)
Study Tip
(cos u, sin u) represents a point (x, y)
on the unit circle.
(
– 1 , √3
2 2
(
2 2
–√ , √
2 2
)
)
(0, 1)
y
(
)
1 √3
2, 2
(
√2 √2
)
,
90º ␲
␲
2 2
2
2␲
␲
3
3␲ 3 120º 60º
3 1
√3 , 1
4
4
–√ ,
45º
␲
135º
5␲
2 2
2 2
30º 6
6 150º
x
(–1, 0)
0º 0
␲ 180º
360º 2␲ (1, 0)
0
330º 11␲
7␲ 210º
3
6
315º
√3 , – 1
225º
6
–√ ,–1
7␲
5␲
300º
2
2
2
2
4 4␲ 240º 3␲ 5␲ 4
3
3
√2 √2
270º 2
√2 √2
(
)
( )
(
)
(
(
–
2
,–
(
2
)
(
3
– 1 ,–√
2
2
)
2
(0, –1)
1 √3
2,– 2
(
,–
2
)
)
)
The following observations are consistent with properties of trigonometric functions
we’ve studied already:
■
■
■
sin u 0 in quadrant I and quadrant II, where y 0.
cos u 0 in quadrant I and quadrant IV, where x 0.
The equation of the unit circle x2 y2 1 leads also to the Pythagorean identity
cos 2 u sin 2 u 1 that we derived in Section 2.4.
Circular Functions
Using the unit circle relationship (x, y) (cos u, sin u) , where u is the central angle whose
terminal side intersects the unit circle at the point (x, y), we can now define the remaining
trigonometric functions using this unit circle approach and the quotient and reciprocal
identities. Because the trigonometric functions are defined in terms of the unit circle, the
s
trigonometric functions are often called circular functions. Recall that u , and since
r
r 1, we know that u s.
y
(0, 1)
(x, y)
sⴝ␪
1
(–1, 0)
x
␪
1
(0, –1)
(1, 0)
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3.4 Definition 3 of Trigonometric Functions: Unit Circle Approach
DEFINITION 3
Trigonometric Functions: Unit Circle Approach
Classroom Example 3.4.1
Compute:
4p
a. cos a b
3
3p
b. cot a b
2
7p
c. sec a b
6
Let (x, y) be any point on the unit circle (x2 ⫹ y2 ⫽ 1) . If u is the real number that
represents the distance from the point (1, 0) along the circumference of the circle
to the point (x, y), then
y
sin u ⫽ y
cos u ⫽ x
tan u ⫽
x⫽0
x
csc u ⫽
1
y
y⫽0
sec u ⫽
1
x
x⫽0
x
y
cot u ⫽
161
y⫽0
Answer:
1
a. ⫺
b. 0
2
The coordinates of the points on the unit circle can be written as (cos u, sin u) , and
since ␪ is a real number, the trigonometric functions are often called circular
functions.
c. ⫺
213
3
Technology Tip
EXAMPLE 1
Finding Exact Trigonometric (Circular) Function Values
Find the exact values for each of the following using the unit circle definition.
7p
a. sin a b
4
5p
b. cos a b
6
3p
c. tan a b
2
Use a TI calculator to
confirm the values for sin a
5p
3p
cos a b, and tan a b.
6
2
7p
b,
4
Solution (a):
12
7p
12
,⫺
b on the unit circle.
The angle
corresponds to the coordinates a
4
2
2
The value of the sine function is the y-coordinate.
sin a
7p
12
b⫽⫺
4
2
Solution (b):
13 1
5p
The angle
corresponds to the coordinates a⫺
, b on the unit circle.
6
2 2
The value of the cosine function is the x-coordinate.
cos a
13
5p
b⫽⫺
6
2
Solution (c):
3p
The angle
corresponds to the coordinates (0, ⫺1) on the unit circle.
2
3p
The value of the cosine function is the x-coordinate.
cos a b ⫽ 0
2
3p
The value of the sine function is the y-coordinate.
sin a b ⫽ ⫺1
2
sin (3p/2)
3p
The tangent function is the ratio of the sine to
tan a b ⫽
cosine functions.
2
cos (3p/2)
3p
3p
3p
⫺1
Substitute cos a b ⫽ 0 and sin a b ⫽ ⫺1.
tan a b ⫽
2
2
2
0
tan a
■ YOUR TURN
Technology Tip
Since tan a
3p
b is undefined, the TI
2
calculator will display an error
message.
3p
b is undefined
2
Find the exact values for each of the following using the unit circle
definition.
5p
7p
2p
a. sin a b
b. cos a b
c. tan a b
6
4
3
■
Answer: a.
1
12
b.
c. ⫺ 13
2
2
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C H A P T E R 3 Radian Measure and the Unit Circle Approach
EXAMPLE 2
Classroom Example 3.4.2
12
Solve cos u on
2
[0, 2p] .
3p 5p
Answer:
,
4 4
Solving Equations Involving Trigonometric
(Circular) Functions
Use the unit circle to find all values of u, 0 u 2p, for which sin u 12.
Solution:
(
Since the value of the sine function is
negative, u must lie in quadrants III or IV.
