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4
Trigonometry
4.1
Radian and Degree Measure
4.2
Trigonometric Functions: The Unit Circle
4.3
Right Triangle Trigonometry
4.4
Trigonometric Functions of Any Angle
4.5
Graphs of Sine and Cosine Functions
4.6
Graphs of Other Trigonometric Functions
4.7
Inverse Trigonometric Functions
4.8
Applications and Models
In Mathematics
Trigonometry is used to find relationships
between the sides and angles of triangles,
and to write trigonometric functions as
models of real-life quantities.
In Real Life
Andre Jenny/Alamy
Trigonometric functions are used to model
quantities that are periodic. For instance,
throughout the day, the depth of water at
the end of a dock in Bar Harbor, Maine
varies with the tides. The depth can be
modeled by a trigonometric function.
(See Example 7, page 325.)
IN CAREERS
There are many careers that use trigonometry. Several are listed below.
• Biologist
Exercise 70, page 308
• Mechanical Engineer
Exercise 95, page 339
• Meteorologist
Exercise 99, page 318
• Surveyor
Exercise 41, page 359
279
280
Chapter 4
Trigonometry
4.1 RADIAN AND DEGREE MEASURE
What you should learn
•
•
•
•
Describe angles.
Use radian measure.
Use degree measure.
Use angles to model and solve
real-life problems.
Why you should learn it
You can use angles to model and solve
real-life problems. For instance, in
Exercise 119 on page 291, you are
asked to use angles to find the speed
of a bicycle.
Angles
As derived from the Greek language, the word trigonometry means “measurement of
triangles.” Initially, trigonometry dealt with relationships among the sides and angles of
triangles and was used in the development of astronomy, navigation, and surveying.
With the development of calculus and the physical sciences in the 17th century, a
different perspective arose—one that viewed the classic trigonometric relationships as
functions with the set of real numbers as their domains. Consequently, the applications
of trigonometry expanded to include a vast number of physical phenomena involving
rotations and vibrations. These phenomena include sound waves, light rays, planetary
orbits, vibrating strings, pendulums, and orbits of atomic particles.
The approach in this text incorporates both perspectives, starting with angles and
their measure.
y
e
m
Ter
id
al s
Terminal
side
in
Vertex
Initial side
Ini
tia
l si
de
© Wolfgang Rattay/Reuters/Corbis
Angle
FIGURE
x
Angle in standard position
FIGURE 4.2
4.1
An angle is determined by rotating a ray (half-line) about its endpoint. The starting
position of the ray is the initial side of the angle, and the position after rotation is the
terminal side, as shown in Figure 4.1. The endpoint of the ray is the vertex of the
angle. This perception of an angle fits a coordinate system in which the origin is the
vertex and the initial side coincides with the positive x-axis. Such an angle is in
standard position, as shown in Figure 4.2. Positive angles are generated by counterclockwise rotation, and negative angles by clockwise rotation, as shown in Figure 4.3.
Angles are labeled with Greek letters ! (alpha), (beta), and " (theta), as well as uppercase letters A, B, and C. In Figure 4.4, note that angles ! and
have the same
initial and terminal sides. Such angles are coterminal.
y
y
Positive angle
(counterclockwise)
y
α
x
FIGURE
4.3
α
x
Negative angle
(clockwise)
β
FIGURE
4.4 Coterminal angles
β
x
Section 4.1
y
Radian and Degree Measure
281
Radian Measure
The measure of an angle is determined by the amount of rotation from the initial side
to the terminal side. One way to measure angles is in radians. This type of measure is
especially useful in calculus. To define a radian, you can use a central angle of a
circle, one whose vertex is the center of the circle, as shown in Figure 4.5.
s=r
r
θ
r
x
Definition of Radian
One radian is the measure of a central angle " that intercepts an arc s equal in
length to the radius r of the circle. See Figure 4.5. Algebraically, this means that
Arc length radius when " 1 radian
FIGURE 4.5
"
s
r
where " is measured in radians.
y
2 radians
Because the circumference of a circle is 2 r units, it follows that a central angle
of one full revolution (counterclockwise) corresponds to an arc length of
r
r
3
radians
r
r
r
4 radians r
FIGURE
s 2 r.
1 radian
6
radians
x
5 radians
4.6
Moreover, because 2 6.28, there are just over six radius lengths in a full circle, as
shown in Figure 4.6. Because the units of measure for s and r are the same, the ratio sr
has no units—it is simply a real number.
Because the radian measure of an angle of one full revolution is 2, you can obtain
the following.
2
1
revolution radians
2
2
1
2 revolution radians
4
4
2
2 1
revolution radians
6
6
3
These and other common angles are shown in Figure 4.7.
One revolution around a circle
of radius r corresponds to an
angle of 2 radians because
s 2r
" 2 radians.
r
r
π
6
π
4
π
2
π
FIGURE
π
3
2π
4.7
Recall that the four quadrants in a coordinate system are numbered I, II, III, and
IV. Figure 4.8 on page 282 shows which angles between 0 and 2 lie in each of the four
quadrants. Note that angles between 0 and 2 are acute angles and angles between
2 and are obtuse angles.
282
Chapter 4
Trigonometry
π
θ=
2
Quadrant II
π < <
θ π
2
Quadrant I
0 <θ < π
2
θ=0
θ =π
Quadrant III Quadrant IV
3π 3π < <
θ 2π
π <θ<
2 2
The phrase “the terminal side
of " lies in a quadrant” is often
abbreviated by simply saying
that “" lies in a quadrant.” The
terminal sides of the “quadrant
angles” 0, 2, , and 32 do
not lie within quadrants.
3π
θ=
2
FIGURE
4.8
Two angles are coterminal if they have the same initial and terminal sides. For
instance, the angles 0 and 2 are coterminal, as are the angles 6 and 136. You can
find an angle that is coterminal to a given angle " by adding or subtracting 2 (one
revolution), as demonstrated in Example 1. A given angle " has infinitely many
coterminal angles. For instance, " 6 is coterminal with
2n
6
where n is an integer.
You can review operations
involving fractions in Appendix
A.1.
Example 1
Sketching and Finding Coterminal Angles
a. For the positive angle 136, subtract 2 to obtain a coterminal angle
13
2 .
6
6
See Figure 4.9.
b. For the positive angle 34, subtract 2 to obtain a coterminal angle
3
5
2 .
4
4
See Figure 4.10.
c. For the negative angle 23, add 2 to obtain a coterminal angle
2
4
2 .
3
3
See Figure 4.11.
π
2
θ = 13π
6
π
2
π
6 0
π
π
2
θ = 3π
4
π
4π
3
0
π
0
− 5π
4
3π
2
FIGURE
4.9
θ = − 2π
3
3π
2
FIGURE
Now try Exercise 27.
4.10
3π
2
FIGURE
4.11
Section 4.1
Radian and Degree Measure
283
Two positive angles ! and are complementary (complements of each other) if
their sum is 2. Two positive angles are supplementary (supplements of each other)
if their sum is . See Figure 4.12.
β
β
α
Complementary angles
FIGURE 4.12
Example 2
α
Supplementary angles
Complementary and Supplementary Angles
If possible, find the complement and the supplement of (a) 25 and (b) 45.
Solution
a. The complement of 25 is
2 5 4
.
2
5
10
10
10
The supplement of 25 is
2 5 2 3
.
5
5
5
5
b. Because 45 is greater than 2, it has no complement. (Remember that complements
are positive angles.) The supplement is
4 5 4 .
5
5
5
5
Now try Exercise 31.
Degree Measure
y
120°
135°
150°
90° = 41 (360°)
60° = 16 (360°)
45° = 18 (360°)
1
30° = 12
(360°)
θ
180°
0°
360°
210°
330°
225°
315°
240° 270° 300°
FIGURE
4.13
x
A second way to measure angles is in terms of degrees, denoted by the symbol #.
1
A measure of one degree (1#) is equivalent to a rotation of 360
of a complete revolution
about the vertex. To measure angles, it is convenient to mark degrees on the
circumference of a circle, as shown in Figure 4.13. So, a full revolution (counterclockwise) corresponds to 360#, a half revolution to 180#, a quarter revolution to 90#, and
so on.
Because 2 radians corresponds to one complete revolution, degrees and radians
are related by the equations
360# 2 rad
and
180# rad.
From the latter equation, you obtain
1# rad
180
and
1 rad 180#
which lead to the conversion rules at the top of the next page.
284
Chapter 4
Trigonometry
Conversions Between Degrees and Radians
rad
.
180#
180#
.
2. To convert radians to degrees, multiply radians by
rad
To apply these two conversion rules, use the basic relationship rad 180#.
(See Figure 4.14.)
1. To convert degrees to radians, multiply degrees by
π
6
30°
π
4
45°
π
2
90°
180°
FIGURE
π
π
3
60°
2π
360°
4.14
When no units of angle measure are specified, radian measure is implied. For
instance, if you write " 2, you imply that " 2 radians.
Example 3
3
rad
radians
180
deg 4
rad
b. 540# 540 deg
3 radians
180 deg rad
3
c. 270# 270 deg
radians
180 deg 2
a. 135# 135 deg
T E C H N O LO G Y
With calculators it is convenient
to use decimal degrees to denote
fractional parts of degrees.
Historically, however, fractional
parts of degrees were expressed
in minutes and seconds, using
the prime (% ) and double prime
(& ) notations, respectively. That is,
1% ! one minute !
Converting from Degrees to Radians
1
60 !1#"
1
1& ! one second ! 3600!1#".
Consequently, an angle of
64 degrees, 32 minutes, and
47 seconds is represented by
! 64# 32% 47&. Many calculators
have special keys for converting
an angle in degrees, minutes,
and seconds D# M% S& to decimal
degree form, and vice versa.
Multiply by 180.
Multiply by 180.
Multiply by 180.
Now try Exercise 57.
Example 4
Converting from Radians to Degrees
180 deg
90#
rad rad
2
2
rad
9
9
180 deg
b.
810#
rad rad
2
2
rad
180 deg
360#
c. 2 rad 2 rad
114.59#
rad
a. Multiply by 180.
Multiply by 180.
Multiply by 180.
Now try Exercise 61.
If you have a calculator with a “radian-to-degree” conversion key, try using it to
verify the result shown in part (b) of Example 4.
Section 4.1
Radian and Degree Measure
285
Applications
The radian measure formula, " sr, can be used to measure arc length along a circle.
Arc Length
For a circle of radius r, a central angle " intercepts an arc of length s given by
s
s r"
θ = 240°
Length of circular arc
where " is measured in radians. Note that if r 1, then s ", and the radian
measure of " equals the arc length.
r=4
Example 5
FIGURE
4.15
Finding Arc Length
A circle has a radius of 4 inches. Find the length of the arc intercepted by a central angle
of 240#, as shown in Figure 4.15.
Solution
To use the formula s r", first convert 240# to radian measure.
240# 240 deg
rad
180
deg 4
radians
3
Then, using a radius of r 4 inches, you can find the arc length to be
s r"
4
43
16
3
16.76 inches.
Note that the units for r" are determined by the units for r because " is given in radian
measure, which has no units.
Now try Exercise 89.
Linear speed measures how fast
the particle moves, and angular
speed measures how fast the
angle changes. By dividing the
formula for arc length by t,
you can establish a relationship
between linear speed v and
angular speed ', as shown.
s r"
s r"
t
t
v r'
The formula for the length of a circular arc can be used to analyze the motion of a
particle moving at a constant speed along a circular path.
Linear and Angular Speeds
Consider a particle moving at a constant speed along a circular arc of radius r. If s
is the length of the arc traveled in time t, then the linear speed v of the particle is
Linear speed v arc length s
.
time
t
Moreover, if " is the angle (in radian measure) corresponding to the arc length s,
then the angular speed ' (the lowercase Greek letter omega) of the particle is
Angular speed ' central angle "
.
time
t
286
Chapter 4
Trigonometry
Example 6
10.2 cm
Finding Linear Speed
The second hand of a clock is 10.2 centimeters long, as shown in Figure 4.16. Find the
linear speed of the tip of this second hand as it passes around the clock face.
Solution
In one revolution, the arc length traveled is
s 2r
2 10.2
FIGURE
Substitute for r.
20.4 centimeters.
4.16
The time required for the second hand to travel this distance is
t 1 minute 60 seconds.
So, the linear speed of the tip of the second hand is
Linear speed 116
ft
s
t
20.4 centimeters
60 seconds
1.068 centimeters per second.
Now try Exercise 111.
Example 7
Finding Angular and Linear Speeds
The blades of a wind turbine are 116 feet long (see Figure 4.17). The propeller rotates
at 15 revolutions per minute.
a. Find the angular speed of the propeller in radians per minute.
b. Find the linear speed of the tips of the blades.
Solution
FIGURE
4.17
a. Because each revolution generates 2 radians, it follows that the propeller turns
152 30 radians per minute. In other words, the angular speed is
Angular speed "
t
30 radians
30 radians per minute.
1 minute
b. The linear speed is
Linear speed s
t
r"
t
11630 feet
1 minute
Now try Exercise 113.
10,933 feet per minute.
Section 4.1
Radian and Degree Measure
287
A sector of a circle is the region bounded by two radii of the circle and their
intercepted arc (see Figure 4.18).
θ
FIGURE
r
4.18
Area of a Sector of a Circle
For a circle of radius r, the area A of a sector of the circle with central angle " is
given by
1
A r 2"
2
where " is measured in radians.
Example 8
Area of a Sector of a Circle
A sprinkler on a golf course fairway sprays water over a distance of 70 feet and rotates
through an angle of 120# (see Figure 4.19). Find the area of the fairway watered by the
sprinkler.
Solution
First convert 120# to radian measure as follows.
120°
70 ft
" 120#
120 deg
FIGURE
4.19
rad
180
deg Multiply by 180.
2
radians
3
Then, using " 23 and r 70, the area is
1
A r 2"
2
Formula for the area of a sector of a circle
2
1
702
2
3
Substitute for r and ".
4900
3
Simplify.
5131 square feet.
Simplify.
Now try Exercise 117.
288
Chapter 4
4.1
Trigonometry
EXERCISES
See www.CalcChat.com for worked-out solutions to odd-numbered exercises.
VOCABULARY: Fill in the blanks.
1. ________ means “measurement of triangles.”
2. An ________ is determined by rotating a ray about its endpoint.
3. Two angles that have the same initial and terminal sides are ________.
4. One ________ is the measure of a central angle that intercepts an arc equal to the radius of the circle.
5. Angles that measure between 0 and 2 are ________ angles, and angles that measure between 2 and are ________ angles.
6. Two positive angles that have a sum of 2 are ________ angles, whereas two positive angles that have a sum
of are ________ angles.
1
7. The angle measure that is equivalent to a rotation of 360
of a complete revolution about an angle’s vertex is
one ________.
8. 180 degrees ________ radians.
9. The ________ speed of a particle is the ratio of arc length to time traveled, and the ________ speed of a particle
is the ratio of central angle to time traveled.
10. The area A of a sector of a circle with radius r and central angle ", where " is measured in radians, is given by
the formula ________.
SKILLS AND APPLICATIONS
In Exercises 11–16, estimate the angle to the nearest one-half
radian.
11.
12.
11
6
26. (a) 4
(b) 3
25. (a)
(b) 7
In Exercises 27–30, determine two coterminal angles (one
positive and one negative) for each angle. Give your answers
in radians.
13.
π
2
27. (a)
14.
π
2
(b)
5
θ= π
6
θ= π
6
π
15.
16.
π
2
In Exercises 17–22, determine the quadrant in which each
angle lies. (The angle measure is given in radians.)
4
6
21. (a) 3.5
19. (a) (b)
5
4
(b) 3
(b) 2.25
18. (a)
11
8
20. (a) 5
6
22. (a) 6.02
(b)
9
8
(b) In Exercises 23–26, sketch each angle in standard position.
23. (a)
3
(b) 2
3
24. (a) 7
4
(b)
5
2
π
2
(b)
7
θ= π
6
π
0
π
29. (a) " 2
3
30. (a) " 9
4
0
3π
2
3π
2
11
9
(b) 4.25
0
3π
2
3π
2
28. (a)
17. (a)
π
0
(b) " θ = − 11π
6
12
(b) " 2
15
Section 4.1
289
Radian and Degree Measure
In Exercises 31–34, find (if possible) the complement and
supplement of each angle.
52. (a) " 390#
31. (a) 3
(b) 4
32. (a) 12
(b) 1112
33. (a) 1
(b) 2
34. (a) 3
(b) 1.5
In Exercises 53–56, find (if possible) the complement and
supplement of each angle.
In Exercises 35–40, estimate the number of degrees in the
angle. Use a protractor to check your answer.
35.
53. (a) 18#
55. (a) 150#
57. (a) 30#
59. (a) 20#
38.
39.
(b) 85#
(b) 79#
54. (a) 46#
56. (a) 130#
(b) 93#
(b) 170#
In Exercises 57–60, rewrite each angle in radian measure as a
multiple of ". (Do not use a calculator.)
36.
37.
(b) " 230#
(b) 45#
(b) 60#
58. (a) 315#
(b) 120#
60. (a) 270# (b) 144#
In Exercises 61–64, rewrite each angle in degree measure.
(Do not use a calculator.)
40.
61. (a)
3
2
(b)
7
6
63. (a)
5
4
(b) 7
3
62. (a) 64. (a)
7
12
11
6
(b)
9
(b)
34
15
In Exercises 41–44, determine the quadrant in which each
angle lies.
In Exercises 65–72, convert the angle measure from degrees
to radians. Round to three decimal places.
41. (a) 130#
42. (a) 8.3#
43. (a) 132# 50%
(b) 285#
(b) 257# 30%
(b) 336#
65. 45#
67. 216.35#
66. 87.4#
68. 48.27#
44. (a) 260#
(b) 3.4#
69. 532#
71. 0.83#
70. 345#
72. 0.54#
In Exercises 45–48, sketch each angle in standard position.
45. (a) 90#
47. (a) 30#
(b) 180# 46. (a) 270#
(b) 135#
48. (a) 750#
(b) 600#
(b) 120#
In Exercises 49–52, determine two coterminal angles (one
positive and one negative) for each angle. Give your answers
in degrees.
49. (a)
(b)
90°
90°
θ = 45°
180°
0
180°
0
θ = − 36°
270°
50. (a)
270°
(b)
90°
θ = 120°
180°
0
270°
51. (a) " 240#
90°
θ = − 420°
180°
0
270°
(b) " 180#
In Exercises 73–80, convert the angle measure from radians
to degrees. Round to three decimal places.