The value of sine is the
y-coordinate. The angles
corresponding to
1
sin ␪ ⴝ ⴚ are
2
11␲
7␲
and
.
6
6
(
Answer: u 2p 4p
,
3 3
■ YOUR TURN
(
)
1 √3
2, 2
)
(
√2 , √2
)
␲
2 2
3␲ 3 120º 60º 3 ␲
3 1
√
√3 , 1
4
– ,
4
45º
␲
135º
5␲
2 2
2 2
30º 6
6 150º
x
(–1, 0)
0º 0
␲ 180º
360º 2␲ (1, 0)
0
330º 11␲
7␲ 210º
3
6
315º
√3 , – 1
225º
6
–√ , – 1
7␲
5␲
300º
2
2
2
2
4 4␲ 240º 3␲ 5␲ 4
3
3
270º 2
√2 √2
√2 √2
(
)
( )
(
)
(
There are two values for u that are greater
than or equal to zero and less than or equal
to 2p that satisfy the equation sin u 12.
■
)
– 1 , √3
2 2
y
(0, 1)
2 √2
√
– ,
␲
90º
2 2
2
2␲
(
–
2
,–
2
(
)
(
3
– 1, –√
2
2
)
u
2
(0, –1)
1 √3
2,– 2
(
,–
2
)
)
)
7p 11p
,
6
6
Find all values of u, 0 u 2p, for which cos u 12.
Properties of Circular Functions
W OR DS
M ATH
The coordinates of any point (x, y)
that lies on the unit circle satisfies
the equation x2 y2 1.
1 x 1 and 1 y 1
Since x cos u and y sin u, the
following trigonometric inequalities hold.
1 cos u 1 and 1 sin u 1
State the domain and range of the
cosine and sine functions.
Domain: (
, )
cos u
1
,
and csc u sin u
sin u
the values for u that make sin u 0 must
be eliminated from the domain of the
cotangent and cosecant functions.
Domain: u np, where n is
an integer
Range: [1, 1]
Since cot u sin u
1
,
and sec u cos u
cos u
the values for u that make cos u 0 must
be eliminated from the domain of the
tangent and secant functions.
Since tan u (2n 1)p
p
np,
2
2
where n is an integer
Domain: u c03.qxd
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3.4 Definition 3 of Trigonometric Functions: Unit Circle Approach
The following box summarizes the domains and ranges of the trigonometric (circular)
functions.
D O MAI N S AN D R AN G E S
( C I R C U LAR ) F U N CTI O N S
O F TH E TR I G O N O M E TR I C
For any real number ␪ and integer n,
FUNCTION
DOMAIN
RANGE
sin u
(
, )
[1, 1]
cos u
(
, )
[1, 1]
tan u
all real numbers such that u cot u
all real numbers such that u np
sec u
all real numbers such that u csc u
all real numbers such that u np
(2n 1)p
p
np
2
2
(
, )
(
, )
(2n 1)p
p
np
2
2
(
, 1冥 ´ 冤1, )
(
, 1冥 ´ 冤1, )
Recall from algebra that even and odd functions have both an algebraic and a graphical
interpretation. Even functions are functions for which f(x) f(x) for all x in the domain
of f, and the graph of an even function is symmetric about the y-axis. Odd functions are
functions for which f(x) f(x) for all x in the domain of f, and the graph of an odd
function is symmetric about the origin.
y
1
(x, y) = (cos␪, sin ␪)
␪
–1
–␪
–1
The cosine function is an even function.
The sine function is an odd function.
x
1
(x, –y) = (cos(–␪), sin(–␪))
= (cos␪, –sin␪)
cos u cos(u)
sin(u) sin u
163
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C H A P T E R 3 Radian Measure and the Unit Circle Approach
Using Properties of Trigonometric
(Circular) Functions
EXAMPLE 3
Technology Tip
Evaluate cos a
Use a TI/scientific calculator
5p
to confirm the value of cos a b.
6
Solution:
5p
b.
6
The cosine function is an even function.
Use the unit circle to evaluate cosine.
cos a
cos a
13
5p
b
6
2
cos a
■
Answer: 12
■ YOUR TURN
Evaluate sin a
5p
5p
b cos a b
6
6
5p
13
b
6
2
5p
b.
6
Study Tip
Set the calculator to radian mode
before evaluating circular functions
in radians. Alternatively, convert the
radian measure to degrees before
evaluating the trigonometric
function value.
It is important to note that although trigonometric (circular) functions can be evaluated
exactly for some special angles, a calculator can be used to approximate trigonometric
(circular) functions for any value.
EXAMPLE 4
Classroom Example 3.4.3
Evaluate exactly:
5p
a. sin a b
3
7p
b. cos a b
4
Answer:
12
13
a.
b.
2
2
Evaluating Trigonometric (Circular)
Functions with a Calculator
Use a calculator to evaluate sin a
C O M M O N M I S TA K E
★
CORRECT
Evaluate with a calculator.
0.965925826
Round to four decimal places.
Classroom Example 3.4.4
8p
Evaluate cos a b using a
11
calculator.
Answer: 0.6549
■
7p
b. Round the answer to four decimal places.
12
sin a
7p
b ⬇ 0.9659
12
INCORRECT
Evaluate with a calculator.
0.031979376
ERROR
(calculator in
degree mode)
Many calculators automatically reset to degree mode after every calculation, so
be sure to always check what mode the calculator indicates.
Answer: 0.7265
■ YOUR TURN
9p
Use a calculator to evaluate tan a b. Round the answer to four
5
decimal places.
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3.4 Definition 3 of Trigonometric Functions: Unit Circle Approach
EXAMPLE 5
Even and Odd Trigonometric (Circular) Functions
Classroom Example 3.4.5
Prove that the cosecant
function is an odd function.
Show that the secant function is an even function.
Solution: Show that sec(u) sec u.
Answer:
1
cos(u)
1
sec(u) cos u
1
sec(u) sec u
cos u
sec(u) Secant is the reciprocal of cosine.
Cosine is an even function, so cos(u) cos u.
Secant is the reciprocal of cosine, sec u 1
.
cos u
165
1
sin(u)
1
csc u
sin u
csc(u) Since sec(u) sec u, the secant function is an even function.
SECTION
3.4
S U M MARY
In this section, we have defined trigonometric functions in
terms of the unit circle. The coordinates of any point (x, y) that
lies on the unit circle satisfy the equation x2 y2 1. The
Pythagorean identity cos 2 u sin 2 u 1 follows immediately
from the unit circle equation if (x, y) (cos u, sin u) , where u is
the central angle whose terminal side intersects the unit circle at
the point (x, y). The cosine function is an even function,
cos(u) cos u, and the sine function is an odd
function, sin(u) sin u.
SECTION
3.4
■
EXERCISES
SKILLS
In Exercises 1–14, find the exact values of the indicated trigonometric functions using the unit circle.
1. sin a
5p
b
3
3. cos a
2. cos a
5p
b
3
7p
b
6
4. sin a
7p
b
6
5. sin a
3p
b
4
6. cos a
7. tan a
7p
b
4
8. cot a
7p
b
4
9. seca
5p
b
6
10. csca
11p
b
6
3p
b
4
11. sec 225°
12. csc 300°
13. tan 240°
14. cot 330°
(
)
– 1 , √3
2 2
y
(0, 1)
2 √2
√
– ,
90º ␲
2 2
2␲
2
(
(
)
)
1 √3
2, 2
(
√2 , √2
)
␲
2 2
␲
3␲ 3 120º 60º 3
3 1
√
√3 , 1
4
– ,
4
45º
␲
135º
5␲
2 2
2 2
6
30º
6 150º
x
(–1, 0)
0º 0
360º 2␲ (1, 0)
␲ 180º
0
330º 11␲
7␲ 210º
3
6
315º
√3 , – 1
225º
6
–√ , – 1
7␲
5␲
300º
2
2
2
2
4 4␲ 240º 3␲ 5␲ 4
3
3
270º 2
√2 √2
√2 √2
(
)
(
( )
(
)
(
–
2
,–
2
(
)
3
– 1, – √
2
2
(
)
2
(0, –1)
1 √3
2,– 2
(
)
,–
2
)
)
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C H A P T E R 3 Radian Measure and the Unit Circle Approach
In Exercises 15–30, use the unit circle and the fact that sine is an odd function and cosine is an even function to find the
exact values of the indicated functions.
15. sin a
16. sin a
2p
b
3
19. cos a
3p
b
4
5p
b
4
20. cos a
5p
b
3
p
17. sin a b
3
18. sin a
21. cos a
22. cos a
5p
b
6
7p
b
6
7p
b
4
23. sin(225°)
24. sin(180°)
25. sin(270°)
26. sin(60°)
27. cos(45°)
28. cos(135°)
29. cos(90°)
30. cos(210°)
In Exercises 31–50, use the unit circle to find all of the exact values of ␪ that make the equation true in the
indicated interval.
31. cos u 13
, 0 u 2p
2
33. sin u 32. cos u 13
, 0 u 2p
2
34. sin u 1
35. cos u , 0 u 2p
2
37. cos u 13
, 0 u 2p
2
13
, 0 u 2p
2
1
36. sin u , 0 u 2p
2
22
, 0 u 2p
2
38. sin u 22
, 0 u 2p
2
39. sin u 0, 0 u 4p
40. sin u 1, 0 u 4p
41. cos u 1, 0 u 4p
42. cos u 0, 0 u 4p
43. tan u 1, 0 u 2p
44. cot u 1, 0 u 2p
45. sec u 12, 0 u 2p
46. csc u 12, 0 u 2p
47. csc u is undefined, 0 u 2p
48. sec u is undefined, 0 u 2p
49. tan u is undefined, 0 u 2p
50. cot u is undefined, 0 u 2p
In Exercises 51–58, approximate the trigonometric function values. Round answers to four decimal places.