73.
75.
77.
79.
7
158
4.2
2
74.
76.
78.
80.
511
132
4.8
0.57
In Exercises 81–84, convert each angle measure to decimal
degree form without using a calculator. Then check your
answers using a calculator.
81. (a) 54# 45%
82. (a) 245# 10%
83. (a) 85# 18% 30&
84. (a) 135# 36&
(b) 128# 30%
(b) 2# 12%
(b) 330# 25&
(b) 408# 16% 20&
In Exercises 85–88, convert each angle measure to degrees,
minutes, and seconds without using a calculator. Then check
your answers using a calculator.
85. (a) 240.6#
86. (a) 345.12#
(b) 145.8#
(b) 0.45#
87. (a) 2.5#
88. (a) 0.36#
(b) 3.58#
(b) 0.79#
290
Chapter 4
Trigonometry
In Exercises 89–92, find the length of the arc on a circle of
radius r intercepted by a central angle ".
Central Angle "
Radius r
89. 15 inches
90. 9 feet
91. 3 meters
120#
60#
150#
92. 20 centimeters
45#
City
106. San Francisco, California
Seattle, Washington
In Exercises 93–96, find the radian measure of the central
angle of a circle of radius r that intercepts an arc of length s.
Radius r
93. 4 inches
Arc Length s
18 inches
94. 14 feet
95. 25 centimeters
96. 80 kilometers
8 feet
10.5 centimeters
150 kilometers
Latitude
37# 47% 36& N
47# 37% 18& N
107. DIFFERENCE IN LATITUDES Assuming that Earth
is a sphere of radius 6378 kilometers, what is the
difference in the latitudes of Syracuse, New York
and Annapolis, Maryland, where Syracuse is about
450 kilometers due north of Annapolis?
108. DIFFERENCE IN LATITUDES Assuming that Earth
is a sphere of radius 6378 kilometers, what is the
difference in the latitudes of Lynchburg, Virginia and
Myrtle Beach, South Carolina, where Lynchburg is
about 400 kilometers due north of Myrtle Beach?
109. INSTRUMENTATION The pointer on a voltmeter is
6 centimeters in length (see figure). Find the angle
through which the pointer rotates when it moves
2.5 centimeters on the scale.
In Exercises 97–100, use the given arc length and radius to
find the angle " (in radians).
97.
98.
θ
2
99.
10 in.
25
1
θ
10
6 cm
100.
28
2 ft
Not drawn to scale
θ
75
θ
7
FIGURE FOR
60
In Exercises 101–104, find the area of the sector of the circle
with radius r and central angle ".
Radius r
101. 6 inches
102. 12 millimeters
103. 2.5 feet
104. 1.4 miles
Central Angle "
3
4
225#
330#
DISTANCE BETWEEN CITIES In Exercises 105 and 106,
find the distance between the cities. Assume that Earth is a
sphere of radius 4000 miles and that the cities are on the
same longitude (one city is due north of the other).
City
105. Dallas, Texas
Omaha, Nebraska
Latitude
32# 47% 39& N
41# 15% 50& N
109
FIGURE FOR
110
110. ELECTRIC HOIST An electric hoist is being used to
lift a beam (see figure). The diameter of the drum on
the hoist is 10 inches, and the beam must be raised
2 feet. Find the number of degrees through which the
drum must rotate.
111. LINEAR AND ANGULAR SPEEDS A circular power
saw has a 714-inch-diameter blade that rotates at
5000 revolutions per minute.
(a) Find the angular speed of the saw blade in radians
per minute.
(b) Find the linear speed (in feet per minute) of one of
the 24 cutting teeth as they contact the wood being
cut.
112. LINEAR AND ANGULAR SPEEDS A carousel with
a 50-foot diameter makes 4 revolutions per minute.
(a) Find the angular speed of the carousel in radians
per minute.
(b) Find the linear speed (in feet per minute) of the
platform rim of the carousel.
Section 4.1
113. LINEAR AND ANGULAR SPEEDS The diameter of
a DVD is approximately 12 centimeters. The drive
motor of the DVD player is controlled to rotate
precisely between 200 and 500 revolutions per minute,
depending on what track is being read.
(a) Find an interval for the angular speed of a DVD as
it rotates.
(b) Find an interval for the linear speed of a point on
the outermost track as the DVD rotates.
114. ANGULAR SPEED A two-inch-diameter pulley on
an electric motor that runs at 1700 revolutions per
minute is connected by a belt to a four-inch-diameter
pulley on a saw arbor.
(a) Find the angular speed (in radians per minute) of
each pulley.
(b) Find the revolutions per minute of the saw.
115. ANGULAR SPEED A car is moving at a rate of
65 miles per hour, and the diameter of its wheels is 2 feet.
(a) Find the number of revolutions per minute the
wheels are rotating.
116.
117.
118.
119.
(b) Find the angular speed of the wheels in radians per
minute.
ANGULAR SPEED A computerized spin balance
machine rotates a 25-inch-diameter tire at 480 revolutions
per minute.
(a) Find the road speed (in miles per hour) at which
the tire is being balanced.
(b) At what rate should the spin balance machine be set
so that the tire is being tested for 55 miles per hour?
AREA A sprinkler on a golf green is set to spray
water over a distance of 15 meters and to rotate through
an angle of 140#. Draw a diagram that shows the region
that can be irrigated with the sprinkler. Find the area of
the region.
AREA A car’s rear windshield wiper rotates 125#.
The total length of the wiper mechanism is 25 inches
and wipes the windshield over a distance of 14 inches.
Find the area covered by the wiper.
SPEED OF A BICYCLE The radii of the pedal
sprocket, the wheel sprocket, and the wheel of the
bicycle in the figure are 4 inches, 2 inches, and
14 inches, respectively. A cyclist is pedaling at a rate
of 1 revolution per second.
14 in.
2 in.
4 in.
Radian and Degree Measure
291
(a) Find the speed of the bicycle in feet per second and
miles per hour.
(b) Use your result from part (a) to write a function
for the distance d (in miles) a cyclist travels in
terms of the number n of revolutions of the pedal
sprocket.
(c) Write a function for the distance d (in miles) a
cyclist travels in terms of the time t (in seconds).
Compare this function with the function from
part (b).
(d) Classify the types of functions you found in parts
(b) and (c). Explain your reasoning.
120. CAPSTONE Write a short paper in your own words
explaining the meaning of each of the following
concepts to a classmate.
(a) an angle in standard position
(b) positive and negative angles
(c) coterminal angles
(d) angle measure in degrees and radians
(e) obtuse and acute angles
(f) complementary and supplementary angles
EXPLORATION
TRUE OR FALSE? In Exercises 121–123, determine
whether the statement is true or false. Justify your answer.
121. A measurement of 4 radians corresponds to two
complete revolutions from the initial side to the
terminal side of an angle.
122. The difference between the measures of two
coterminal angles is always a multiple of 360# if
expressed in degrees and is always a multiple of 2
radians if expressed in radians.
123. An angle that measures 1260# lies in Quadrant III.
124. THINK ABOUT IT A fan motor turns at a given
angular speed. How does the speed of the tips of the
blades change if a fan of greater diameter is installed
on the motor? Explain.
125. THINK ABOUT IT Is a degree or a radian the larger
unit of measure? Explain.
126. WRITING If the radius of a circle is increasing and
the magnitude of a central angle is held constant, how
is the length of the intercepted arc changing? Explain
your reasoning.
127. PROOF Prove that the area of a circular sector of
1
radius r with central angle " is A 2" r 2, where " is
measured in radians.
292
Chapter 4
Trigonometry
4.2 TRIGONOMETRIC FUNCTIONS: THE UNIT CIRCLE
What you should learn
• Identify a unit circle and describe
its relationship to real numbers.
• Evaluate trigonometric functions
using the unit circle.
• Use the domain and period to
evaluate sine and cosine functions.
• Use a calculator to evaluate
trigonometric functions.
The Unit Circle
The two historical perspectives of trigonometry incorporate different methods for
introducing the trigonometric functions. Our first introduction to these functions is
based on the unit circle.
Consider the unit circle given by
x2 y 2 1
Unit circle
as shown in Figure 4.20.
Why you should learn it
y
(0, 1)
Trigonometric functions are used to
model the movement of an oscillating
weight. For instance, in Exercise 60
on page 298, the displacement from
equilibrium of an oscillating weight
suspended by a spring is modeled as
a function of time.
(−1, 0)
(1, 0)
x
(0, −1)
Richard Megna/Fundamental Photographs
FIGURE
4.20
Imagine that the real number line is wrapped around this circle, with positive numbers
corresponding to a counterclockwise wrapping and negative numbers corresponding to
a clockwise wrapping, as shown in Figure 4.21.
y
y
(x , y )
t
t>0
θ
(1, 0)
t<0
t
x
(1, 0)
t
(x , y)
FIGURE
x
θ
t
4.21
As the real number line is wrapped around the unit circle, each real number t
corresponds to a point x, y on the circle. For example, the real number 0 corresponds
to the point 1, 0. Moreover, because the unit circle has a circumference of 2, the real
number 2 also corresponds to the point 1, 0.
In general, each real number t also corresponds to a central angle " (in standard
position) whose radian measure is t. With this interpretation of t, the arc length
formula s r" (with r 1) indicates that the real number t is the (directional) length
of the arc intercepted by the angle ", given in radians.
Section 4.2
Trigonometric Functions: The Unit Circle
293
The Trigonometric Functions
From the preceding discussion, it follows that the coordinates x and y are two functions
of the real variable t. You can use these coordinates to define the six trigonometric
functions of t.
sine
cosecant
cosine
secant
tangent
cotangent
These six functions are normally abbreviated sin, csc, cos, sec, tan, and cot, respectively.
Definitions of Trigonometric Functions
Let t be a real number and let x, y be the point on the unit circle corresponding
to t.
Note in the definition at the
right that the functions in the
second row are the reciprocals
of the corresponding functions
in the first row.
sin t y
1
csc t ,
y
y
(0, 1)
(−
2
,
2
2
2
(−1, 0)
(−
2
,
2
FIGURE
2
2
−
(
)
)
2
,
2
2
2
x
(1, 0)
(
(0, −1)
2
,
2
)
−
2
2
)
4.22
y0
cos t x
y
tan t ,
x
1
sec t , x 0
x
x
cot t , y 0
y
In the definitions of the trigonometric functions, note that the tangent and secant
are not defined when x 0. For instance, because t 2 corresponds to
x, y 0, 1, it follows that tan2 and sec2 are undefined. Similarly, the
cotangent and cosecant are not defined when y 0. For instance, because t 0
corresponds to x, y 1, 0, cot 0 and csc 0 are undefined.
In Figure 4.22, the unit circle has been divided into eight equal arcs, corresponding
to t-values of
3
5 3 7
0, , , , , , , , and 2.
4 2 4
4 2 4
Similarly, in Figure 4.23, the unit circle has been divided into 12 equal arcs,
corresponding to t-values of
2 5
7 4 3 5 11
0, , , , , , , , , , ,
, and 2.
6 3 2 3 6
6 3 2 3 6
To verify the points on the unit circle in Figure 4.22, note that
y
(
− 21 ,
(−
3 1
, 2
2
3
2
)
(−1, 0)
(−
3
,
2
− 21
( 21 , 23 )
( 23 , 21 )
(1, 0)
4.23
22, 22 also lies
on the line y x. So, substituting x for y in the equation of the unit circle produces
the following.
x
x2 x2 1
( 21 , − 23 )
3 (0, −1)
2 )
( 23 , − 21 )
2x2 1
x2 Because the point is in the first quadrant, x have y )
(− 21 , −
FIGURE
)
(0, 1)
x0
2
1
2
2
2
x±
2
2
and because y x, you also
. You can use similar reasoning to verify the rest of the points in
2
Figure 4.22 and the points in Figure 4.23.
Using the x, y coordinates in Figures 4.22 and 4.23, you can evaluate the trigonometric functions for common t-values. This procedure is demonstrated in Examples 1,
2, and 3. You should study and learn these exact function values for common t-values
because they will help you in later sections to perform calculations.
294
Chapter 4
Trigonometry
Example 1
You can review dividing
fractions and rationalizing
denominators in Appendix A.1
and Appendix A.2, respectively.
Evaluating Trigonometric Functions
Evaluate the six trigonometric functions at each real number.
5
a. t b. t c. t 0
d. t 6
4
Solution
For each t-value, begin by finding the corresponding point x, y on the unit circle. Then
use the definitions of trigonometric functions listed on page 293.
a. t b. t 3 1
corresponds to the point x, y , .
6
2 2
sin
1
y
6
2
csc
1
1
2
6
y
12
cos
3
x
6
2
sec
1
23
2
3
6
x
3
tan
3
y
12
1
6
x 32 3
3
cot
x 32
3
6
y
12
2
2
5
corresponds to the point x, y ,
.
4
2
2
sin
2
5
y
4
2
csc
5 1
2
2
2
4
y
cos
2
5
x
4
2
sec
2
5 1
2
2
4
x
tan
5 y 22
1
4
x 22
cot
5 x 22
1
4
y 22
c. t 0 corresponds to the point x, y 1, 0.
sin 0 y 0
csc 0 1
is undefined.
y
cos 0 x 1
sec 0 1 1
1
x
1
cot 0 x
is undefined.
y
y 0
0
x 1
tan 0 d. t corresponds to the point x, y 1, 0.
sin y 0
csc 1
is undefined.
y
cos x 1
sec 1
1
1
x
1
cot x
is undefined.
y
tan y
0
0
x 1
Now try Exercise 23.
Section 4.2
Example 2
295
Trigonometric Functions: The Unit Circle
Evaluating Trigonometric Functions
Evaluate the six trigonometric functions at t .
3
Solution
Moving clockwise around the unit circle, it follows that t 3 corresponds to the
point x, y 12, 32.
3 2
csc 3 2
sec 3 cot sin cos tan 3 3 3
2
23
3
3 2
1
32
3
12
3 32 3 3
12
1
3
Now try Exercise 33.
Domain and Period of Sine and Cosine
y
(0, 1)
(1, 0)
(−1, 0)
x
−1 ≤ y ≤ 1
The domain of the sine and cosine functions is the set of all real numbers. To determine
the range of these two functions, consider the unit circle shown in Figure 4.24. By
definition, sin t y and cos t x. Because x, y is on the unit circle, you know that
1 ( y ( 1 and 1 ( x ( 1. So, the values of sine and cosine also range between
1 and 1.
1 (
(1
1 ( sin t ( 1
(0, −1)
and
1 (
x
(1
1 ( cos t ( 1
Adding 2 to each value of t in the interval 0, 2 completes a second revolution
around the unit circle, as shown in Figure 4.25. The values of sint 2 and
cost 2 correspond to those of sin t and cos t. Similar results can be obtained for
repeated revolutions (positive or negative) on the unit circle. This leads to the general
result
−1 ≤ x ≤ 1
FIGURE
y
4.24
sint 2 n sin t
and
t=
t=
3π 3π
4, 4
π π
,
2 2
+ 2π , π2 + 4π, ...
y
+ 2π , ...
t=
π π
,
4 4
+ 2π , ...
cost 2 n cos t
for any integer n and real number t. Functions that behave in such a repetitive (or cyclic)
manner are called periodic.
t = π, 3π, ...
x
t = 0, 2π, ...
t=
5π 5π
4, 4
+ 2π , ...
t=
FIGURE
4.25
3π 3π
,
2 2
t = 74π , 74π + 2π , ...
+ 2π , 32π + 4π, ...
Definition of Periodic Function
A function f is periodic if there exists a positive real number c such that
f t c f t
for all t in the domain of f. The smallest number c for which f is periodic is
called the period of f.
296
Chapter 4
Trigonometry
Recall from Section 1.5 that a function f is even if f t f t, and is odd if
f t f t.
Even and Odd Trigonometric Functions
The cosine and secant functions are even.
cost cos t
sect sec t
The sine, cosecant, tangent, and cotangent functions are odd.
sint sin t
csct csc t
tant tan t
cott cot t
Example 3
1
13
13
sin .
2 , you have sin
sin 2 6
6
6
6
6
2
a. Because
From the definition of periodic
function, it follows that the
sine and cosine functions are
periodic and have a period of
2. The other four trigonometric
functions are also periodic, and
will be discussed further in
Section 4.6.
Using the Period to Evaluate the Sine and Cosine
b. Because 7
4 , you have
2
2
cos 7
cos 4 cos 0.
2
2
2
4
4
c. For sin t , sint because the sine function is odd.
5
5
Now try Exercise 37.
Evaluating Trigonometric Functions with a Calculator
T E C H N O LO G Y
When evaluating trigonometric
functions with a calculator,
remember to enclose all fractional
angle measures in parentheses.
For instance, if you want to
evaluate sin t for t ! "/6, you
should enter
SIN
)
6
ENTER
.
These keystrokes yield the correct
value of 0.5. Note that some
calculators automatically place a
left parenthesis after trigonometric
functions. Check the user’s guide
for your calculator for specific
keystrokes on how to evaluate
trigonometric functions.
When evaluating a trigonometric function with a calculator, you need to set the
calculator to the desired mode of measurement (degree or radian).
Most calculators do not have keys for the cosecant, secant, and cotangent
functions. To evaluate these functions, you can use the x 1 key with their respective
reciprocal functions sine, cosine, and tangent. For instance, to evaluate csc8, use
the fact that
csc
1
8
sin8
and enter the following keystroke sequence in radian mode.
SIN
Example 4
Function
2
a. sin
3
b. cot 1.5
)
8
x 1
ENTER
Display 2.6131259
Using a Calculator
Mode
Calculator Keystrokes
Radian
SIN
Radian
TAN
Now try Exercise 55.
2 3
)
1.5
Display
ENTER
0.8660254
x 1
0.0709148
ENTER
Section 4.2
EXERCISES
4.2
Trigonometric Functions: The Unit Circle
297
See www.CalcChat.com for worked-out solutions to odd-numbered exercises.
VOCABULARY: Fill in the blanks.