51. cos a
7p
b
11
55. sin 4
■
52. sin a
5p
b
9
56. cos 7
53. cot a
11p
b
5
57. tan(2.5)
54. tan a
12p
b
7
58. csc 1
A P P L I C AT I O N S
For Exercises 59 and 60, refer to the following:
For Exercises 61 and 62, refer to the following:
The average daily temperature in Peoria, Illinois, can be
2p(x 31)
predicted by the formula T 50 28 cos
, where
365
x is the number of the day in a nonleap year (January 1 1,
February 1 32, etc.) and T is in degrees Fahrenheit.
The human body temperature normally fluctuates during the day.
Assume a person’s body temperature can be predicted by the
p
formula T 99.1 0.5 sin ax b, where x is the number of
12
hours since midnight and T is in degrees Fahrenheit.
59. Atmospheric Temperature. What is the expected
temperature on February 15?
61. Body Temperature. What is the person’s temperature at
6:00 A.M.?
60. Atmospheric Temperature. What is the expected
temperature on August 15?
62. Body Temperature. What is the person’s temperature at
9:00 P.M.?
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3.4 Definition 3 of Trigonometric Functions: Unit Circle Approach
For Exercises 63 and 64, refer to the following:
The height of the water in a harbor changes with the tides.
On a particular day, it can be determined by the formula
p
h(x) 5 4.8 sin c (x 4)d , where x is the number of hours
6
since midnight and h is the height of the tide in feet.
167
65. Yo-Yo Dieting. A woman has been yo-yo dieting for years.
Her weight changes throughout the year as she gains and
loses weight. Her weight in a particular month can be
p
determined by the formula w(x) 145 10 cos a xb,
6
where x is the month and w is in pounds. If x 1
corresponds to January, how much does she weigh in June?
66. Yo-Yo Dieting. How much does the woman in Exercise 65
weigh in December?
Bill Brooks/Alamy
67. Seasonal Sales. The average number of guests visiting the
Magic Kingdom at Walt Disney World per day is given by
p
n(x) 30,000 20,000 sin c (x 1) d , where n is the
2
number of guests and x is the month. If January
corresponds to x 1, how many people, on average, are
visiting the Magic Kingdom per day in February?
68. Seasonal Sales. How many guests are visiting the Magic
Kingdom in Exercise 67 in December?
69. Temperature. The average high temperature for a certain
p
city is given by the equation T 60 20 cos a tb, where
6
T is degrees Fahrenheit and t is time in months. What is
the average temperature in June (t 6)?
70. Temperature. The average high temperature for a certain
p
city is given by the equation T 65 25 cos a tb, where
6
T is degrees Fahrenheit and t is time in months. What is
the average temperature in October (t 10)?
71. Gear. The vertical position in centimeters of a tooth on a
gear is given by the function y 3 sin(10t) , where t is time
in seconds. Find the vertical position after 2.5 seconds.
72. Gear. The vertical position in centimeters of a tooth on
a gear is given by the equation y 5 sin(3.6t) , where t is
time in seconds. Find the vertical position after 10 seconds.
73. Oscillating Spring. A weight is attached to a spring and
then pulled down and let go to begin a vertical motion. The
position of the weight in inches from equilibrium is given by
7
7p
the equation y 15 sin a t b, where t is time in
2
2
seconds after the spring is let go. Find the position of the
weight 3.5 seconds after being let go.
63. Tides. What is the height of the tide at 3:00 P.M.?
64. Tides. What is the height of the tide at 5:00 A.M.?
74. Oscillating Spring. A weight is attached to a spring and
then pulled down and let go to begin a vertical motion. The
position of the weight in inches from equilibrium is given by
p
the equation y 15 sin a4.6t b, where t is time in
2
seconds after the spring is let go. Find the position of the
weight 5 seconds after being let go.
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C H A P T E R 3 Radian Measure and the Unit Circle Approach
For Exercises 75 and 76, refer to the following:
In Exercises 77 and 78, refer to the following:
During the course of treatment of an illness, the concentration of
a drug in the bloodstream in micrograms per microliter fluctuates
during the dosing period of 8 hours according to the model
By analyzing available empirical data, it has been determined
that the body temperature of a particular species fluctuates
during a 24-hour day according to the model
C(t) 15.4 4.7 sin a
p
p
t b,
4
2
Note: This model does not apply to the first dose of the
medication.
75. Health/Medicine. Find the concentration of the drug in the
bloodstream at the beginning of a dosing period.
76. Health/Medicine. Find the concentration of the drug in the
bloodstream 6 hours after taking a dose of the drug.
■
T(t) 36.3 1.4 cos c
0t8
p
(t 2)d ,
12
0 t 24
where T represents temperature in degrees Celsius and t
represents time in hours measured from 12:00 a.m.
(midnight).
77. Biology. Find the approximate body temperature at
midnight. Round your answer to the nearest degree.
78. Biology. Find the approximate body temperature at
2:45 p.m. Round your answer to the nearest degree.
C AT C H T H E M I S TA K E
In Exercises 79 and 80, explain the mistake that is made.
79. Use the unit circle to evaluate tan a
5p
b exactly.
6
Solution:
Solution:
Tangent is the ratio
of sine to cosine.
Use the unit circle
to identify sine
and cosine.
tan a
sin a
5p
b
6
5p
sin a b
6
cos a
5p
b
6
5p
13
5p
1
and cos a b b
6
2
6
2
13
Substitute values for
2
5p
sine and cosine.
tan a b 6
1
2
Simplify.
80. Use the unit circle to evaluate sec a
11p
b exactly.
6
sec a
11p
b
6
Use the unit circle
to evaluate cosine.
cos a
11p
1
b
6
2
Substitute the value
for cosine.
sec a
11p
1
b
6
1
2
Simplify.
sec a
11p
b 2
6
Secant is the reciprocal
of cosine.
1
11p
cos a
b
6
This is incorrect. What mistake was made?
5p
tan a b 13
6
This is incorrect. What mistake was made?
■
CONCEPTUAL
In Exercises 81–84, determine whether each statement is true or false.
81. sin(2np u) sin u, for n an integer.
82. cos(2np u) cos u, for n an integer.
83. sin u 1 when u (2n 1)p
, for n an integer.
2
84. cos u 1 when u np, for n an integer.
85. Is y csc x an even or an odd function? Justify your answer.
86. Is y tan x an even or an odd function? Justify your answer.
87. Find all the values of u, 0 u 2p, for which the
equation is true: sin u cos u.
88. Find all the values of u (u is any real number) for which
the equation is true: sin u cos u.
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3.4 Definition 3 of Trigonometric Functions: Unit Circle Approach
■
169
CHALLENGE
89. How many times is the expression ƒ cos(2pt) ƒ 1 true for
0 t 12?
92. For what values of x, such that 0 x 2p, is the
expression ƒ sec t ƒ ƒ cos t ƒ true?
p
90. How many times is the expression ` sin a tb ` 1 true for
2
0 t 10?
93. Find values of x such that 0 x 2p and both of the
following are true: sin x 12 and cos x 12.
91. For what values of x, such that 0 x 2p, is the
expression ƒ cos t ƒ ƒ sin t ƒ true?
■
94. Find values of x such that 0 x 2p and both of the
following are true: tan x 1 and sec x 0.
TECH NOLOGY
p
, 2 X 2,
15
95. Use a calculator to approximate sin 423°. What do you
expect sin(423°) to be? Verify your answer with a
calculator.
Set the window so that 0 t 2p, step 96. Use a calculator to approximate cos 227°. What do you
expect cos(227°) to be? Verify your answer with a
calculator.
p
97. To approximate cos a b, use the trace function to move
3
p
5 steps aof
eachb to the right of t 0 and read the
15
x-coordinate.
For Exercises 97 and 98, refer to the following:
A graphing calculator can be used to graph the unit circle with
parametric equations (these will be covered in more detail in
Section 8.5). For now, set the calculator in parametric and radian
modes and let
X1 cos T
Y1 sin T
and 2 Y 2.
p
98. To approximate sin a b, use the trace function to move
3
p
5 steps aof
eachb to the right of t 0 and read the
15
y-coordinate.
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C H A P T E R 3 I N Q U I R Y- B A S E D L E A R N I N G P R O J E C T
Mr. Wilson is looking to expand his watering trough for his horses. His neighbor,
Dr. Parkinson, suggests considering something other than the “square” trough he
currently has. Jokingly, she says, “Mr. Wilson, think ‘outside the box.’” Upon the advice
of his mathematics professor neighbor, Mr. Wilson decides to pitch out the sides of
his troughs, forming a trapezoidal cross section using his current barn as one of the
sides. (Reread the Inquiry-Based Learning Project in Chapter 1.)
Your goal is to maximize the cross section of his trough. To do this, you will first
use theta (u) as your variable and look at how the area changes as u changes.
Barn
Current
trough
2 ft
Barn
2 ft ␪
2 ft
Future
trough
␪ 2 ft
2 ft
2 ft
1. Fill in the chart (use two decimals). To get started, when u is 0, the trapezoid is
just the original square trough. As u increases, the original square becomes two
triangles and one rectangle. Use right triangle trigonometry to calculate the various
bases and height. Note: The base for the triangles differs from the base of the
rectangles. Also, you do not need to do every single trapezoid by hand. Do as
many as you think is necessary to understand how to write the area as a function
of u. The ability to write out the area function is the primary goal.
Theta (u)
0
5
15
25
35
45
55
65
75
85
2. From your chart values, describe what happens to the area of the trough as u
increases.
3. Is the maximum area for the trough necessarily included in this chart? Explain.
4. Write the area A(u) as a function of u using sin u and cos u. Again, look to how
you calculated areas in the chart for direction. Also be sure to use your calculator’s
table to check your problems done by hand and vice versa.
5. Graph this function on a reasonable domain and be sure to indicate what the
domain is.
6. Explain the meaning of the y-intercept in this scenario.
7. Summarize your findings for Mr. Wilson. Remember, you were given a charge to
build the biggest trough possible. How are you going to do it and what is the new
and improved area?
8. After looking at your results from Chapter 1’s Inquiry-Based Learning Project,
explain why many people consider the optimum u to be a counterintuitive result.