1. Each real number t corresponds to a point x, y on the ________ ________.
2. A function f is ________ if there exists a positive real number c such that f t c f t for all t in the domain of f.
3. The smallest number c for which a function f is periodic is called the ________ of f.
4. A function f is ________ if f t f t and ________ if f t f t.
SKILLS AND APPLICATIONS
In Exercises 5–8, determine the exact values of the six
trigonometric functions of the real number t.
y
5.
6.
(
12 5
,
13 13
(
y
(− 178 , 1715 (
t
θ
θ t x
11
6
3
25. t 2
24. t 23. t 26. t 2
In Exercises 27–34, evaluate (if possible) the six trigonometric
functions of the real number.
x
2
3
4
29. t 3
5
6
7
30. t 4
28. t 27. t y
7.
y
8.
t
t
θ
θ
x
(
(− 45 , − 35(
12
,
13
5
− 13
(
In Exercises 9–16, find the point x, y on the unit circle that
corresponds to the real number t.
2
11. t 4
5
13. t 6
4
15. t 3
9. t 3
4
31. t x
33. t 32. t 2
34. t In Exercises 35– 42, evaluate the trigonometric function
using its period as an aid.
35. sin 4
36. cos 3
37. cos
7
3
38. sin
9
4
3
3
14. t 4
5
16. t 3
39. cos
17
4
40. sin
19
6
12. t 41. sin 8
3
6
7
21. t 4
19. t 18. t 3
4
4
22. t 3
20. t 42. cos 9
4
In Exercises 43–48, use the value of the trigonometric
function to evaluate the indicated functions.
43. sin t 2
(a) sint
44. sint 8
(a) sin t
(b) csct
1
45. cost 5
(a) cos t
(b) sect
(b) csc t
3
46. cos t 4
(a) cost
(b) sect
1
4
3
2
10. t In Exercises 17–26, evaluate (if possible) the sine, cosine, and
tangent of the real number.
17. t 5
3
4
47. sin t 5
(a) sin t
(b) sint 3
4
48. cos t 5
(a) cos t
(b) cost 298
Chapter 4
Trigonometry
In Exercises 49–58, use a calculator to evaluate the trigonometric function. Round your answer to four decimal places.
(Be sure the calculator is set in the correct angle mode.)
49. sin
4
50. tan
3
51. cot
4
52. csc
2
3
53. cos1.7
55. csc 0.8
57. sec22.8
66.
67.
54. cos2.5
56. sec 1.8
58. cot0.9
68.
59. HARMONIC MOTION The displacement from
equilibrium of an oscillating weight suspended by a
spring is given by yt 14 cos 6t, where y is the
displacement (in feet) and t is the time (in seconds). Find
the displacements when (a) t 0, (b) t 14, and (c) t 12.
60. HARMONIC MOTION The displacement from
equilibrium of an oscillating weight suspended by a
spring and subject to the damping effect of friction is
given by y t 14et cos 6t, where y is the displacement
(in feet) and t is the time (in seconds).
(a) Complete the table.
t
0
1
4
1
2
3
4
1
y
(b) Use the table feature of a graphing utility to approximate the time when the weight reaches equilibrium.
(c) What appears to happen to the displacement as t
increases?
69.
70.
71.
(b) Make a conjecture about any relationship between
sin t1 and sin t1.
(c) Make a conjecture about any relationship between
cos t1 and cos t1.
Use the unit circle to verify that the cosine and secant
functions are even and that the sine, cosecant, tangent,
and cotangent functions are odd.
Verify that cos 2t 2 cos t by approximating cos 1.5
and 2 cos 0.75.
Verify that sint1 t2 sin t1 sin t2 by approximating
sin 0.25, sin 0.75, and sin 1.
THINK ABOUT IT Because f t sin t is an odd
function and gt cos t is an even function, what can
be said about the function ht f tgt?
THINK ABOUT IT Because f t sin t and
gt tan t are odd functions, what can be said about
the function ht f tgt?
GRAPHICAL ANALYSIS With your graphing utility in
radian and parametric modes, enter the equations
X1T cos T and Y1T sin T
and use the following settings.
Tmin 0, Tmax 6.3, Tstep 0.1
Xmin 1.5, Xmax 1.5, Xscl 1
Ymin 1, Ymax 1, Yscl 1
(a) Graph the entered equations and describe the graph.
(b) Use the trace feature to move the cursor around the
graph. What do the t-values represent? What do the
x- and y-values represent?
(c) What are the least and greatest values of x and y?
EXPLORATION
TRUE OR FALSE? In Exercises 61– 64, determine whether
the statement is true or false. Justify your answer.
61. Because sint sin t, it can be said that the sine of
a negative angle is a negative number.
62. tan a tana 6
63. The real number 0 corresponds to the point 0, 1 on the
unit circle.
64. cos 7
cos 2
2
65. Let x1, y1 and x2, y2 be points on the unit circle
corresponding to t t1 and t t1, respectively.
(a) Identify the symmetry of the points x1, y1 and
x2, y2.
72. CAPSTONE A student you are tutoring has used a
unit circle divided into 8 equal parts to complete the
table for selected values of t. What is wrong?
t
0
x
1
y
0
sin t
1
cos t
0
tan t
Undef.
4
2
2
2
2
2
2
2
2
1
2
0
1
0
1
0
3
4
2
2
2
2
2
2
2
2
1
1
0
1
0
Undef.
Section 4.3
Right Triangle Trigonometry
299
4.3 RIGHT TRIANGLE TRIGONOMETRY
• Evaluate trigonometric functions
of acute angles.
• Use fundamental trigonometric
identities.
• Use a calculator to evaluate
trigonometric functions.
• Use trigonometric functions to
model and solve real-life problems.
The Six Trigonometric Functions
Our second look at the trigonometric functions is from a right triangle perspective.
Consider a right triangle, with one acute angle labeled ", as shown in Figure 4.26.
Relative to the angle ", the three sides of the triangle are the hypotenuse, the
opposite side (the side opposite the angle "), and the adjacent side (the side adjacent to
the angle ").
ten
us
e
Why you should learn it
Hy
po
Trigonometric functions are often used
to analyze real-life situations. For
instance, in Exercise 76 on page 309,
you can use trigonometric functions
to find the height of a helium-filled
balloon.
Side opposite θ
What you should learn
θ
Side adjacent to θ
FIGURE
4.26
Using the lengths of these three sides, you can form six ratios that define the six
trigonometric functions of the acute angle ".
sine cosecant cosine secant tangent cotangent
Joseph Sohm/Visions of America/Corbis
In the following definitions, it is important to see that 0# < " < 90# " lies in the first
quadrant) and that for such angles the value of each trigonometric function is positive.
Right Triangle Definitions of Trigonometric Functions
Let be an acute angle of a right triangle. The six trigonometric functions of the
angle are defined as follows. (Note that the functions in the second row are the
reciprocals of the corresponding functions in the first row.)
sin
opp
hyp
cos
adj
hyp
tan
opp
adj
csc
hyp
opp
sec
hyp
adj
cot
adj
opp
The abbreviations opp, adj, and hyp represent the lengths of the three sides of a
right triangle.
opp the length of the side opposite
adj the length of the side adjacent to
hyp the length of the hypotenuse
300
Chapter 4
Trigonometry
Example 1
Evaluating Trigonometric Functions
ote
n
us
e
Use the triangle in Figure 4.27 to find the values of the six trigonometric functions of .
Solution
Hy
p
4
By the Pythagorean Theorem, hyp2 opp2 adj2, it follows that
hyp 42 32
θ
25
3
FIGURE
5.
4.27
So, the six trigonometric functions of
opp 4
hyp 5
csc
hyp 5
opp 4
cos adj
3
hyp 5
sec
hyp 5
adj
3
tan opp 4
adj
3
cot
adj
3
.
opp 4
sin
You can review the Pythagorean
Theorem in Section 1.1.
HISTORICAL NOTE
Georg Joachim Rhaeticus
(1514–1574) was the leading
Teutonic mathematical
astronomer of the 16th century.
He was the first to define the
trigonometric functions as ratios
of the sides of a right triangle.
are
Now try Exercise 7.
In Example 1, you were given the lengths of two sides of the right triangle, but not
the angle . Often, you will be asked to find the trigonometric functions of a given acute
angle . To do this, construct a right triangle having as one of its angles.
Example 2
Evaluating Trigonometric Functions of 45
Find the values of sin 45", cos 45", and tan 45".
Solution
45°
2
1
Construct a right triangle having 45" as one of its acute angles, as shown in Figure 4.28.
Choose the length of the adjacent side to be 1. From geometry, you know that the other
acute angle is also 45". So, the triangle is isosceles and the length of the opposite side
is also 1. Using the Pythagorean Theorem, you find the length of the hypotenuse to
be 2.
sin 45" 2
1
opp
hyp 2
2
cos 45" 2
adj
1
hyp 2
2
tan 45" opp 1
1
adj
1
45°
1
FIGURE
4.28
Now try Exercise 23.
Section 4.3
Example 3
Because the angles 30", 45", and
60" 6, 4, and 3 occur
frequently in trigonometry, you
should learn to construct the
triangles shown in Figures 4.28
and 4.29.
Right Triangle Trigonometry
301
Evaluating Trigonometric Functions of 30 and 60
Use the equilateral triangle shown in Figure 4.29 to find the values of sin 60",
cos 60", sin 30", and cos 30".
30°
2
2
3
60°
1
FIGURE
1
4.29
Solution
T E C H N O LO G Y
You can use a calculator to
convert the answers in Example 3
to decimals. However, the
radical form is the exact value
and in most cases, the exact
value is preferred.
Use the Pythagorean Theorem and the equilateral triangle in Figure 4.29 to verify the
lengths of the sides shown in the figure. For 60", you have adj 1, opp 3, and
hyp 2. So,
sin 60" For
opp 3
hyp
2
cos 60" and
adj
1
.
hyp 2
30", adj 3, opp 1, and hyp 2. So,
sin 30" opp 1
hyp 2
cos 30" and
3
adj
.
hyp
2
Now try Exercise 27.
Sines, Cosines, and Tangents of Special Angles
sin 30" sin
1
6
2
cos 30" cos
3
6
2
tan 30" tan
3
6
3
sin 45" sin
2
4
2
cos 45" cos
2
4
2
tan 45" tan
1
4
sin 60" sin
3
3
2
cos 60" cos
1
3
2
tan 60" tan
3
3
1
In the box, note that sin 30" 2 cos 60". This occurs because 30" and 60" are
complementary angles. In general, it can be shown from the right triangle definitions
that cofunctions of complementary angles are equal. That is, if is an acute angle, the
following relationships are true.
sin90" cos
cos90" sin
tan90" cot
cot90" tan
sec90" csc
csc90" sec
302
Chapter 4
Trigonometry
Trigonometric Identities
In trigonometry, a great deal of time is spent studying relationships between trigonometric functions (identities).
Fundamental Trigonometric Identities
Reciprocal Identities
sin
1
csc
cos
1
sec
tan
1
cot
csc
1
sin
sec
1
cos
cot
1
tan
cot
cos
sin
Quotient Identities
tan
sin
cos
Pythagorean Identities
sin2
cos2
1
1 tan2
sec2
1 cot2 csc2
Note that sin2
Example 4
represents sin 2, cos2
represents cos 2, and so on.
Applying Trigonometric Identities
Let
be an acute angle such that sin
(b) tan using trigonometric identities.
0.6. Find the values of (a) cos
and
Solution
a. To find the value of cos , use the Pythagorean identity
sin2 cos2 1.
So, you have
0.6 2 cos2 1
Substitute 0.6 for sin .
cos2 1 0.6 2 0.64
cos
0.64 0.8.
Subtract 0.62 from each side.
Extract the positive square root.
b. Now, knowing the sine and cosine of , you can find the tangent of
tan
1
0.6
sin
cos
0.6
0.8
to be
0.75.
θ
0.8
FIGURE
4.30
Use the definitions of cos
these results.
and tan , and the triangle shown in Figure 4.30, to check
Now try Exercise 33.
Section 4.3
Example 5
Right Triangle Trigonometry
303
Applying Trigonometric Identities
3. Find the values of (a) cot
Let
be an acute angle such that tan
(b) sec using trigonometric identities.
and
Solution
a. cot
1
tan
cot
1
3
b. sec2 1 tan2
10
3
Pythagorean identity
sec2 1 32
sec2
10
10
sec
Use the definitions of cot
these results.
θ
FIGURE
Reciprocal identity
1
4.31
and sec , and the triangle shown in Figure 4.31, to check
Now try Exercise 35.
Evaluating Trigonometric Functions with a Calculator
You can also use the reciprocal
identities for sine, cosine, and
tangent to evaluate the cosecant,
secant, and cotangent functions
with a calculator. For instance,
you could use the following
keystroke sequence to evaluate
sec 28".
1
%
COS
28
ENTER
The calculator should display
1.1325701.
To use a calculator to evaluate trigonometric functions of angles measured in degrees,
first set the calculator to degree mode and then proceed as demonstrated in Section 4.2.
For instance, you can find values of cos 28" and sec 28" as follows.
Function
a. cos 28"
b. sec 28"
Mode
Calculator Keystrokes
Degree
Degree
COS
28
COS
Display
ENTER
28
x 1
ENTER
0.8829476
1.1325701
Throughout this text, angles are assumed to be measured in radians unless noted
otherwise. For example, sin 1 means the sine of 1 radian and sin 1" means the sine of
1 degree.
Example 6
Using a Calculator
Use a calculator to evaluate sec5" 40# 12$ .
Solution
1
1
Begin by converting to decimal degree form. [Recall that 1# 60
1" and 1$ 3600
1".
5" 40# 12$ 5" 60" 3600" 5.67"
40
12
Then, use a calculator to evaluate sec 5.67".
Function
sec5" 40# 12$ sec 5.67"
Calculator Keystrokes
COS
Now try Exercise 51.
5.67
x 1
Display
ENTER
1.0049166
304
Chapter 4
Trigonometry
Applications Involving Right Triangles
Object
Observer
Observer
Angle of
elevation
Horizontal
Horizontal
Angle of
depression
Many applications of trigonometry involve a process called solving right triangles. In
this type of application, you are usually given one side of a right triangle and one of the
acute angles and are asked to find one of the other sides, or you are given two sides and
are asked to find one of the acute angles.
In Example 7, the angle you are given is the angle of elevation, which represents
the angle from the horizontal upward to an object. For objects that lie below the
horizontal, it is common to use the term angle of depression, as shown in Figure 4.32.
Example 7
Object
FIGURE
4.32
Using Trigonometry to Solve a Right Triangle
A surveyor is standing 115 feet from the base of the Washington Monument, as shown
in Figure 4.33. The surveyor measures the angle of elevation to the top of the monument
as 78.3". How tall is the Washington Monument?
Solution
From Figure 4.33, you can see that
y
Angle of
elevation
78.3°
x = 115 ft
FIGURE
tan 78.3" opp y
adj
x
where x 115 and y is the height of the monument. So, the height of the Washington
Monument is
y x tan 78.3" 1154.82882 555 feet.
Not drawn to scale
Now try Exercise 67.
4.33
Example 8
Using Trigonometry to Solve a Right Triangle
A historic lighthouse is 200 yards from a bike path along the edge of a lake. A walkway to the lighthouse is 400 yards long. Find the acute angle between the bike path
and the walkway, as illustrated in Figure 4.34.
θ
200 yd
FIGURE
400 yd
4.34
Solution
From Figure 4.34, you can see that the sine of the angle
sin
opp 200 1
.
hyp 400 2
Now you should recognize that
30".
Now try Exercise 69.
is
Section 4.3
Right Triangle Trigonometry
305
By now you are able to recognize that 30" is the acute angle that satisfies the
equation sin 12. Suppose, however, that you were given the equation sin 0.6
and were asked to find the acute angle . Because
sin 30" 1
2
0.5000
and
sin 45" 1
2
0.7071
you might guess that lies somewhere between 30" and 45". In a later section, you will
study a method by which a more precise value of can be determined.
Example 9
Solving a Right Triangle
Find the length c of the skateboard ramp shown in Figure 4.35.
c
18.4°
FIGURE
4.35
Solution
From Figure 4.35, you can see that
sin 18.4" opp
hyp
4
.
c
So, the length of the skateboard ramp is
c
4
sin 18.4"
4
0.3156
12.7 feet.
Now try Exercise 71.
4 ft
306
Chapter 4
Trigonometry
EXERCISES
4.3
See www.CalcChat.com for worked-out solutions to odd-numbered exercises.
VOCABULARY
1. Match the trigonometric function with its right triangle definition.
(a) Sine
(b) Cosine
(c) Tangent
(d) Cosecant
(i)
hypotenuse
adjacent
(ii)
adjacent
opposite
hypotenuse
opposite
(iii)
(iv)
(e) Secant
adjacent
hypotenuse
(v)
(f) Cotangent
opposite
hypotenuse
(vi)
opposite
adjacent
In Exercises 2–4, fill in the blanks.
2. Relative to the angle , the three sides of a right triangle are the ________ side, the ________ side, and the ________.
3. Cofunctions of ________ angles are equal.
4. An angle that measures from the horizontal upward to an object is called the angle of ________, whereas an angle that
measures from the horizontal downward to an object is called the angle of ________.
SKILLS AND APPLICATIONS
In Exercises 5–8, find the exact values of the six trigonometric functions of the angle ! shown in the figure. (Use the
Pythagorean Theorem to find the third side of the triangle.)
5.
6.
13
6
5
θ
θ
8
7.
41
θ
9
8.
θ
4
In Exercises 9 –12, find the exact values of the six
trigonometric functions of the angle ! for each of the two
triangles. Explain why the function values are the same.
10. 1.25
8
θ 1
θ
5
15
θ
7.5
11.
3
4
12.
1
θ
4
sec
2
14.
16.
18.
20.
3
1
5
sin cot 3
1
45"
23. sec
4
24. tan
3
25. cot
6
4
29. cot
30. tan
6
4
tan 5
17
sec 7
csc 9
22. cos
28. sin
3
θ
5
6
30"
27. csc
2
2
cos
Function
21. sin
(deg)
θ
θ
6
3
tan
26. csc
θ
4
13.
15.
17.
19.
In Exercises 21–30, construct an appropriate triangle to
complete the table. !0 & ! & 90 , 0 & ! & " / 2"
4
9.
In Exercises 13–20, sketch a right triangle corresponding to
the trigonometric function of the acute angle !. Use the
Pythagorean Theorem to determine the third side and then
find the other five trigonometric functions of !.
(rad)
Function Value
3
3
2
1
3
3
Section 4.3
In Exercises 31–36, use the given function value(s), and
trigonometric identities (including the cofunction identities),
to find the indicated trigonometric functions.
31. sin 60" 3
2
1
cos 60" 2
(b) cos 30"
(d) cot 60"
,
(a) sin 30"
(c) tan 60"
1
32. sin 30" ,
2
tan 30" 3
3
(a) csc 30"
(c) cos 30"
1
33. cos 3
(b) cot 60"
(d) cot 30"
(a) sin
(c) sec
34. sec 5
(a) cos
(c) cot90" 35. cot ) 5
(a) tan )
(c) cot90" )
(b) tan
(d) csc90" 36. cos ( 48. (a) tan 23.5"
49. (a) sin 16.35"
(b) cot 66.5"
(b) csc 16.35"
50. (a) cot 79.56"
51. (a) cos 4" 50# 15$
52. (a) sec 42" 12#
(b) sec 79.56"
(b) sec 4" 50# 15$
(b) csc 48" 7#
53. (a) cot 11" 15#
54. (a) sec 56" 8# 10$
55. (a) csc 32" 40# 3$
(b) tan 11" 15#
(b) cos 56" 8# 10$
(b) tan 44" 28# 16$
56. (a) sec 95
(b) csc )
(d) cos )
7
57. (a) sin
61. (a) csc
62. (a) cot
(b) sin (
(d) sin90" (
In Exercises 37– 46, use trigonometric identities to transform
the left side of the equation into the right side
!0 < ! < " / 2".
37. tan cot 1
38. cos sec 1
39. tan ) cos ) sin )
1
2
2
2
59. (a) sec 2
60. (a) tan 3
4
(a) sec (
(c) cot (
' 20 32" (b) cot 95 ' 30 32"
In Exercises 57– 62, find the values of ! in degrees
!0 < ! < 90 " and radians !0 < ! < " / 2" without the aid
of a calculator.
58. (a) cos
(b) cot
(d) sin
307
Right Triangle Trigonometry
23
3
3
3
(b) csc
2
(b) tan
1
(b) cot 1
(b) cos 12
2
(b) sin
(b) sec
2
2
In Exercises 63–66, solve for x, y, or r as indicated.
63. Solve for y.
64. Solve for x.
30
18
y
30°
x
60°
65. Solve for x.
40. cot ) sin ) cos )
66. Solve for r.
41. 1 sin 1 sin cos2
42. 1 cos 1 cos sin2
43. sec tan sec tan 1
44. sin2 cos2 2 sin2 1
sin
cos
csc sec
45.
cos
sin
tan ( cot (
csc2 (
46.
tan (
In Exercises 47–56, use a calculator to evaluate each function.
Round your answers to four decimal places. (Be sure the
calculator is in the correct angle mode.)
47. (a) sin 10"
(b) cos 80"
r
32
60°
x
20
45°
67. EMPIRE STATE BUILDING You are standing 45 meters
from the base of the Empire State Building. You
estimate that the angle of elevation to the top of the 86th
floor (the observatory) is 82". If the total height of the
building is another 123 meters above the 86th floor,
what is the approximate height of the building? One of
your friends is on the 86th floor. What is the distance
between you and your friend?
308
Chapter 4
Trigonometry
68. HEIGHT A six-foot person walks from the base of a
broadcasting tower directly toward the tip of the
shadow cast by the tower. When the person is 132 feet
from the tower and 3 feet from the tip of the shadow, the
person’s shadow starts to appear beyond the tower’s
shadow.
(a) Draw a right triangle that gives a visual
representation of the problem. Show the known
quantities of the triangle and use a variable to
indicate the height of the tower.
(b) Use a trigonometric function to write an equation
involving the unknown quantity.
(c) What is the height of the tower?
69. ANGLE OF ELEVATION You are skiing down a
mountain with a vertical height of 1500 feet. The
distance from the top of the mountain to the base is
3000 feet. What is the angle of elevation from the base
to the top of the mountain?
70. WIDTH OF A RIVER A biologist wants to know the
width w of a river so that instruments for studying the
pollutants in the water can be set properly. From point
A, the biologist walks downstream 100 feet and sights to
point C (see figure). From this sighting, it is determined
that 54". How wide is the river?
72. HEIGHT OF A MOUNTAIN In traveling across flat
land, you notice a mountain directly in front of you. Its
angle of elevation (to the peak) is 3.5". After you drive
13 miles closer to the mountain, the angle of elevation
is 9". Approximate the height of the mountain.
3.5°
13 mi
d
y
60
56
3°
(x2, y2)
15 cm
(x1, y1)
30°
30°
56 60
w
FIGURE FOR
θ = 54°
A 100 ft
71. LENGTH A guy wire runs from the ground to a cell
tower. The wire is attached to the cell tower 150 feet
above the ground. The angle formed between the wire
and the ground is 43" (see figure).
Not drawn to scale
73. MACHINE SHOP CALCULATIONS A steel plate has
the form of one-fourth of a circle with a radius of
60 centimeters. Two two-centimeter holes are to be
drilled in the plate positioned as shown in the figure.
Find the coordinates of the center of each hole.
30°
C
9°
73
5 cm
x
FIGURE FOR
74. MACHINE SHOP CALCULATIONS A tapered shaft
has a diameter of 5 centimeters at the small end and is
15 centimeters long (see figure). The taper is 3". Find
the diameter d of the large end of the shaft.
75. GEOMETRY Use a compass to sketch a quarter of a
circle of radius 10 centimeters. Using a protractor,
construct an angle of 20" in standard position (see
figure). Drop a perpendicular line from the point of
intersection of the terminal side of the angle and the
arc of the circle. By actual measurement, calculate
the coordinates x, y of the point of intersection
and use these measurements to approximate the six
trigonometric functions of a 20" angle.
y
150 ft
10
(x, y)
θ = 43°
(a) How long is the guy wire?
(b) How far from the base of the tower is the guy wire
anchored to the ground?
74
m
10 c
20°
10
x
Section 4.3
76. HEIGHT A 20-meter line is used to tether a
helium-filled balloon. Because of a breeze, the line
makes an angle of approximately 85" with the ground.
(a) Draw a right triangle that gives a visual representation of the problem. Show the known quantities of
the triangle and use a variable to indicate the height
of the balloon.
(b) Use a trigonometric function to write an equation
involving the unknown quantity.
(c) What is the height of the balloon?
(d) The breeze becomes stronger and the angle the
balloon makes with the ground decreases. How
does this affect the triangle you drew in part (a)?
(e) Complete the table, which shows the heights
(in meters) of the balloon for decreasing angle
measures .
Angle,
80"
70"
60"
50"
40"
30"
20"
10"
Height
Angle,
Height
(f) As the angle the balloon makes with the ground
approaches 0", how does this affect the height of the
balloon? Draw a right triangle to explain your
reasoning.
EXPLORATION
Right Triangle Trigonometry
309
84. THINK ABOUT IT
(a) Complete the table.
0"
18"
36"
54"
72"
90"
sin
cos
(b) Discuss the behavior of the sine function for in the
range from 0" to 90".
(c) Discuss the behavior of the cosine function for in
the range from 0" to 90".
(d) Use the definitions of the sine and cosine functions
to explain the results of parts (b) and (c).
85. WRITING In right triangle trigonometry, explain why
sin 30" 12 regardless of the size of the triangle.
86. GEOMETRY Use the equilateral triangle shown in
Figure 4.29 and similar triangles to verify the points in
Figure 4.23 (in Section 4.2) that do not lie on the axes.
87. THINK ABOUT IT You are given only the value tan .
Is it possible to find the value of sec without finding
the measure of ? Explain.
88. CAPSTONE The Johnstown Inclined Plane in
Pennsylvania is one of the longest and steepest hoists
in the world. The railway cars travel a distance of
896.5 feet at an angle of approximately 35.4", rising to
a height of 1693.5 feet above sea level.
TRUE OR FALSE? In Exercises 77–82, determine whether
the statement is true or false. Justify your answer.
77. sin 60" csc 60" 1
78. sec 30" csc 60"
79. sin 45" cos 45" 1 80. cot2 10" csc2 10" 1
sin 60"
81.
82. tan5"2 tan2 5"
sin 2"
sin 30"
83. THINK ABOUT IT
(a) Complete the table.
0.1
896.5 ft
1693.5 feet
above sea level
35.4°
Not drawn to scale
(a) Find the vertical rise of the inclined plane.
0.2
0.3
0.4
0.5
sin
(b) Is or sin greater for in the interval 0, 0.5?
(c) As approaches 0, how do and sin compare?
Explain.
(b) Find the elevation of the lower end of the inclined
plane.
(c) The cars move up the mountain at a rate of
300 feet per minute. Find the rate at which they rise
vertically.
310
Chapter 4
Trigonometry
4.4 TRIGONOMETRIC FUNCTIONS OF ANY ANGLE
What you should learn
• Evaluate trigonometric functions
of any angle.
• Find reference angles.
• Evaluate trigonometric functions of
real numbers.
Why you should learn it
You can use trigonometric functions
to model and solve real-life problems.
For instance, in Exercise 99 on page
318, you can use trigonometric
functions to model the monthly
normal temperatures in New York City
and Fairbanks, Alaska.
Introduction
In Section 4.3, the definitions of trigonometric functions were restricted to acute angles.
In this section, the definitions are extended to cover any angle. If is an acute angle,
these definitions coincide with those given in the preceding section.
Definitions of Trigonometric Functions of Any Angle
Let be an angle in standard position with x, y a point on the terminal side of
and r x2 y2 0.
y
r
sin
tan
y
,
x
sec
r
,
x
x
r
y
cos
x0
cot
x
, y0
y
x0
csc
r
,
y
(x, y)
r
θ
y0
x
Because r x 2 y 2 cannot be zero, it follows that the sine and cosine
functions are defined for any real value of . However, if x 0, the tangent and secant
of are undefined. For example, the tangent of 90" is undefined. Similarly, if y 0, the
cotangent and cosecant of are undefined.
James Urbach/SuperStock
Example 1
Evaluating Trigonometric Functions
Let 3, 4 be a point on the terminal side of . Find the sine, cosine, and tangent of .
Solution
Referring to Figure 4.36, you can see that x 3, y 4, and
r x 2 y 2 3 2 42 25 5.
So, you have the following.
sin
y 4
r
5
cos
x
3
r
5
tan
y
(−3, 4)
4
3
r
2
1
The formula r x2 y2 is
a result of the Distance Formula.
You can review the Distance
Formula in Section 1.1.
θ
x
−3
FIGURE
Now try Exercise 9.
−2
4.36
−1
1
y
4
x
3
Section 4.4
y
π <θ<π
2
x<0
y>0
x
x>0
y<0
π < θ < 3π
2
sin θ : +
cos θ : −
tan θ : −
sin θ : +
cos θ : +
tan θ : +
5
4 and cos
Quadrant IV
sin θ : −
cos θ : −
tan θ : +
sin θ : −
cos θ : +
tan θ : −
FIGURE
> 0, find sin
and sec .
Note that lies in Quadrant IV because that is the only quadrant in which the tangent
is negative and the cosine is positive. Moreover, using
y
x
tan
x
Quadrant III
Evaluating Trigonometric Functions
Solution
y
Quadrant I
Example 2
Given tan
3π < θ < 2π
2
Quadrant II
311
The signs of the trigonometric functions in the four quadrants can be determined
from the definitions of the functions. For instance, because cos xr, it follows that
cos is positive wherever x > 0, which is in Quadrants I and IV. (Remember, r is
always positive.) In a similar manner, you can verify the results shown in Figure 4.37.
0<θ < π
2
x>0
y>0
x<0
y<0
Trigonometric Functions of Any Angle
5
4
and the fact that y is negative in Quadrant IV, you can let y 5 and x 4. So,
r 16 25 41 and you have
sin
y
5
41
r
4.37
0.7809
sec
41
r
x
4
1.6008.
Now try Exercise 23.
Example 3
Trigonometric Functions of Quadrant Angles
3
Evaluate the cosine and tangent functions at the four quadrant angles 0, , , and .
2
2
Solution
To begin, choose a point on the terminal side of each angle, as shown in Figure 4.38.
For each of the four points, r 1, and you have the following.
y
π
2
cos 0 (0, 1)
cos
(−1, 0)
(1, 0)
π
0
3π
2
FIGURE
4.38
(0, −1)
x 1
1
r
1
x 0
0
2
r
1
tan 0 tan
y 0
0
x 1
y 1
⇒ undefined
2
x 0
x, y 1, 0
x, y 0, 1
x
cos cos
x 1
1
r
1
3 x 0
0
2
r
1
tan tan
Now try Exercise 37.
0
y
0
x 1
3 y 1
⇒ undefined
2
x
0
x, y 1, 0
x, y 0, 1
312
Chapter 4
Trigonometry
Reference Angles
The values of the trigonometric functions of angles greater than 90" (or less than 0") can
be determined from their values at corresponding acute angles called reference angles.
Definition of Reference Angle
Let be an angle in standard position. Its reference angle is the acute angle #
formed by the terminal side of and the horizontal axis.
Figure 4.39 shows the reference angles for
in Quadrants II, III, and IV.
Quadrant II
Reference
angle: θ ′
θ
Reference
angle: θ ′
θ ′ = π − θ (radians)
θ ′ = 180° − θ (degrees)
FIGURE
θ
Quadrant III
θ ′ = θ − π (radians)
θ ′ = θ − 180° (degrees)
θ
Reference
angle: θ ′
Quadrant
IV
θ ′ = 2π − θ (radians)
θ ′ = 360° − θ (degrees)
4.39
y
Example 4
θ = 300°
Finding Reference Angles
Find the reference angle #.
x
θ ′ = 60°
a.
300"
b.
2.3
c.
135"
Solution
FIGURE
4.40
a. Because 300" lies in Quadrant IV, the angle it makes with the x-axis is
# 360" 300"
y
60".
Figure 4.40 shows the angle
θ = 2.3
θ ′ = π − 2.3
x
Degrees
300" and its reference angle # 60".
b. Because 2.3 lies between 2 1.5708 and Quadrant II and its reference angle is
3.1416, it follows that it is in
# 2.3
FIGURE
4.41
0.8416.
y
225° and −135°
225° are coterminal.
x
θ ′ = 45°
θ = −135°
Figure 4.41 shows the angle 2.3 and its reference angle # 2.3.
c. First, determine that 135" is coterminal with 225", which lies in Quadrant III. So,
the reference angle is
# 225" 180"
45".
Figure 4.42 shows the angle
FIGURE
4.42
Radians
Degrees
135" and its reference angle # 45".
Now try Exercise 45.
Section 4.4
y
Trigonometric Functions of Any Angle
313
Trigonometric Functions of Real Numbers
(x, y)
To see how a reference angle is used to evaluate a trigonometric function, consider the
point x, y on the terminal side of , as shown in Figure 4.43. By definition, you know
that
r=
p
hy
opp
sin
y
r
and
tan
y
.
x
For the right triangle with acute angle # and sides of lengths x and y , you have
θ
θ′
x
adj
sin # y
opp
hyp
r
tan # y
opp
.
adj
x
and
opp y , adj x
FIGURE 4.43
So, it follows that sin and sin # are equal, except possibly in sign. The same is true
for tan and tan # and for the other four trigonometric functions. In all cases, the sign
of the function value can be determined by the quadrant in which lies.
Evaluating Trigonometric Functions of Any Angle
To find the value of a trigonometric function of any angle :
1. Determine the function value for the associated reference angle #.
2. Depending on the quadrant in which
function value.
Learning the table of values
at the right is worth the effort
because doing so will increase
both your efficiency and your
confidence. Here is a pattern
for the sine function that may
help you remember the values.
0"
sin
30" 45" 60" 90"
By using reference angles and the special angles discussed in the preceding
section, you can greatly extend the scope of exact trigonometric values. For instance,
knowing the function values of 30" means that you know the function values of all
angles for which 30" is a reference angle. For convenience, the table below shows the
exact values of the trigonometric functions of special angles and quadrant angles.
Trigonometric Values of Common Angles
(degrees)
0"
30"
45"
60"
90"
180"
270"
0
6
4
3
2
3
2
sin
0
1
2
2
3
2
2
1
0
1
cos
1
3
2
2
2
1
2
0
1
0
tan
0
1
3
Undef.
0
Undef.
(radians)
0 1 2 3 4
2
2
2
2
2
Reverse the order to get cosine
values of the same angles.
lies, affix the appropriate sign to the
3
3
314
Chapter 4
Trigonometry
Example 5
Using Reference Angles
Evaluate each trigonometric function.
a. cos
4
3
b. tan210"
c. csc
11
4
Solution
43 lies in Quadrant III, the reference angle is
a. Because
#
4
3
3
as shown in Figure 6.41. Moreover, the cosine is negative in Quadrant III, so
cos
4
cos
3
3
1
.
2
b. Because 210" 360" 150", it follows that 210" is coterminal with the
second-quadrant angle 150". So, the reference angle is # 180" 150" 30", as
shown in Figure 4.45. Finally, because the tangent is negative in Quadrant II, you
have
tan210" tan 30"
3
3
.
c. Because 114 2 34, it follows that 114 is coterminal with the
second-quadrant angle 34. So, the reference angle is # 34 4, as
shown in Figure 4.46. Because the cosecant is positive in Quadrant II, you have
csc
11
csc
4
4
1
sin4
2.
y
y
y
θ ′ = 30°
θ = 4π
3
x
x
θ′ = π
3
FIGURE
4.44
θ′ = π
4
θ = −210°
FIGURE
Now try Exercise 59.
4.45
FIGURE
4.46
θ = 11π
4
x
Section 4.4
Example 6
315
Trigonometric Functions of Any Angle
Using Trigonometric Identities
13. Find (a) cos
Let be an angle in Quadrant II such that sin
using trigonometric identities.
and (b) tan
by
Solution
a. Using the Pythagorean identity sin2 cos2 1, you obtain
13
2
cos2
cos 2 1 Because cos
1 8
.
9 9
< 0 in Quadrant II, you can use the negative root to obtain
cos
Substitute 13 for sin .
1
8
9
22
.
3
b. Using the trigonometric identity tan
tan
sin
, you obtain
cos
13
223
Substitute for sin
and cos .
1
22
2
4
.
Now try Exercise 69.
You can use a calculator to evaluate trigonometric functions, as shown in the next
example.