170
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MODELING OUR WORLD
Tire selection affects fuel economy in automobiles. The more miles per gallon consumers
can obtain in their automobiles, the less gasoline we consume (money) and hence burn
(pollution/greenhouse gases). Tire size (both diameter and tread width) can affect gas
mileage, depending on what kind of driving you do (highway vs. city and flat vs. hilly).
Go back and reread the Chapter 3 opener about the Ford Expedition and the
consequences (speedometer and odometer) of altering the tires. Assume the original
tires have a diameter of 26 inches and the new tires have a diameter of 28 inches.
1. If you know you drove 15,000 miles in a year (according to your GPS Navigation
System), what would your odometer actually read (assume the onboard computer
was not adjusted when the new tires were put on the Expedition)?
2. If your speedometer reads 85 miles per hour, what is your actual speed?
3. If your onboard computer is saying you are getting 16 miles per gallon, what is
your actual gas mileage?
4. Assuming gasoline costs $4 per gallon, how much money would you be saving by
increasing your tires 2 inches in diameter?
5. Find a function that models your gasoline savings per year as a function of
increase in diameter of tires.
6. Do the gasoline savings seem worth the investment in larger tires?
171
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CHAPTER 3
SECTION
CONCEPT
3.1
Radian measure
The radian measure of an angle
Converting between degrees
and radians
3.2
6.1 Inverse Trigonometric Functions
REVIEW
KEY IDEAS/FORMULAS
s
r
s (arc length) and r (radius) must have
the same units.
u (in radians) Degrees to radians: ur ud a
p
b
180°
Radians to degrees: ud ur a
180°
b
p
Arc length and area of a
circular sector
CHAPTER REVIEW
Arc length
r
r
s r ur , where ur is in radians, or
s r ud a
Area of circular sector
␪
s
r
␪
s
r
p
b, where ud is in degrees
180°
1
A r 2ur , where ur is in radians, or
2
1
p
b, where ud is in degrees
A r 2ud a
2
180°
3.3
Linear and angular speeds
Uniform circular motion
■ Linear speed: speed around the circumference of a circle
■ Angular speed: rotation speed of angle
Linear speed
Linear speed v is given by
v
s
t
where s is the arc length and t is time.
Angular speed
Angular speed v is given by
v
Relationship between linear and
angular speeds
u
t
where u is given in radians.
v
or
v
v rv
r
It is important to note that these formulas hold true only when
angular speed is given in radians per unit of time.
172
172
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SECTION
CONCEPT
3.4
Definition 3 of trigonometric functions:
Unit circle approach
KEY IDEAS/FORMULAS
y
Trigonometric functions and the unit circle
(circular functions)
(0, 1)
(cos␪, sin␪)
r=1
(–1, 0)
␪
x
␪
(1, 0)
(0, –1)
(
)
– 1 , √3
2 2
y
(0, 1)
2 √2
√
– ,
90º ␲
2 2
2␲
2
(
(
)
)
1 √3
2, 2
(
√2 , √2
)
␲
2 2
␲
3␲ 3 120º 60º 3
3 1
√
√3 , 1
4
4
– ,
45º
␲
135º
5␲
2 2
2 2
30º 6
6 150º
x
(–1, 0)
0º 0
␲ 180º
360º 2␲ (1, 0)
0
330º 11␲
7␲ 210º
3
6
315º
√3 , – 1
225º
6
–√ , – 1
7␲
5␲
300º
2
2
2
2
4 4␲ 240º 3␲ 5␲ 4
3
3
270º 2
√2 √2
√2 √2
(
)
(
)
(
–
2
,–
2
)
(
(
3
–1 , – √
2
2
Properties of circular functions
)
2
(0, –1)
1 √3
2,– 2
(
Cosine is an even function:
Sine is an odd function:
FUNCTION
,–
2
CHAPTER REVIEW
(
( )
)
)
)
cos(u) cos u
sin(u) sin u
DOMAIN
RANGE
sin u
(
, )
[1, 1]
cos u
(
, )
[1, 1]
tan u
u
u np
cot u
sec u
csc u
(2n 1)p
p
np
2
2
u
(2n 1)p
p
np
2
2
u np
(
, )
(
, )
(
, 1冥 ´ 冤1, )
(
, 1冥 ´ 冤1, )
* n is an ineger.
173
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REVIEW EXERCISES
CHAPTER 3
Find the area of the circular sector given the indicated
radius and central angle.
3.1 Radian Measure
Convert from degrees to radians. Leave your answers exact
in terms of ␲.
1. 135°
2. 240°
3. 330°
4. 180°
5. 216°
6. 108°
7. 504°
8. 600°
9. 150°
p
3
12.
11p
6
13.
5p
4
14.
2p
3
15.
5p
9
16.
17p
10
17.
13p
4
18.
11p
3
20. 13p
6
Find the reference angle of each angle given (in radians).
REVIEW EXERCISES
21.
7p
4
22.
5p
6
23.
7p
6
24.
2p
3
3.2 Arc Length and Area of a
Circular Sector
Find the arc length intercepted by the indicated central
angle of the circle with the given radius. Round to two
decimal places.
25. u p
, r 5 cm
3
27. u 100°, r 5 in.
26. u 5p
, r 10 in.