Example 7
Using a Calculator
Use a calculator to evaluate each trigonometric function.
a. cot 410"
b. sin7
9
c. sec
Solution
Function
a. cot 410"
b. sin7
c. sec
9
Mode
Calculator Keystrokes
Degree
Radian
SIN
Radian
TAN
COS
Now try Exercise 79.
410
7 x 1
Display
ENTER
ENTER
9
x 1
ENTER
0.8390996
0.6569866
1.0641778
316
Chapter 4
Trigonometry
EXERCISES
4.4
See www.CalcChat.com for worked-out solutions to odd-numbered exercises.
VOCABULARY: Fill in the blanks.
In Exercises 1–6, let ! be an angle in standard position, with x, y! a point on the terminal side of ! and r " x2 1 y2 # 0.
r
________
y
1. sin
________
2.
3. tan
________
4. sec
5.
x
________
r
6.
________
x
________
y
7. Because r x2 y2 cannot be ________, the sine and cosine functions are ________ for any real value of .
8. The acute positive angle that is formed by the terminal side of the angle and the horizontal axis is called
the ________ angle of and is denoted by #.
SKILLS AND APPLICATIONS
In Exercises 9–12, determine the exact values of the six
trigonometric functions of the angle !.
y
9. (a)
y
(b)
(4, 3)
θ
θ
x
x
In Exercises 19–22, state the quadrant in which ! lies.
19. sin
20. sin
21. sin
> 0 and cos
> 0
< 0 and cos
< 0
> 0 and cos
< 0
22. sec
> 0 and cot
< 0
(− 8, 15)
In Exercises 23–32, find the values of the six trigonometric
functions of ! with the given constraint.
y
10. (a)
y
(b)
Function Value
15
23. tan 8
θ
θ
x
x
(−12, −5)
(1, − 1)
y
11. (a)
4
θ
θ
x
x
(−
3, −1)
12. (a)
(4, − 1)
y
y
(b)
θ
(3, 1)
θ
x
x
(− 4, 4)
In Exercises 13–18, the point is on the terminal side of an
angle in standard position. Determine the exact values of the
six trigonometric functions of the angle.
13. 5, 12
15. 5, 2
17. 5.4, 7.2
14. 8, 15
16. 4, 10
18. 32, 74 1
3
tan
26. cos 5
27. cot 3
28. csc 4
y
(b)
8
24. cos 17
3
25. sin 5
Constraint
sin > 0
29. sec 2
30. sin 0
31. cot is undefined.
32. tan is undefined.
< 0
lies in Quadrant II.
lies in Quadrant III.
cos > 0
cot < 0
sin
< 0
sec 1
2 ! ! 32
! ! 2
In Exercises 33–36, the terminal side of ! lies on the given
line in the specified quadrant. Find the values of the six
trigonometric functions of ! by finding a point on the line.
Line
33. y x
1
34. y 3x
35. 2x y 0
Quadrant
36. 4x 3y 0
IV
II
III
III
Section 4.4
In Exercises 37–44, evaluate the trigonometric function of
the quadrant angle.
37. sin 38. csc
3
2
39. sec
3
2
40. sec 41. sin
2
42. cot 43. csc 44. cot
2
In Exercises 45–52, find the reference angle !&, and sketch !
and !& in standard position.
45.
47.
49.
51.
160$
125$
46.
48.
2
3
4.8
50.
52.
309$
215$
7
6
11.6
In Exercises 53–68, evaluate the sine, cosine, and tangent of
the angle without using a calculator.
53. 225$
54. 300$
55. 750$
57. 150$
2
59.
3
56. 405$
58. 840$
3
60.
4
5
61.
4
7
62.
6
6
9
65.
4
67. 68. 75. sin 10$
77. cos110$
79. tan 304$
81. sec 72$
76. sec 225$
78. csc330$
80. cot 178$
82. tan188$
83. tan 4.5
85. tan
9
84. cot 1.35
87. sin0.65
88. sec 0.29
9 86. tan 11
89. cot 8
90. csc 1
91. (a) sin
2
92. (a) cos
93. (a) csc
23
3
2
2
Quadrant
72. csc 2
5
73. cos 8
9
74. sec 4
IV
I
III
IV
II
III
1
(b) sin
2
(b) cos
(b) cot
1
2
2
(b) sec 2
(b) cot 3
3
(b) sin 2
3
2
97. DISTANCE An airplane, flying at an altitude of 6 miles,
is on a flight path that passes directly over an observer
(see figure). If is the angle of elevation from the
observer to the plane, find the distance d from the
observer to the plane when (a) 30$, (b) 90$,
and (c) 120$.
23
4
d
In Exercises 69–74, find the indicated trigonometric value in
the specified quadrant.
Function
3
69. sin 5
70. cot 3
3
71. tan 2
15
14
In Exercises 91–96, find two solutions of the equation. Give
your answers in degrees 0% ! ! < 360%! and in radians
0 ! ! < 2$!. Do not use a calculator.
96. (a) sin
2
10
66.
3
3
2
In Exercises 75–90, use a calculator to evaluate the
trigonometric function. Round your answer to four decimal
places. (Be sure the calculator is set in the correct angle mode.)
94. (a) sec 2
95. (a) tan 1
64. 63. 317
Trigonometric Functions of Any Angle
Trigonometric Value
cos
sin
sec
cot
sec
tan
6 mi
θ
Not drawn to scale
98. HARMONIC MOTION The displacement from
equilibrium of an oscillating weight suspended by a
spring is given by yt 2 cos 6t, where y is the
displacement (in centimeters) and t is the time (in
seconds). Find the displacement when (a) t 0,
(b) t 14, and (c) t 12.
318
Chapter 4
Trigonometry
99. DATA ANALYSIS: METEOROLOGY The table
shows the monthly normal temperatures (in degrees
Fahrenheit) for selected months in New York City N and Fairbanks, Alaska F. (Source: National
Climatic Data Center)
Month
New York
City, N
Fairbanks,
F
January
April
July
October
December
33
52
77
58
38
10
32
62
24
6
(a) Use the regression feature of a graphing utility to
find a model of the form y a sinbt c d
for each city. Let t represent the month, with t 1
corresponding to January.
(b) Use the models from part (a) to find the monthly
normal temperatures for the two cities in February,
March, May, June, August, September, and
November.
(c) Compare the models for the two cities.
100. SALES A company that produces snowboards, which are
seasonal products, forecasts monthly sales over the next
2 years to be S 23.1 0.442t 4.3 cost6,
where S is measured in thousands of units and t is
the time in months, with t 1 representing January
2010. Predict sales for each of the following months.
(a) February 2010
(b) February 2011
(c) June 2010
(d) June 2011
101. HARMONIC MOTION The displacement from
equilibrium of an oscillating weight suspended by a
spring and subject to the damping effect of friction
is given by y t 2et cos 6t, where y is the
displacement (in centimeters) and t is the time (in
seconds). Find the displacement when (a) t 0,
1
1
(b) t 4, and (c) t 2.
102. ELECTRIC CIRCUITS The current I (in amperes)
when 100 volts is applied to a circuit is given by
I 5e2t sin t, where t is the time (in seconds) after
the voltage is applied. Approximate the current at
t 0.7 second after the voltage is applied.
EXPLORATION
TRUE OR FALSE? In Exercises 103 and 104, determine
whether the statement is true or false. Justify your answer.
103. In each of the four quadrants, the signs of the secant
function and sine function will be the same.
104. To find the reference angle for an angle (given in
degrees), find the integer n such that
0 ! 360$n ! 360$. The difference 360$n is
the reference angle.
105. WRITING Consider an angle in standard position
with r 12 centimeters, as shown in the figure. Write
a short paragraph describing the changes in the values
of x, y, sin , cos , and tan as
increases
continuously from 0$ to 90$.
y
(x, y)
12 cm
θ
x
106. CAPSTONE Write a short paper in your own words
explaining to a classmate how to evaluate the six
trigonometric functions of any angle in standard
position. Include an explanation of reference angles
and how to use them, the signs of the functions in
each of the four quadrants, and the trigonometric
values of common angles. Be sure to include figures
or diagrams in your paper.
107. THINK ABOUT IT The figure shows point Px, y on
a unit circle and right triangle OAP.
y
P(x, y)
t
r
θ
O
A
x
(a) Find sin t and cos t using the unit circle definitions
of sine and cosine (from Section 4.2).
(b) What is the value of r? Explain.
(c) Use the definitions of sine and cosine given in this
section to find sin and cos . Write your answers
in terms of x and y.
(d) Based on your answers to parts (a) and (c), what
can you conclude?
Section 4.5
319
Graphs of Sine and Cosine Functions
4.5 GRAPHS OF SINE AND COSINE FUNCTIONS
What you should learn
• Sketch the graphs of basic sine and
cosine functions.
• Use amplitude and period to help
sketch the graphs of sine and cosine
functions.
• Sketch translations of the graphs
of sine and cosine functions.
• Use sine and cosine functions to
model real-life data.
Why you should learn it
Basic Sine and Cosine Curves
In this section, you will study techniques for sketching the graphs of the sine and cosine
functions. The graph of the sine function is a sine curve. In Figure 4.47, the black
portion of the graph represents one period of the function and is called one cycle of the
sine curve. The gray portion of the graph indicates that the basic sine curve repeats
indefinitely in the positive and negative directions. The graph of the cosine function is
shown in Figure 4.48.
Recall from Section 4.2 that the domain of the sine and cosine functions is the set
of all real numbers. Moreover, the range of each function is the interval 1, 1, and
each function has a period of 2. Do you see how this information is consistent with
the basic graphs shown in Figures 4.47 and 4.48?
Sine and cosine functions are
often used in scientific calculations.
For instance, in Exercise 87 on page 328,
you can use a trigonometric function
to model the airflow of your
respiratory cycle.
y
y = sin x
1
Range:
−1 ≤ y ≤ 1
x
− 3π
2
−π
−π
2
π
2
π
3π
2
2π
5π
2
−1
Period: 2π
FIGURE
4.47
© Karl Weatherly/Corbis
y
y = cos x
1
Range:
−1 ≤ y ≤ 1
x
− 3π
2
−π
π
2
π
3π
2
2π
5π
2
−1
Period: 2 π
FIGURE
4.48
Note in Figures 4.47 and 4.48 that the sine curve is symmetric with respect to
the origin, whereas the cosine curve is symmetric with respect to the y-axis. These
properties of symmetry follow from the fact that the sine function is odd and the cosine
function is even.
320
Chapter 4
Trigonometry
To sketch the graphs of the basic sine and cosine functions by hand, it helps to note
five key points in one period of each graph: the intercepts, maximum points, and
minimum points (see Figure 4.49).
y
y
Maximum Intercept Minimum
π,1
Intercept
y = sin x
2
(
)
(π , 0)
(0, 0)
Quarter
period
(32π , −1)
Half
period
Period: 2π
FIGURE
Intercept Minimum
Maximum
(0, 1)
y = cos x
Intercept
Three-quarter
period
(2π, 0)
Full
period
Quarter
period
(2π, 1)
( 32π , 0)
( π2 , 0)
x
Intercept Maximum
x
Full
period
(π , −1)
Period: 2π
Half
period
Three-quarter
period
4.49
Example 1
Using Key Points to Sketch a Sine Curve
Sketch the graph of y 2 sin x on the interval , 4.
Solution
Note that
y 2 sin x 2sin x
indicates that the y-values for the key points will have twice the magnitude of those on
the graph of y sin x. Divide the period 2 into four equal parts to get the key points
for y 2 sin x.
Intercept
Maximum
0, 0,
,2 ,
2
Intercept
, 0,
Minimum
Intercept
3
, 2 , and 2, 0
2
By connecting these key points with a smooth curve and extending the curve in both
directions over the interval , 4, you obtain the graph shown in Figure 4.50.
y
T E C H N O LO G Y
3
When using a graphing utility to
graph trigonometric functions,
pay special attention to the viewing
window you use. For instance,
try graphing y " [sin 10x!]/10
in the standard viewing window
in radian mode. What do you
observe? Use the zoom feature
to find a viewing window that
displays a good view of the graph.
2
y = 2 sin x
1
x
− π2
y = sin x
−2
FIGURE
4.50
Now try Exercise 39.
3π
2
5π
2
7π
2
Section 4.5
Graphs of Sine and Cosine Functions
321
Amplitude and Period
In the remainder of this section you will study the graphic effect of each of the
constants a, b, c, and d in equations of the forms
y d a sinbx c
and
y d a cosbx c.
A quick review of the transformations you studied in Section 1.7 should help in this
investigation.
The constant factor a in y a sin x acts as a scaling factor—a vertical stretch or
vertical shrink of the basic sine curve. If a > 1, the basic sine curve is stretched, and
if a < 1, the basic sine curve is shrunk. The result is that the graph of y a sin x
ranges between a and a instead of between 1 and 1. The absolute value of a is the
amplitude of the function y a sin x. The range of the function y a sin x for a > 0
is a ! y ! a.
Definition of Amplitude of Sine and Cosine Curves
The amplitude of y a sin x and y a cos x represents half the distance
between the maximum and minimum values of the function and is given by
Amplitude a .
Example 2
Scaling: Vertical Shrinking and Stretching
On the same coordinate axes, sketch the graph of each function.
a. y 1
cos x
2
b. y 3 cos x
Solution
y
y = 3 cos x
3
y = cos x
a. Because the amplitude of y 12 cos x is 12, the maximum value is 12 and the minimum
1
value is 2. Divide one cycle, 0 ! x ! 2, into four equal parts to get the key
points
Maximum Intercept
x
2π
−2
FIGURE
4.51
2 , 0, , 21, 32, 0,
Maximum
and
2, 12.
b. A similar analysis shows that the amplitude of y 3 cos x is 3, and the key points
are
−1
−3
0, 21,
Minimum Intercept
y=
1
cos
2
x
Maximum Intercept Minimum
Intercept
2 , 0,
32, 0,
0, 3,
, 3,
Maximum
and
2, 3.
The graphs of these two functions are shown in Figure 4.51. Notice that the graph
of y 12 cos x is a vertical shrink of the graph of y cos x and the graph of y 3 cos x
is a vertical stretch of the graph of y cos x.
Now try Exercise 41.
322
Chapter 4
Trigonometry
You know from Section 1.7 that the graph of y f x is a reflection in the
x-axis of the graph of y f x. For instance, the graph of y 3 cos x is a reflection
of the graph of y 3 cos x, as shown in Figure 4.52.
Because y a sin x completes one cycle from x 0 to x 2, it follows that
y a sin bx completes one cycle from x 0 to x 2b.
y
y = −3 cos x
y = 3 cos x
3
1
x
−π
π
2π
Period of Sine and Cosine Functions
Let b be a positive real number. The period of y a sin bx and y a cos bx is
given by
−3
FIGURE
Period 4.52
2
.
b
Note that if 0 < b < 1, the period of y a sin bx is greater than 2 and
represents a horizontal stretching of the graph of y a sin x. Similarly, if b > 1, the
period of y a sin bx is less than 2 and represents a horizontal shrinking of the graph
of y a sin x. If b is negative, the identities sinx sin x and cosx cos x
are used to rewrite the function.
Example 3
Scaling: Horizontal Stretching
x
Sketch the graph of y sin .
2
Solution
1
The amplitude is 1. Moreover, because b 2, the period is
2 2
1 4.
b
2
Substitute for b.
Now, divide the period-interval 0, 4 into four equal parts with the values , 2, and
3 to obtain the key points on the graph.
In general, to divide a
period-interval into four equal
parts, successively add
“period4,” starting with the
left endpoint of the interval. For
instance, for the period-interval
6, 2 of length 23,
you would successively add
Intercept
0, 0,
Maximum
, 1,
Minimum
Intercept
3, 1, and 4, 0
The graph is shown in Figure 4.53.
y
y = sin x
2
y = sin x
1
x
−π
23 4
6
to get 6, 0, 6, 3, and
2 as the x-values for the key
points on the graph.
Intercept
2, 0,
π
−1
Period: 4π
FIGURE
4.53
Now try Exercise 43.
Section 4.5
Graphs of Sine and Cosine Functions
323
Translations of Sine and Cosine Curves
The constant c in the general equations
y a sinbx c
You can review the techniques
for shifting, reflecting, and
stretching graphs in Section 1.7.
and
y a cosbx c
creates a horizontal translation (shift) of the basic sine and cosine curves. Comparing
y a sin bx with y a sinbx c, you find that the graph of y a sinbx c
completes one cycle from bx c 0 to bx c 2. By solving for x, you can find
the interval for one cycle to be
Left endpoint Right endpoint
c
c
2
.
! x ! b
b
b
Period
This implies that the period of y a sinbx c is 2b, and the graph of y a sin bx
is shifted by an amount cb. The number cb is the phase shift.
Graphs of Sine and Cosine Functions
The graphs of y a sinbx c and y a cosbx c have the following
characteristics. (Assume b > 0.)
Period Amplitude a
2
b
The left and right endpoints of a one-cycle interval can be determined by solving
the equations bx c 0 and bx c 2.
Example 4
Horizontal Translation
Analyze the graph of y 1
.
sin x 2
3
Algebraic Solution
Graphical Solution
1
The amplitude is 2 and the period is 2. By solving the equations
x
0
3
x
2
3
x
3
and
x
7
3
1
you see that the interval 3, 73 corresponds to one cycle of the
graph. Dividing this interval into four equal parts produces the key
points
Intercept Maximum Intercept
5 1
4
,0 ,
, ,
,0 ,
3
6 2
3
Now try Exercise 49.
Use a graphing utility set in radian mode to graph
y 12 sinx 3, as shown in Figure 4.54.
Use the minimum, maximum, and zero or root
features of the graphing utility to approximate the
key points 1.05, 0, 2.62, 0.5, 4.19, 0,
5.76, 0.5, and 7.33, 0.
Minimum
Intercept
11 1
7
, , and
,0 .
6
2
3
−
1
π
sin x −
2
3
( (
5
2
2
−1
FIGURE
y=
4.54
324
Chapter 4
Trigonometry
y = −3 cos(2 πx + 4 π)
Example 5
Horizontal Translation
y
Sketch the graph of
3
y 3 cos2x 4.
2
Solution
x
−2
The amplitude is 3 and the period is 22 1. By solving the equations
1
2 x 4 0
2 x 4
x 2
−3
Period 1
FIGURE
and
4.55
2 x 4 2
2 x 2
x 1
you see that the interval 2, 1 corresponds to one cycle of the graph. Dividing this
interval into four equal parts produces the key points
Minimum
2, 3,
Intercept
7
,0 ,
4
Maximum
3
,3 ,
2
Intercept
5
,0 ,
4
Minimum
and
1, 3.
The graph is shown in Figure 4.55.
Now try Exercise 51.
The final type of transformation is the vertical translation caused by the constant
d in the equations
y d a sinbx c
and
y d a cosbx c.
The shift is d units upward for d > 0 and d units downward for d < 0. In other words,
the graph oscillates about the horizontal line y d instead of about the x-axis.
y
Example 6
y = 2 + 3 cos 2x
5
Vertical Translation
Sketch the graph of
y 2 3 cos 2x.
Solution
The amplitude is 3 and the period is . The key points over the interval 0, are
1
−π
0, 5,
π
−1
Period π
FIGURE
4.56
x
4 , 2,
2 , 1,
34, 2,
and
, 5.
The graph is shown in Figure 4.56. Compared with the graph of f x 3 cos 2x, the
graph of y 2 3 cos 2x is shifted upward two units.
Now try Exercise 57.
Section 4.5
Graphs of Sine and Cosine Functions
325
Mathematical Modeling
Sine and cosine functions can be used to model many real-life situations, including
electric currents, musical tones, radio waves, tides, and weather patterns.
Time, t
Depth, y
Midnight
2 A.M.
4 A.M.
6 A.M.
8 A.M.
10 A.M.
Noon
3.4
8.7
11.3
9.1
3.8
0.1
1.2
Example 7
Finding a Trigonometric Model
Throughout the day, the depth of water at the end of a dock in Bar Harbor, Maine varies
with the tides. The table shows the depths (in feet) at various times during the
morning. (Source: Nautical Software, Inc.)
a. Use a trigonometric function to model the data.
b. Find the depths at 9 A.M. and 3 P.M.
c. A boat needs at least 10 feet of water to moor at the dock. During what times in the
afternoon can it safely dock?
Solution
y
a. Begin by graphing the data, as shown in Figure 4.57. You can use either a sine or a
cosine model. Suppose you use a cosine model of the form
Changing Tides
Depth (in feet)
12
y a cosbt c d.
10
The difference between the maximum height and the minimum height of the graph
is twice the amplitude of the function. So, the amplitude is
8
6
1
1
a maximum depth minimum depth 11.3 0.1 5.6.
2
2
4
2
t
4 A.M.
8 A.M.
Noon
Time
FIGURE
The cosine function completes one half of a cycle between the times at which the
maximum and minimum depths occur. So, the period is
p 2time of min. depth time of max. depth 210 4 12
4.57
which implies that b 2p " 0.524. Because high tide occurs 4 hours after
midnight, consider the left endpoint to be cb 4, so c " 2.094. Moreover,
because the average depth is 12 11.3 0.1 5.7, it follows that d 5.7. So, you
can model the depth with the function given by
y 5.6 cos0.524t 2.094 5.7.
b. The depths at 9 A.M. and 3 P.M. are as follows.
y 5.6 cos0.524 % 9 2.094 5.7
12
(14.7, 10) (17.3, 10)
" 0.84 foot
9 A.M.
y 5.6 cos0.524 % 15 2.094 5.7
y = 10
0
24
0
y = 5.6 cos(0.524t − 2.094) + 5.7
FIGURE
4.58
" 10.57 feet
3 P.M.
c. To find out when the depth y is at least 10 feet, you can graph the model with the
line y 10 using a graphing utility, as shown in Figure 4.58. Using the intersect
feature, you can determine that the depth is at least 10 feet between 2:42 P.M.
t " 14.7 and 5:18 P.M. t " 17.3.
Now try Exercise 91.
326
Chapter 4
Trigonometry
EXERCISES
4.5
See www.CalcChat.com for worked-out solutions to odd-numbered exercises.
VOCABULARY: Fill in the blanks.
1. One period of a sine or cosine function is called one ________ of the sine or cosine curve.
2. The ________ of a sine or cosine curve represents half the distance between the maximum and minimum values
of the function.
c
3. For the function given by y a sinbx c, represents the ________ ________ of the graph of the function.
b
4. For the function given by y d a cosbx c, d represents a ________ ________ of the graph of the function.
SKILLS AND APPLICATIONS
In Exercises 5–18, find the period and amplitude.
5. y 2 sin 5x
In Exercises 19–26, describe the relationship between the
graphs of f and g. Consider amplitude, period, and shifts.
6. y 3 cos 2x
y
19. f x sin x
gx sinx y
3
2
1
x
x
π
10
−3
7. y π
2
−2
−3
3
x
cos
4
2
8. y 3 sin
20. f x cos x
gx cosx 22. f x sin 3x
gx sin3x
21. f x cos 2x
gx cos 2x
23. f x cos x
gx cos 2x
25. f x sin 2x
x
3
24. f x sin x
gx sin 3x
26. f x cos 4x
gx 2 cos 4x
gx 3 sin 2x
y
y
In Exercises 27–30, describe the relationship between the
graphs of f and g. Consider amplitude, period, and shifts.
4
1
x
−π
−2
π 2π
−1
x
π
y
27.
3
−2
−3
y
π
2
−1
x
−π
π
3
2
1
x
−2π
2x
3
11. y 4 sin x
12. y cos
13. y 3 sin 10x
5
4x
15. y cos
3
5
1
17. y sin 2 x
4
1
14. y 5 sin 6x
x
5
16. y cos
2
4
2
x
18. y cos
3
10
y
30.
f
4
3
2
g
2π
x
f
−2
−3
g
−2
−3
−2
π
x
y
29.
2
1
g 2
π
x
3
10. y cos
2
2
y
3
f
−4
1
x
9. y sin
2
3
y
28.
x
−2π
g
f
2π
x
−2
In Exercises 31–38, graph f and g on the same set of
coordinate axes. (Include two full periods.)
31. f x 2 sin x
gx 4 sin x
33. f x cos x
gx 2 cos x
32. f x sin x
x
gx sin
3
34. f x 2 cos 2x
gx cos 4x
Section 4.5
1
x
35. f x sin
2
2
GRAPHICAL REASONING In Exercises 73–76, find a and
d for the function f x! " a cos x ( d such that the graph of
f matches the figure.
36. f x 4 sin x
1
x
sin
2
2
37. f x 2 cos x
gx 3 gx 4 sin x 3
y
73.
38. f x cos x
gx 2 cosx 2
4
f
−π
1
In Exercises 39– 60, sketch the graph of the function. (Include
two full periods.)
40. y 14 sin x
42. y 4 cos x
x
2
43. y cos
47. y sin
46. y sin
2 x
3
49. y sin x 2
2 x
3
1
55. y 2 10 cos 60 x
x
2
cos 3
2
4
4
54. y 3 5 cos
t
12
61. gx sin4x 62. gx sin2x 63. gx cosx 2 64. gx 1 cosx 65. gx 2 sin4x 3 66. gx 4 sin2x In Exercises 67–72, use a graphing utility to graph the
function. Include two full periods. Be sure to choose an
appropriate viewing window.
2
67. y 2 sin4x 68. y 4 sin x 3
3
1
69. y cos 2 x 2
x 70. y 3 cos
2
2
2
1
x
72. y sin 120 t
71. y 0.1 sin
10
100
−5
y
78.
3
2
1
1
60. y 3 cos6x x
f
π
π
y
77.
In Exercises 61– 66, g is related to a parent function
f x! " sin x! or f x! " cos x!. (a) Describe the sequence of
transformations from f to g. (b) Sketch the graph of g. (c) Use
function notation to write g in terms of f.
x
π
−1
−2
GRAPHICAL REASONING In Exercises 77–80, find a, b,
and c for the function f x! " a sin bx c! such that the
graph of f matches the figure.
56. y 2 cos x 3
4
58. y 4 cos x 4
57. y 3 cosx 3
59. y x
6
52. y 4 cos x 53. y 2 sin
f
−π −2
50. y sinx 2
51. y 3 cosx 1
−π
f
48. y 10 cos
y
76.
10
8
6
4
x
4
f
−3
−4
y
75.
x
π
x
π
2
−1
−2
44. y sin 4x
45. y cos 2 x
y
74.
gx cosx 39. y 5 sin x
41. y 13 cos x
327
Graphs of Sine and Cosine Functions
x
−π
−3
3
2
π
−2
−3
y
80.
3
2
1
f
x
π
−3
y
79.
f
f
x
x
2
4
−2
−3
In Exercises 81 and 82, use a graphing utility to graph y1 and
y2 in the interval ['2$, 2$]. Use the graphs to find real
numbers x such that y1 " y2.
81. y1 sin x
y2 12
82. y1 cos x
y2 1
In Exercises 83–86, write an equation for the function that is
described by the given characteristics.
83. A sine curve with a period of , an amplitude of 2,
a right phase shift of 2, and a vertical translation up
1 unit
328
Chapter 4
Trigonometry
84. A sine curve with a period of 4, an amplitude of 3,
a left phase shift of 4, and a vertical translation down
1 unit
85. A cosine curve with a period of , an amplitude of 1,
a left phase shift of , and a vertical translation down
3
2 units
86. A cosine curve with a period of 4, an amplitude of 3,
a right phase shift of 2, and a vertical translation up
2 units
87. RESPIRATORY CYCLE For a person at rest, the velocity
v (in liters per second) of airflow during a respiratory
cycle (the time from the beginning of one breath
to the beginning of the next) is given by
t
v 0.85 sin , where t is the time (in seconds). (Inhalation
3
occurs when v > 0, and exhalation occurs when v < 0.)
(a) Find the time for one full respiratory cycle.
(b) Find the number of cycles per minute.
(c) Sketch the graph of the velocity function.
88. RESPIRATORY CYCLE After exercising for a
few minutes, a person has a respiratory cycle for
which the velocity of airflow is approximated
t
by v 1.75 sin , where t is the time (in seconds).
2
(Inhalation occurs when v > 0, and exhalation occurs
when v < 0.)
(a) Find the time for one full respiratory cycle.
(b) Find the number of cycles per minute.
(c) Sketch the graph of the velocity function.
89. DATA ANALYSIS: METEOROLOGY The table shows
the maximum daily high temperatures in Las Vegas L
and International Falls I (in degrees Fahrenheit) for
month t, with t 1 corresponding to January. (Source:
National Climatic Data Center)
Month, t
Las Vegas, L
International Falls, I
1
2
3
4
5
6
7
8
9
10
11
12
57.1
63.0
69.5
78.1
87.8
98.9
104.1
101.8
93.8
80.8
66.0
57.3
13.8
22.4
34.9
51.5
66.6
74.2
78.6
76.3
64.7
51.7
32.5
18.1
(a) A model for the temperature in Las Vegas is given
by
Lt 80.60 23.50 cos
6
t
3.67 .
Find a trigonometric model for International Falls.
(b) Use a graphing utility to graph the data points and
the model for the temperatures in Las Vegas. How
well does the model fit the data?
(c) Use a graphing utility to graph the data points and
the model for the temperatures in International
Falls. How well does the model fit the data?
(d) Use the models to estimate the average maximum
temperature in each city. Which term of the models
did you use? Explain.
(e) What is the period of each model? Are the periods
what you expected? Explain.
(f) Which city has the greater variability in
temperature throughout the year? Which factor
of the models determines this variability? Explain.
90. HEALTH The function given by
P 100 20 cos
5 t
3
approximates the blood pressure P (in millimeters of
mercury) at time t (in seconds) for a person at rest.
(a) Find the period of the function.
(b) Find the number of heartbeats per minute.
91. PIANO TUNING When tuning a piano, a technician
strikes a tuning fork for the A above middle C and sets
up a wave motion that can be approximated by
y 0.001 sin 880 t, where t is the time (in seconds).
(a) What is the period of the function?
(b) The frequency f is given by f 1p. What is the
frequency of the note?
92. DATA ANALYSIS: ASTRONOMY The percents y (in
decimal form) of the moon’s face that was illuminated
on day x in the year 2009, where x 1 represents
January 1, are shown in the table. (Source: U.S. Naval
Observatory)x
x
y
4
11
18
26
33
40
0.5
1.0
0.5
0.0
0.5
1.0
Section 4.5
(a) Create a scatter plot of the data.
(b) Find a trigonometric model that fits the data.
(c) Add the graph of your model in part (b) to the
scatter plot. How well does the model fit the data?
(d) What is the period of the model?
(e) Estimate the moon’s percent illumination for
March 12, 2009.
93. FUEL CONSUMPTION The daily consumption C (in
gallons) of diesel fuel on a farm is modeled by
C 30.3 21.6 sin
365 10.9
2 t
where t is the time (in days), with t 1 corresponding
to January 1.
(a) What is the period of the model? Is it what you
expected? Explain.
(b) What is the average daily fuel consumption? Which
term of the model did you use? Explain.
(c) Use a graphing utility to graph the model. Use the
graph to approximate the time of the year when
consumption exceeds 40 gallons per day.
94. FERRIS WHEEL A Ferris wheel is built such that the
height h (in feet) above ground of a seat on the wheel at
time t (in seconds) can be modeled by
ht 53 50 sin
10 t 2 .
(a) Find the period of the model. What does the period
tell you about the ride?
(b) Find the amplitude of the model. What does the
amplitude tell you about the ride?
(c) Use a graphing utility to graph one cycle of the
model.
EXPLORATION
TRUE OR FALSE? In Exercises 95–97, determine whether
the statement is true or false. Justify your answer.
95. The graph of the function given by f x sinx 2
translates the graph of f x sin x exactly one period
to the right so that the two graphs look identical.
96. The function given by y 12 cos 2x has an amplitude
that is twice that of the function given by y cos x.
97. The graph of y cos x is a reflection of the graph of
y sinx 2 in the x-axis.
98. WRITING Sketch the graph of y cos bx for b 12,
2, and 3. How does the value of b affect the graph? How
many complete cycles occur between 0 and 2 for each
value of b?
Graphs of Sine and Cosine Functions
329
99. WRITING Sketch the graph of y sinx c for
c 4, 0, and 4. How does the value of c affect
the graph?
100. CAPSTONE Use a graphing utility to graph the
function given by y d a sinbx c, for several
different values of a, b, c, and d. Write a paragraph
describing the changes in the graph corresponding to
changes in each constant.
CONJECTURE In Exercises 101 and 102, graph f and g on
the same set of coordinate axes. Include two full periods.
Make a conjecture about the functions.
2
101. f x sin x,
gx cos x 102. f x sin x,
gx cos x 2
103. Using calculus, it can be shown that the sine and cosine
functions can be approximated by the polynomials
sin x
x
x3 x5
and cos x
3! 5!
1
x4
x2
2!
4!
where x is in radians.
(a) Use a graphing utility to graph the sine function
and its polynomial approximation in the same
viewing window. How do the graphs compare?
(b) Use a graphing utility to graph the cosine function
and its polynomial approximation in the same
viewing window. How do the graphs compare?
(c) Study the patterns in the polynomial approximations of the sine and cosine functions and predict
the next term in each. Then repeat parts (a) and (b).
How did the accuracy of the approximations
change when an additional term was added?
104. Use the polynomial approximations of the sine and
cosine functions in Exercise 103 to approximate
the following function values. Compare the results
with those given by a calculator. Is the error in the
approximation the same in each case? Explain.
6
(d) cos0.5 (e) cos 1 (f) cos
4
PROJECT: METEOROLOGY To work an extended
application analyzing the mean monthly temperature and
mean monthly precipitation in Honolulu, Hawaii, visit this
text’s website at academic.cengage.com. (Data Source:
National Climatic Data Center)
(a) sin
1
2
(b) sin 1
(c) sin
330
Chapter 4
Trigonometry
4.6 GRAPHS OF OTHER TRIGONOMETRIC FUNCTIONS
What you should learn
• Sketch the graphs of tangent
functions.
• Sketch the graphs of cotangent
functions.
• Sketch the graphs of secant and
cosecant functions.
• Sketch the graphs of damped
trigonometric functions.
Why you should learn it
Recall that the tangent function is odd. That is, tanx tan x. Consequently, the
graph of y tan x is symmetric with respect to the origin. You also know from the
identity tan x sin xcos x that the tangent is undefined for values at which cos x 0.
Two such values are x ± 2 ± 1.5708.
x
tan x
2
Undef.
1.57
1.5
4
0
4
1.5
1.57
2
1255.8
14.1
1
0
1
14.1
1255.8
Undef.
As indicated in the table, tan x increases without bound as x approaches 2 from the
left, and decreases without bound as x approaches 2 from the right. So, the graph
of y tan x has vertical asymptotes at x 2 and x 2, as shown in
Figure 4.59. Moreover, because the period of the tangent function is , vertical
asymptotes also occur when x 2 n, where n is an integer. The domain of the
tangent function is the set of all real numbers other than x 2 n, and the range
is the set of all real numbers.
Alan Pappe/Photodisc/Getty Images
Graphs of trigonometric functions can
be used to model real-life situations
such as the distance from a television
camera to a unit in a parade, as in
Exercise 92 on page 339.
Graph of the Tangent Function
y
y = tan x
PERIOD: DOMAIN: ALL x 2 n
RANGE: ( , )
VERTICAL ASYMPTOTES: x 2 n
SYMMETRY: ORIGIN
3
2
1
x
− 3π
2
−π
2
π
2
π
3π
2
−3
• You can review odd and even
functions in Section 1.5.
• You can review symmetry of
a graph in Section 1.2.
• You can review trigonometric
identities in Section 4.3.
• You can review asymptotes in
Section 2.6.
• You can review domain and
range of a function in Section
1.4.
• You can review intercepts of
a graph in Section 1.2.
FIGURE
4.59
Sketching the graph of y a tanbx c is similar to sketching the graph of
y a sinbx c in that you locate key points that identify the intercepts and asymptotes.
Two consecutive vertical asymptotes can be found by solving the equations
bx c 2
and
bx c .
2
The midpoint between two consecutive vertical asymptotes is an x-intercept of the
graph. The period of the function y a tanbx c is the distance between two
consecutive vertical asymptotes. The amplitude of a tangent function is not defined.
After plotting the asymptotes and the x-intercept, plot a few additional points between
the two asymptotes and sketch one cycle. Finally, sketch one or two additional cycles
to the left and right.