6
28. u 36°, r 12 ft
Find the measure of the angle whose intercepted arc and
radius of a circle are given.
29. r 12 in., s 6 in.
30. r 10 ft, s 27 in.
31. r 6 ft, s 4p ft
32. r 8 m, s 2p m
33. r 5 ft, s 10 in.
34. r 4 km, s 4 m
Find the measure of each radius given the arc length and
central angle of each circle.
35. u 5p
, s p in.
8
36. u 2p
, s 3p km
3
37. u 150°, s 14p m
38. u 63°, s 14p in.
5p
39. u 10°, s yd
18
5p
40. u 80°, s ft
3
174
43. u 60°, r 60 m
42. u 5p
, r 9 in.
12
44. u 81°, r 36 cm
Find the linear speed of a point that moves with constant speed
in a circular motion if the point travels arc length s in time t.
11.
5p
18
p
, r 24 mi
3
3.3 Linear and Angular Speeds
10. 15°
Convert from radians to degrees.
19. 41. u 45. s 3 ft, t 9 sec
46. s 5280 ft, t 4 min
47. s 15 mi, t 3 min
48. s 12 cm, t 0.25 sec
Find the distance traveled by a point that moves with
constant speed v along a circle in time t.
49. v 15 mi/hr, t 1 day
50. v 16 ft/sec, t 1 min
51. v 80 mi/hr, t 15 min
52. v 1.5 cm/hr, t 6 sec
Find the angular speed (radians/second) associated with
rotating a central angle ␪ in time t.
53. u 6p, t 9 sec
54. u p, t 0.05 sec
55. u 225°, t 20 sec
56. u 330°, t 22 sec
Find the linear speed of a point traveling at a constant
speed along the circumference of a circle with radius r and
angular speed ␻.
57. v 5p rad
, r 12 m
6 sec
58. v p rad
, r 30 in.
20 sec
Find the distance s a point travels along a circle over a time t,
given the angular speed ␻ and radius of the circle r.
p rad
, t 30 sec
4 sec
3p rad
60. r 6 in., v , t 6 sec
4 sec
2p rad
61. r 12 yd, v , t 30 sec
3 sec
p rad
62. r 100 in., v , t 3 min
18 sec
59. r 10 ft, v Applications
63. A ladybug is clinging to the outer edge of a child’s spinning
disk. The disk is 4 inches in diameter and is spinning at
60 revolutions per minute. How fast is the ladybug traveling?
64. How fast is a motorcyclist traveling in miles per hour if
his tires are 30 inches in diameter and his angular speed is
10p radians per second?
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Review Exercises
3.4 Definition 3 of Trigonometric
Functions: Unit Circle Approach
Find the exact values of the indicated trigonometric functions.
5p
65. tan a b
6
5p
66. cos a b
6
67. sin a
68. sec a
11p
b
6
11p
b
6
5p
69. cot a b
4
5p
70. csc a b
4
71. sin a
72. cos a
3p
b
2
3p
b
2
73. cos p
74. tan 315°
75. cos 60°
76. sin 330°
77. sin a
78. cos a
5p
b
6
80. sin(135°)
Find all of the exact values of ␪ that make the equation true
in the indicated interval.
81. sin u 12, 0 u 2p
82. cos u 12, 0 u 2p
83. tan u 0, 0 u 4p
84. sin u 1, 0 u 4p
Technology Exercises
Section 3.1
Find the measure (in degrees, minutes, and nearest seconds) of
a central angle ␪ that intercepts an arc on a circle with radius
r with indicated arc length s. Use the TI calculator commands
ANGLE and DMS to change to degrees, minutes, and seconds.
85. r 11.2 ft, s 19.7 ft
86. r 56.9 cm, s 139.2 cm
Section 3.4
For Exercises 87 and 88, refer to the following:
A graphing calculator can be used to graph the unit circle with
parametric equations (these will be covered in more detail in
Section 8.3). For now, set the calculator in parametric and radian
modes and let
X1 cos T
Y1 sin T
p
, 2 X 2,
15
and 2 Y 2. To approximate the sine or cosines of a T value,
use the TRACE key, enter the T value, and read the corresponding
coordinates from the screen.
13p
87. Use the above steps to approximate cos a
b to four
12
decimal places.
5p
88. Use the above steps to approximate sin a b to four decimal
6
places.
Set the window so that 0 T 2p, step REVIEW EXERCISES
79. cos(240°)
5p
b
4
175
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CHAPTER 3
Page 176
P R ACTI C E TE ST
1. Find the measure (in radians) of a central angle u that
intercepts an arc on a circle with radius r 20 centimeters
and arc length s 4 millimeters.
2. Convert
13p
to degree measure.
4
3. Convert 260° to radian measure. Leave the answer exact in
terms of p.
4. Convert 217° to radian measure. Round to two decimal
places.
7p
5. What is the reference angle to u ?
12
6. Find the radius of the minute hand on a clock if a point on
the end travels 10 centimeters in 20 minutes.
7. Betty is walking around a circular walking path. If the
radius of the path is 0.50 miles and she has walked
through an angle of 120°, how far has she walked?