Section 4.6
y = tan
y
x
2
Example 1
331
Sketching the Graph of a Tangent Function
Sketch the graph of y tanx2.
3
2
Solution
1
By solving the equations
x
−π
π
3π
x
2
2
x
2
2
and
x x
you can see that two consecutive vertical asymptotes occur at x and x .
Between these two asymptotes, plot a few points, including the x-intercept, as shown in
the table. Three cycles of the graph are shown in Figure 4.60.
−3
FIGURE
Graphs of Other Trigonometric Functions
4.60
tan
x
2
2
0
2
1
0
1
Undef.
x
Undef.
Now try Exercise 15.
Example 2
Sketching the Graph of a Tangent Function
Sketch the graph of y 3 tan 2x.
Solution
y
By solving the equations
y = −3 tan 2x
6
x
− 3π − π
4
2
−π
4 −2
−4
π
4
π
2
3π
4
2x 2
x
4
and
2x 2
x
4
you can see that two consecutive vertical asymptotes occur at x 4 and x 4.
Between these two asymptotes, plot a few points, including the x-intercept, as shown in
the table. Three cycles of the graph are shown in Figure 4.61.
−6
FIGURE
4.61
x
3 tan 2x
4
Undef.
8
3
0
8
4
0
3
Undef.
By comparing the graphs in Examples 1 and 2, you can see that the graph of
y a tanbx c increases between consecutive vertical asymptotes when a > 0, and
decreases between consecutive vertical asymptotes when a < 0. In other words, the graph
for a < 0 is a reflection in the x-axis of the graph for a > 0.
Now try Exercise 17.
332
Chapter 4
Trigonometry
Graph of the Cotangent Function
The graph of the cotangent function is similar to the graph of the tangent function. It
also has a period of . However, from the identity
y cot x T E C H N O LO G Y
Some graphing utilities have
difficulty graphing trigonometric
functions that have vertical
asymptotes. Your graphing utility
may connect parts of the graphs
of tangent, cotangent, secant,
and cosecant functions that are
not supposed to be connected.
To eliminate this problem,
change the mode of the graphing
utility to dot mode.
you can see that the cotangent function has vertical asymptotes when sin x is zero,
which occurs at x n, where n is an integer. The graph of the cotangent function is
shown in Figure 4.62. Note that two consecutive vertical asymptotes of the graph of
y a cotbx c can be found by solving the equations bx c 0 and bx c .
y
1
x
−π
−π
2
π
2
Sketching the Graph of a Cotangent Function
1
Solution
x
π
3π 4π
6π
By solving the equations
x
0
3
x
3
3
and
x 3
x0
4.63
2π
4.62
2
−2π
FIGURE
3π
2
π
x
Sketch the graph of y 2 cot .
3
3
PERIOD: DOMAIN: ALL x n
RANGE: ( , )
VERTICAL ASYMPTOTES: x n
SYMMETRY: ORIGIN
2
Example 3
y = 2 cot x
3
y = cot x
3
FIGURE
y
cos x
sin x
you can see that two consecutive vertical asymptotes occur at x 0 and x 3.
Between these two asymptotes, plot a few points, including the x-intercept, as shown in
the table. Three cycles of the graph are shown in Figure 4.63. Note that the period is
3, the distance between consecutive asymptotes.
x
2 cot
x
3
0
3
4
3
2
9
4
3
Undef.
2
0
2
Undef.
Now try Exercise 27.
Section 4.6
333
Graphs of Other Trigonometric Functions
Graphs of the Reciprocal Functions
The graphs of the two remaining trigonometric functions can be obtained from the
graphs of the sine and cosine functions using the reciprocal identities
csc x 1
sin x
1
.
cos x
sec x and
For instance, at a given value of x, the y-coordinate of sec x is the reciprocal of the
y-coordinate of cos x. Of course, when cos x 0, the reciprocal does not exist. Near
such values of x, the behavior of the secant function is similar to that of the tangent
function. In other words, the graphs of
tan x sin x
cos x
sec x and
1
cos x
have vertical asymptotes at x 2 n, where n is an integer, and the cosine is zero
at these x-values. Similarly,
cot x cos x
sin x
csc x and
1
sin x
have vertical asymptotes where sin x 0 —that is, at x n.
To sketch the graph of a secant or cosecant function, you should first make a sketch
of its reciprocal function. For instance, to sketch the graph of y csc x, first sketch the
graph of y sin x. Then take reciprocals of the y-coordinates to obtain points on the
graph of y csc x. This procedure is used to obtain the graphs shown in Figure 4.64.
y
y
y = csc x
3
2
y = sin x
−π
−1
y = sec x
3
π
2
π
x
x
−π
−1
−2
π
2
π
2π
y = cos x
−3
PERIOD: 2
DOMAIN: ALL x n
RANGE: ( , 1 1, )
VERTICAL ASYMPTOTES: x n
SYMMETRY: ORIGIN
FIGURE 4.64
y
Cosecant:
relative
minimum
Sine:
minimum
4
3
2
1
x
−1
−2
−3
−4
FIGURE
Sine:
π
maximum
Cosecant:
relative
maximum
4.65
2π
PERIOD: 2
DOMAIN: ALL x 2 n
RANGE: ( , 1 1, )
VERTICAL ASYMPTOTES: x 2 n
SYMMETRY: y-AXIS
In comparing the graphs of the cosecant and secant functions with those of the sine
and cosine functions, note that the “hills” and “valleys” are interchanged. For example,
a hill (or maximum point) on the sine curve corresponds to a valley (a relative
minimum) on the cosecant curve, and a valley (or minimum point) on the sine curve
corresponds to a hill (a relative maximum) on the cosecant curve, as shown in
Figure 4.65. Additionally, x-intercepts of the sine and cosine functions become vertical
asymptotes of the cosecant and secant functions, respectively (see Figure 4.65).
334
Chapter 4
Trigonometry
y = 2 csc x + π y y = 2 sin x + π
4
4
(
)
(
)
Example 4
Sketching the Graph of a Cosecant Function
4
.
4
Sketch the graph of y 2 csc x 3
Solution
1
x
π
Begin by sketching the graph of
2π
.
4
y 2 sin x For this function, the amplitude is 2 and the period is 2. By solving the equations
FIGURE
x
4.66
0
4
x
x
and
4
2
4
x
7
4
you can see that one cycle of the sine function corresponds to the interval from
x 4 to x 74. The graph of this sine function is represented by the gray
curve in Figure 4.66. Because the sine function is zero at the midpoint and endpoints of
this interval, the corresponding cosecant function
y 2 csc x 2
4
sinx 1 4
has vertical asymptotes at x 4, x 34, x 74, etc. The graph of the
cosecant function is represented by the black curve in Figure 4.66.
Now try Exercise 33.
Example 5
Sketching the Graph of a Secant Function
Sketch the graph of y sec 2x.
Solution
y = sec 2x
y
y = cos 2x
Begin by sketching the graph of y cos 2x, as indicated by the gray curve in Figure
4.67. Then, form the graph of y sec 2x as the black curve in the figure. Note that the
x-intercepts of y cos 2x
3
4 , 0,
−π
−π
2
−1
π
2
π
x
4 , 0,
34, 0, . . .
correspond to the vertical asymptotes
x ,
4
x
,
4
x
3
,. . .
4
−2
−3
FIGURE
4.67
of the graph of y sec 2x. Moreover, notice that the period of y cos 2x and
y sec 2x is .
Now try Exercise 35.
Section 4.6
Graphs of Other Trigonometric Functions
335
Damped Trigonometric Graphs
A product of two functions can be graphed using properties of the individual functions.
For instance, consider the function
f x x sin x
as the product of the functions y x and y sin x. Using properties of absolute value
and the fact that sin x
1, you have 0
x sin x
x . Consequently,
y
y = −x 3π
x
y=x
x sin x
x
which means that the graph of f x x sin x lies between the lines y x and y x.
Furthermore, because
2π
π
f x x sin x ± x
x
π
−π
x
n
2
and
f x x sin x 0
−2π
x n
at
the graph of f touches the line y x or the line y x at x 2 n and has
x-intercepts at x n. A sketch of f is shown in Figure 4.68. In the function
f x x sin x, the factor x is called the damping factor.
−3π
f(x) = x sin x
FIGURE
at
4.68
Example 6
Damped Sine Wave
Sketch the graph of f x ex sin 3x.
Do you see why the graph of
f x x sin x touches the lines
y ± x at x 2 n and
why the graph has x-intercepts
at x n? Recall that the sine
function is equal to 1 at 2,
32, 52, . . . odd multiples
of 2 and is equal to 0 at ,
2, 3, . . . multiples of .
Solution
Consider f x as the product of the two functions
y ex
y sin 3x
and
each of which has the set of real numbers as its domain. For any real number x, you
1. So, ex sin 3x
ex, which means that
know that ex 0 and sin 3x
ex
ex sin 3x
ex.
Furthermore, because
f(x) = e−x sin 3x y
f x ex sin 3x ± ex at
6
y=
e−x
π
3
2π
3
f x ex sin 3x 0 at x x
y = −e−x
−6
FIGURE
4.69
n
6
3
and
4
−4
x
π
n
3
the graph of f touches the curves y ex and y ex at x 6 n3 and has
intercepts at x n3. A sketch is shown in Figure 4.69.
Now try Exercise 65.
336
Chapter 4
Trigonometry
Figure 4.70 summarizes the characteristics of the six basic trigonometric functions.
y
y
2
2
y = sin x
y
y = tan x
3
y = cos x
2
1
1
x
−π
−π
2
π
2
π
x
−π
3π
2
π
−2
DOMAIN: ( , )
RANGE: 1, 1
PERIOD: 2
DOMAIN: ( , )
RANGE: 1, 1
PERIOD: 2
y = csc x =
1
sin x
y
3
π
2
y = sec x =
1
cos x
y
2
2
1
1
π
2π
y = cot x = tan1 x
3
3
π
2
x
x
−π
−π
2
5π
2
3π
2
π
DOMAIN: ALL x 2 n
RANGE: ( , )
PERIOD: x
−π
x
−π
2
−1
−2
y
2π
π
2
π
3π
2
π
2π
2π
−2
−3
DOMAIN: ALL x n
RANGE: ( , 1 1, )
PERIOD: 2
FIGURE 4.70
DOMAIN: ALL x 2 n
RANGE: ( , 1 1, )
PERIOD: 2
DOMAIN: ALL x n
RANGE: ( , )
PERIOD: CLASSROOM DISCUSSION
Combining Trigonometric Functions Recall from Section 1.8 that functions can
be combined arithmetically. This also applies to trigonometric functions. For each of
the functions
h!x" ! x
sin x
and
h!x" ! cos x " sin 3x
(a) identify two simpler functions f and g that comprise the combination, (b) use a
table to show how to obtain the numerical values of h!x" from the numerical values
of f !x" and g!x", and (c) use graphs of f and g to show how the graph of h may be
formed.
Can you find functions
f !x" ! d
such that f !x"
a sin!bx
c"
and
g!x" ! 0 for all x?
g!x" ! d
a cos!bx
c"
Section 4.6
EXERCISES
4.6
337
Graphs of Other Trigonometric Functions
See www.CalcChat.com for worked-out solutions to odd-numbered exercises.
VOCABULARY: Fill in the blanks.
1. The tangent, cotangent, and cosecant functions are ________ , so the graphs of these functions have symmetry
with respect to the ________.
2. The graphs of the tangent, cotangent, secant, and cosecant functions all have ________ asymptotes.
3. To sketch the graph of a secant or cosecant function, first make a sketch of its corresponding ________ function.
4. For the functions given by f x gx ! sin x, gx is called the ________ factor of the function f x.
5. The period of y tan x is ________.
6. The domain of y cot x is all real numbers such that ________.
7. The range of y sec x is ________.
8. The period of y csc x is ________.
SKILLS AND APPLICATIONS
In Exercises 9–14, match the function with its graph. State
the period of the function. [The graphs are labeled (a), (b),
(c), (d), (e), and (f).]
y
(a)
2
1
1
tan x
3
17. y 2 tan 3x
19. y 12 sec x
15. y y
(b)
In Exercises 15–38, sketch the graph of the function. Include
two full periods.
1
x
x
1
2
y
4
3
2
1
3
2
x
−π
2
−3
−4
3π
2
−3
y
4
29. y 2 sec 3x
x
31. y tan
4
33. y 2 cscx 3
37. y π
2
x
1
11. y cot x
2
x
1
13. y sec
2
2
32. y tanx 34. y csc2x 36. y sec x 1
1
csc x 4
4
38. y 2 cot x 2
x
1
9. y sec 2x
x
2
35. y 2 secx y
(f )
22. y 3 csc 4x
24. y 2 sec 4x 2
x
26. y csc
3
x
28. y 3 cot
2
1
30. y 2 tan x
27. y 3 cot 2x
x
π
2
− 3π
2
(e)
y
(d)
18. y 3 tan x
20. y 14 sec x
21. y csc x
23. y 12 sec x
25. y csc
(c)
16. y tan 4x
x
10. y tan
2
12. y csc x
x
14. y 2 sec
2
In Exercises 39–48, use a graphing utility to graph the
function. Include two full periods.
39. y tan
x
3
40. y tan 2x
41. y 2 sec 4x
43. y tan x 4
45. y csc4x x 47. y 0.1 tan
4
4
42. y sec x
1
44. y cot x 4
2
46. y 2 sec2x 1
x 48. y sec
3
2
2
338
Chapter 4
Trigonometry
In Exercises 49–56, use a graph to solve the equation on the
interval ["2#, 2#].
50. tan x 3
49. tan x 1
51. cot x 3
3
52. cot x 1
53. sec x 2
54. sec x 2
55. csc x 2
56. csc x 23
3
70. y1 tan x cot2 x, y2 cot x
71. y1 1 cot2 x, y2 csc2 x
72. y1 sec2 x 1, y2 tan2 x
In Exercises 73–76, match the function with its graph.
Describe the behavior of the function as x approaches zero.
[The graphs are labeled (a), (b), (c), and (d).]
y
(a)
2
f x sec x
gx cot x
f x x tan x
gx x csc x
58.
60.
62.
64.
f x tan x
gx csc x
f x x2 sec x
4
x
In Exercises 57– 64, use the graph of the function to
determine whether the function is even, odd, or neither.
Verify your answer algebraically.
57.
59.
61.
63.
y
(b)
x
π
2
gx x2 cot x
y
(d)
4
3
2
1
4
2
65. GRAPHICAL REASONING
given by
f x 2 sin x and gx Consider the functions
1
csc x
2
on the interval 0, .
(a) Graph f and g in the same coordinate plane.
(b) Approximate the interval in which f > g.
(c) Describe the behavior of each of the functions as x
approaches . How is the behavior of g related to
the behavior of f as x approaches ?
66. GRAPHICAL REASONING Consider the functions
given by
f x tan
x
x
1
and gx sec
2
2
2
on the interval 1, 1.
(a) Use a graphing utility to graph f and g in the same
viewing window.
(b) Approximate the interval in which f < g.
(c) Approximate the interval in which 2f < 2g. How
does the result compare with that of part (b)?
Explain.
In Exercises 67–72, use a graphing utility to graph the two
equations in the same viewing window. Use the graphs to
determine whether the expressions are equivalent. Verify the
results algebraically.
67. y1 sin x csc x, y2 1
68. y1 sin x sec x, y2 tan x
cos x
, y2 cot x
69. y1 sin x
x
−π
3π
2
−4
y
(c)
2
π
2
−1
−2
−3
−4
−5
−6
π
−2
−π
−4
73. f x x cos x
75. gx x sin x
π
−1
−2
x
74. f x x sin x
76. gx x cos x
CONJECTURE In Exercises 77– 80, graph the functions f
and g. Use the graphs to make a conjecture about the
relationship between the functions.
2 ,
78. f x sin x cosx ,
2
77. f x sin x cos x gx 0
gx 2 sin x
79. f x sin2 x, gx 12 1 cos 2x
1
x
80. f x cos2
, gx 1 cos x
2
2
In Exercises 81–84, use a graphing utility to graph the
function and the damping factor of the function in the same
viewing window. Describe the behavior of the function as x
increases without bound.
81. gx ex
83. f x 22
2x4
sin x
cos x
82. f x ex cos x
84. hx 2x 4 sin x
2
In Exercises 85–90, use a graphing utility to graph the
function. Describe the behavior of the function as x
approaches zero.
85. y 6
cos x, x > 0
x
86. y 4
sin 2x, x > 0
x
87. gx sin x
x
88. f x 1
89. f x sin
x
1 cos x
x
1
90. hx x sin
x
91. DISTANCE A plane flying at an altitude of 7 miles
above a radar antenna will pass directly over the radar
antenna (see figure). Let d be the ground distance from
the antenna to the point directly under the plane and
let x be the angle of elevation to the plane from the
antenna. (d is positive as the plane approaches the
antenna.) Write d as a function of x and graph the
function over the interval 0 < x < .
7 mi
x
d
Not drawn to scale
92. TELEVISION COVERAGE A television camera is on a
reviewing platform 27 meters from the street on which a
parade will be passing from left to right (see figure).
Write the distance d from the camera to a particular unit
in the parade as a function of the angle x, and graph the
function over the interval 2 < x < 2. (Consider
x as negative when a unit in the parade approaches from
the left.)
Temperature
(in degrees Fahrenheit)
Section 4.6
Graphs of Other Trigonometric Functions
80
339
H(t)
60
40
L(t)
20
t
1
2
3
4
5
6
7
8
9
10 11 12
Month of year
(a) What is the period of each function?
(b) During what part of the year is the difference
between the normal high and normal low
temperatures greatest? When is it smallest?
(c) The sun is northernmost in the sky around June 21,
but the graph shows the warmest temperatures at a
later date. Approximate the lag time of the
temperatures relative to the position of the sun.
94. SALES The projected monthly sales S (in thousands of
units) of lawn mowers (a seasonal product) are modeled
by S 74 3t 40 cost6, where t is the time (in
months), with t 1 corresponding to January. Graph
the sales function over 1 year.
95. HARMONIC MOTION An object weighing W pounds
is suspended from the ceiling by a steel spring (see
figure). The weight is pulled downward (positive
direction) from its equilibrium position and released.
The resulting motion of the weight is described by the
1
function y 2 et4 cos 4t, t > 0, where y is the distance (in feet) and t is the time (in seconds).