P R ACTI C E TE ST
8. Calculate the arc length on a circle with central angle
p
and radius r 8 yards.
u
15
13. Layla is building an ornamental wall that is in the shape of
a piece of a circle 12 feet in diameter. If the central angle
of the circle is 40, how long is the rock wall?
14. A blueberry pie is made in a 9-inch-diameter pie pan. If a
1-inch-radius circle is cut out of the middle for decoration,
what is the area of each piece of pie if the pie is cut into
8 equal pieces?
15. Tom’s hands go in a 9-inch-radius circular pattern as he
rows his boat across a lake. If his hands make a complete
rotation every 1.5 seconds, what are the angular speed and
linear speed of his hands?
In Exercises 16–20, if possible, find the exact value of the
indicated trigonometric function using the unit circle.
16. sin a
7p
b
6
19. cot a
3p
b
2
17. tan a
7p
b
4
20. sec a
18. csc a
3p
b
4
7p
b
2
21. What is the measure in radians of the smaller angle
between the hour and minute hands at 10:10?
9. A sprinkler has a 25-foot spray and it covers an angle of
30°. What is the area that the sprinkler waters? Round to
the nearest square foot.
22. Find all of the exact values of u that make the equation
10. A bicycle with tires of radius r 15 inches is being
ridden by a boy at a constant speed—the tires are making
five rotations per second. How many miles will he ride in
15 minutes? (1 mi 5280 ft)
23. Find all of the exact values of u that make the equation
11. The smaller gear in the diagram below has a radius of
2 centimeters, and the larger gear has a radius of
5.2 centimeters. If the smaller gear rotates 135°, how
many degrees has the larger gear rotated? Round
answer to the nearest degree.
sin u tan u 23
true in the interval 0 u 2p.
2
23
true in the interval 0 u 2p.
3
24. Sales of a seasonal product s vary according to the time of
year sold given as t. If the equation that models sales is
s 500 125 cos a
pt
b, what were the sales in March
6
(t 3) ?
12. Samuel rides 55 feet on a merry-go-round that is 10 feet in
diameter in a clockwise direction. Through what angle has
Samuel rotated?
176
25. The manager of a 24-hour plant tracks productivity
throughout the day and finds that the equation
p
p
p 50 12 cos a t b accurately models output
12
4
p from his workers at time t, where p is the number of
units produced by the workers and t is the time in hours
after midnight. What is the plant’s output at 5:00 in the
evening?
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C U M U L AT I V E T E S T
C H A P T E R S 1–3
1. In a triangle with angles a, b, and g, and
a b g 180°, if a 115°, b 25°, find g.
10. Given the angle 99.99° in standard position, state the
quadrant of this angle.
2. In a 30°-60°-90° triangle, if the hypotenuse is 24 meters,
what are the lengths of the two legs?
11. Given the angle 270° in standard position, find the axis
of this angle.
3. If D (9x 6)° and G (7x 2)° in the diagram
below, find the measures D and G.
12. The angle u in standard position has the terminal side
defined by the line 3x 2y 0, x 0. Calculate the
values for the six trigonometric functions of u.
m||n
A B
C D
E F
G H
13. Given the angle u 900° in standard position, calculate,
if possible, the values for the six trigonometric functions
of u.
m
9
14. If cos u 41
, and the terminal side of u lies in
quadrant III, find csc u.
n
15. Evaluate the expression sin 540° sec(540°) .
4. Height of a Woman. If a 6-foot volleyball player has a
1-foot 4-inch shadow, how long a shadow will her 4-foot
6-inch daughter cast?
16. Find the positive measure of u (rounded to the nearest
degree) if tan u 1.4285 and the terminal side of u lies in
quadrant III.
5. Use the triangle below to find cos u.
17. Given cot u 25
, use the reciprocal identity to find tan u.
3
15
12
19. Find sin u and cos u if tan u 6 and the terminal side
of u lies in quadrant III.
␪
9
6. Write csc 30° in terms of its cofunction.
7. Perform the operation ⬔B ⬔A, where ⬔A 9° 24r 15s
and ⬔B 74° 13r 29s .
8. Use a calculator to approximate sec (78° 25r) . Round the
answer to four decimal places.
9. Given a 37.4° and a 132 miles, use the right triangle
diagram to solve the right triangle. Write the answer for
angle measures in decimal degrees.
␣
c
b
20. Find the measure (in radians) of a central angle u that
intercepts an arc on a circle of radius r 1.6 centimeters
with arc length s 4 millimeters.
21. Clock. How many radians does the second hand of a clock
turn in 1 minute, 45 seconds?
9p
22. Find the exact length of the radius with arc length s 7
2p
meters and central angle u .
7
23. Find the distance traveled (arc length) of a point that moves
with constant speed v 2.6 meters per second along a circle
in 3.3 seconds.
24. Bicycle. How fast is a bicyclist traveling in miles per hour
if his tires are 24 inches in diameter and his angular speed
is 5p radians per second?
25. Find all of the exact values of u, when tan u 1 and
0 u 2p.
␤
a
177
C U M U L AT I V E T E S T
18. If cos u 16 and the terminal side of u lies in quadrant IV,
find sin u.