Not drawn to scale
27 m
Equilibrium
d
y
x
Camera
93. METEOROLOGY The normal monthly high temperatures H (in degrees Fahrenheit) in Erie, Pennsylvania
are approximated by
(a) Use a graphing utility to graph the function.
(b) Describe the behavior of the displacement function
for increasing values of time t.
Ht 56.94 20.86 cos t6 11.58 sin t6
EXPLORATION
and the normal monthly low temperatures L are
approximated by
TRUE OR FALSE? In Exercises 96 and 97, determine
whether the statement is true or false. Justify your answer.
Lt 41.80 17.13 cos t6 13.39 sin t6
96. The graph of y csc x can be obtained on a calculator
by graphing the reciprocal of y sin x.
97. The graph of y sec x can be obtained on a
calculator by graphing a translation of the reciprocal of
y sin x.
where t is the time (in months), with t 1 corresponding
to January (see figure). (Source: National Climatic
Data Center)
340
Chapter 4
Trigonometry
98. CAPSTONE Determine which function is represented by the graph. Do not use a calculator. Explain
your reasoning.
(a)
(b)
y
y
x
π
4
π
2
(i) f x tan 2x
(ii) f x tanx2
(iii) f x 2 tan x
(iv) f x tan 2x
(v) f x tanx2
x
−π −π
2
4
(i) f x (ii) f x (iii) f x (iv) f x (v) f x π
4
π
2
sec 4x
cscx4
secx4
csc4x as x approaches #2 from the right
#
#
as x approaches from the left
(b) x →
2 2
#
#
(c) x → "
as x approaches " from the right
2 2
#
#
(d) x → "
as x approaches " from the left
2 2
#
2
"
"
99. f x tan x
As x → 0 , the value of f !x" → .
As x → 0", the value of f !x" → .
As x → # , the value of f !x" → .
As x → # ", the value of f !x" → .
101. f x cot x
What value does the sequence approach?
104. APPROXIMATION Using calculus, it can be shown
that the tangent function can be approximated by the
polynomial
tan x
x
2x 3 16x 5
3!
5!
where x is in radians. Use a graphing utility to graph
the tangent function and its polynomial approximation
in the same viewing window. How do the graphs
compare?
105. APPROXIMATION Using calculus, it can be shown
that the secant function can be approximated by the
polynomial
sec x
1
x 2 5x 4
2!
4!
where x is in radians. Use a graphing utility to graph
the secant function and its polynomial approximation
in the same viewing window. How do the graphs
compare?
106. PATTERN RECOGNITION
(a) Use a graphing utility to graph each function.
4
1
sin x sin 3 x
3
y2 4
1
1
sin x sin 3 x sin 5 x
3
5
(b) Identify the pattern started in part (a) and find a
function y3 that continues the pattern one more
term. Use a graphing utility to graph y3.
(c) The graphs in parts (a) and (b) approximate the
periodic function in the figure. Find a function y4
that is a better approximation.
y
102. f x csc x
103. THINK ABOUT IT Consider the function given by
f x x cos x.
(a) Use a graphing utility to graph the function and
verify that there exists a zero between 0 and 1. Use
the graph to approximate the zero.
y1 100. f x sec x
In Exercises 101 and 102, use a graphing utility to graph the
function. Use the graph to determine the behavior of the
function as x → c.
(a)
(b)
(c)
(d)
x1 cosx0
x2 cosx1
csc 4x
In Exercises 99 and 100, use a graphing utility to graph the
function. Use the graph to determine the behavior of the
function as x → c.
(a) x →
x0 1
x3 cosx2
3
2
1
− π4
(b) Starting with x0 1, generate a sequence x1, x2,
x3, . . . , where xn cosxn1. For example,
1
x
3
Section 4.7
Inverse Trigonometric Functions
341
4.7 INVERSE TRIGONOMETRIC FUNCTIONS
What you should learn
• Evaluate and graph the inverse
sine function.
• Evaluate and graph the other
inverse trigonometric functions.
• Evaluate and graph the
compositions of trigonometric
functions.
Inverse Sine Function
Recall from Section 1.9 that, for a function to have an inverse function, it must be
one-to-one—that is, it must pass the Horizontal Line Test. From Figure 4.71, you can
see that y sin x does not pass the test because different values of x yield the same
y-value.
y
y = sin x
1
Why you should learn it
You can use inverse trigonometric
functions to model and solve
real-life problems. For instance,
in Exercise 106 on page 349, an
inverse trigonometric function can be
used to model the angle of elevation
from a television camera to a space
shuttle launch.
−π
π
x
−1
sin x has an inverse function
on this interval.
FIGURE
4.71
However, if you restrict the domain to the interval 2 x 2
(corresponding to the black portion of the graph in Figure 4.71), the following properties
hold.
1. On the interval 2, 2, the function y sin x is increasing.
2. On the interval 2, 2, y sin x takes on its full range of values,
1 sin x 1.
3. On the interval 2, 2, y sin x is one-to-one.
So, on the restricted domain 2 x 2, y sin x has a unique inverse
function called the inverse sine function. It is denoted by
y arcsin x
or
y sin1 x.
NASA
The notation sin1 x is consistent with the inverse function notation f 1x. The arcsin x
notation (read as “the arcsine of x”) comes from the association of a central angle with
its intercepted arc length on a unit circle. So, arcsin x means the angle (or arc) whose
sine is x. Both notations, arcsin x and sin1 x, are commonly used in mathematics, so
remember that sin1 x denotes the inverse sine function rather than 1sin x. The values
of arcsin x lie in the interval 2 arcsin x 2. The graph of y arcsin x is
shown in Example 2.
Definition of Inverse Sine Function
When evaluating the inverse sine
function, it helps to remember
the phrase “the arcsine of x is
the angle (or number) whose
sine is x.”
The inverse sine function is defined by
y arcsin x
if and only if
sin y x
where 1 x 1 and 2 y 2. The domain of y arcsin x is
1, 1, and the range is 2, 2.
342
Chapter 4
Trigonometry
Example 1
As with the trigonometric
functions, much of the work
with the inverse trigonometric
functions can be done by exact
calculations rather than by
calculator approximations.
Exact calculations help to
increase your understanding of
the inverse functions by relating
them to the right triangle
definitions of the trigonometric
functions.
Evaluating the Inverse Sine Function
If possible, find the exact value.
2
a. arcsin 1
b. sin1
3
c. sin1 2
2
Solution
6 2 for 2
a. Because sin 1
2 6 .
arcsin b. Because sin
sin1
1
2
, it follows that
2
Angle whose sine is 21
3
for 3
2
2
3
y
.
3
, it follows that
2
y
Angle whose sine is 32
c. It is not possible to evaluate y sin1 x when x 2 because there is no angle whose
sine is 2. Remember that the domain of the inverse sine function is 1, 1.
Now try Exercise 5.
Example 2
Graphing the Arcsine Function
Sketch a graph of
y arcsin x.
Solution
By definition, the equations y arcsin x and sin y x are equivalent for
2 y 2. So, their graphs are the same. From the interval 2, 2, you
can assign values to y in the second equation to make a table of values. Then plot the
points and draw a smooth curve through the points.
y
(1, π2 )
π
2
( 22 , π4 )
( 21 , π6 )
(0, 0)
− 1, −π
2 6
(
(
FIGURE
)
4.72
x sin y
1
1
)
−1, − π
2
x
2
y
4
2
2
6
0
6
4
1
2
0
1
2
2
2
2
1
y = arcsin x
−π
2
(
2 π
−
,−
2
4
)
The resulting graph for y arcsin x is shown in Figure 4.72. Note that it is the
reflection (in the line y x) of the black portion of the graph in Figure 4.71. Be sure
you see that Figure 4.72 shows the entire graph of the inverse sine function. Remember
that the domain of y arcsin x is the closed interval 1, 1 and the range is the closed
interval 2, 2.
Now try Exercise 21.
Section 4.7
343
Inverse Trigonometric Functions
Other Inverse Trigonometric Functions
The cosine function is decreasing and one-to-one on the interval 0
in Figure 4.73.
, as shown
x
y
y = cos x
x
−π
π
2
−1
π
2π
cos x has an inverse function
on this interval.
FIGURE
4.73
Consequently, on this interval the cosine function has an inverse function—the inverse
cosine function—denoted by
y arccos x
or
y cos1 x.
Similarly, you can define an inverse tangent function by restricting the domain of
y tan x to the interval 2, 2. The following list summarizes the definitions of
the three most common inverse trigonometric functions. The remaining three are
defined in Exercises 115–117.
Definitions of the Inverse Trigonometric Functions
Function
Domain
Range
y arcsin x if and only if sin y x
1
x
1
y arccos x if and only if cos y x
1
x
1
0
y arctan x if and only if tan y x
< x < 2
2
y
y
< y <
2
2
The graphs of these three inverse trigonometric functions are shown in Figure 4.74.
y
y
π
2
y
π
2
π
y = arcsin x
π
2
x
−1
1
−
π
2
DOMAIN: 1, 1
RANGE: 2 , 2 FIGURE 4.74
y = arctan x
y = arccos x
x
−1
DOMAIN: 1, 1
RANGE: 0, x
−2
1
−1
−
1
π
2
DOMAIN: , RANGE: 2 , 2 2
344
Chapter 4
Trigonometry
Example 3
Evaluating Inverse Trigonometric Functions
Find the exact value.
a. arccos
2
b. cos11
2
d. tan11
c. arctan 0
Solution
a. Because cos4 22, and 4 lies in 0, , it follows that
arccos
2
2
.
4
Angle whose cosine is 22
b. Because cos 1, and lies in 0, , it follows that
cos11 .
Angle whose cosine is 1
c. Because tan 0 0, and 0 lies in 2, 2, it follows that
arctan 0 0.
Angle whose tangent is 0
d. Because tan 4 1, and 4 lies in 2, 2, it follows that
tan11 .
4
Angle whose tangent is 1
Now try Exercise 15.
Example 4
Calculators and Inverse Trigonometric Functions
Use a calculator to approximate the value (if possible).
a. arctan8.45
b. sin1 0.2447
c. arccos 2
Solution
Function
WARNING / CAUTION
Remember that the domain of the
inverse sine function and the
inverse cosine function is 1, 1,
as indicated in Example 4(c).
Mode
Radian
Calculator Keystrokes
TAN1 8.45 ENTER
Radian
COS1
a. arctan8.45
From the display, it follows that arctan8.45 1.453001.
SIN1 0.2447 ENTER
b. sin1 0.2447
Radian
1
From the display, it follows that sin 0.2447 0.2472103.
c. arccos 2
2
ENTER
In real number mode, the calculator should display an error message because the
domain of the inverse cosine function is 1, 1.
Now try Exercise 29.
In Example 4, if you had set the calculator to degree mode, the displays would have
been in degrees rather than radians. This convention is peculiar to calculators. By
definition, the values of inverse trigonometric functions are always in radians.
Section 4.7
Inverse Trigonometric Functions
345
Compositions of Functions
Recall from Section 1.9 that for all x in the domains of f and f 1, inverse functions have
the properties
You can review the composition
of functions in Section 1.8.
f f 1x x
f 1 f x x.
and
Inverse Properties of Trigonometric Functions
If 1
1 and 2
x
sinarcsin x x
If 1
, then
y
cosarccos x x
2, then
arcsinsin y y.
and
1 and 0
x
y
arccoscos y y.
and
If x is a real number and 2 < y < 2, then
tanarctan x x
arctantan y y.
and
Keep in mind that these inverse properties do not apply for arbitrary values of x and
y. For instance,
arcsin sin
3
3
arcsin1 .
2
2
2
In other words, the property
arcsinsin y y
is not valid for values of y outside the interval 2, 2.
Example 5
Using Inverse Properties
If possible, find the exact value.
a. tanarctan5
b. arcsin sin
5
3
c. coscos1 Solution
a. Because 5 lies in the domain of the arctan function, the inverse property applies,
and you have
tanarctan5 5.
b. In this case, 53 does not lie within the range of the arcsine function,
2 y 2. However, 53 is coterminal with
5
2 3
3
which does lie in the range of the arcsine function, and you have
arcsin sin
3 .
5
arcsin sin 3
3
c. The expression coscos1 is not defined because cos1 is not defined.
Remember that the domain of the inverse cosine function is 1, 1.
Now try Exercise 49.
346
Chapter 4
Trigonometry
Example 6 shows how to use right triangles to find exact values of
compositions of inverse functions. Then, Example 7 shows how to use right triangles to
convert a trigonometric expression into an algebraic expression. This conversion
technique is used frequently in calculus.
y
32− 22=
3
Example 6
Evaluating Compositions of Functions
5
Find the exact value.
u = arccos
2
3
x
2
Angle whose cosine is 23
FIGURE 4.75
a. tan arccos
2
3
3
Solution
2
2
a. If you let u arccos 3, then cos u 3. Because cos u is positive, u is a first-quadrant
angle. You can sketch and label angle u as shown in Figure 4.75. Consequently,
y
5 2 − (−3) 2 = 4
x
( (
u = arcsin −
3
5
−3
5
tan arccos
2
opp 5
tan u .
3
adj
2
b. If you let u arcsin 35 , then sin u 35. Because sin u is negative, u is a fourthquadrant angle. You can sketch and label angle u as shown in Figure 4.76.
Consequently,
5 cos u hyp 5.
cos arcsin Angle whose sine is
FIGURE 4.76
5 b. cos arcsin 3
5
adj
3
4
Now try Exercise 57.
Example 7
Some Problems from Calculus
Write each of the following as an algebraic expression in x.
a. sinarccos 3x,
1
1 − (3x)2
0
x
1
3
b. cotarccos 3x,
0
3x
Angle whose cosine is 3x
FIGURE 4.77
1
3
Solution
If you let u arccos 3x, then cos u 3x, where 1
u = arccos 3x
x <
cos u 3x
1. Because
adj
3x
hyp
1
you can sketch a right triangle with acute angle u, as shown in Figure 4.77. From this
triangle, you can easily convert each expression to algebraic form.
opp
1 9x 2, 0
hyp
adj
3x
b. cotarccos 3x cot u , 0
opp 1 9x 2
a. sinarccos 3x sin u 1
3
1
x <
3
x
Now try Exercise 67.
In Example 7, similar arguments can be made for x-values lying in the interval
13, 0.
Section 4.7
EXERCISES
4.7
347
Inverse Trigonometric Functions
See www.CalcChat.com for worked-out solutions to odd-numbered exercises.
VOCABULARY: Fill in the blanks.
Function
Alternative Notation
Domain
1. y arcsin x
__________
__________
2. __________
3. y arctan x
y cos1 x
1 x 1
__________
__________
Range
2
2
y
__________
__________
4. Without restrictions, no trigonometric function has a(n) __________ function.
SKILLS AND APPLICATIONS
In Exercises 5–20, evaluate the expression without using a
calculator.
5. arcsin 12
7. arccos 12
6. arcsin 0
8. arccos 0
3
9. arctan
11. cos1 2
13. arctan3 2
15. arccos 17. sin1 y
41.
π
2
π
4
10. arctan1
3
3
In Exercises 41 and 42, determine the missing coordinates of
the points on the graph of the function.
1
12. sin1 2
14. arctan 3
16. arcsin
2 3
3
(
1 2
(− 3, )
(
)
π
(−1, )
(− 12 , ) π
x
3
π
,−6
4
)
y = arccos x
(
π
,6
1
2
)
x
−2
43.
−1
44.
x
x
20. cos1 1
In Exercises 21 and 22, use a graphing utility to graph f, g, and
y ! x in the same viewing window to verify geometrically that
g is the inverse function of f. (Be sure to restrict the domain
of f properly.)
21. f x sin x,
−3 −2
π
,4
In Exercises 43– 48, use an inverse trigonometric function to
write as a function of x.
2
18. tan1 19. tan1 0
2
3
2
y
42.
y = arctan x
θ
θ
4
4
45.
46.
5
gx arcsin x
22. f x tan x, gx arctan x
x+1
x+2
θ
θ
10
In Exercises 23– 40, use a calculator to evaluate the
expression. Round your result to two decimal places.
23. arccos 0.37
25. arcsin0.75
27. arctan3
24. arcsin 0.65
26. arccos0.7
29. sin1 0.31
31. arccos0.41
28. arctan 25
30. cos1 0.26
32. arcsin0.125
33. arctan 0.92
7
35. arcsin 8
19
37. tan1 4
34. arctan 2.8
1
36. arccos 3 95
38. tan1 7 39. tan1 372 40. tan1 2165 47.
48.
2x
θ
x−1
θ
x+3
x2 − 1
In Exercises 49–54, use the properties of inverse trigonometric
functions to evaluate the expression.
49. sinarcsin 0.3
51. cosarccos0.1
53. arcsinsin 3
50. tanarctan 45
52. sinarcsin0.2
7
54. arccos cos
2
348
Chapter 4
Trigonometry
In Exercises 55–66, find the exact value of the expression.
(Hint: Sketch a right triangle.)
55. sinarctan 4 56. secarcsin 5 57. costan1 2
58. sin cos1
59. cosarcsin 3
61. sec arctan 5 2
63. sin arccos 3 60. csc 62. tan
64. cot 3
5
5
2
83. g x arcsinx 1
66. sec sin1 2
2
84. gx arcsin
In Exercises 67–76, write an algebraic expression that is
equivalent to the expression. (Hint: Sketch a right triangle, as
demonstrated in Example 7.)
67. cotarctan x
68. sinarctan x
69. cosarcsin 2x
70. secarctan 3x
71. sinarccos x
72. secarcsinx 1
x
2
In Exercises 85–90, sketch a graph of the function.
85. y 2 arccos x
86. gt arccost 2
87. f x) arctan 2x
88. f x arctan x
2
89. hv tanarccos v
x
90. f x arccos
4
In Exercises 91–96, use a graphing utility to graph the
function.
3
1
74. cotarctan x
x
75. cscarctan
2 xh
76. cosarcsin
r x
91. f x 2 arccos2x
92. f x arcsin4x
93. f x arctan2x 3
94. f x 3 arctan x
95. f x sin1
96. f x In Exercises 77 and 78, use a graphing utility to graph f
and g in the same viewing window to verify that the two
functions are equal. Explain why they are equal. Identify any
asymptotes of the graphs.
2x
77. f x sinarctan 2x, gx 1 4x2
4 x 2
x
78. f x tan arccos , gx 2
x
x 2
In Exercises 83 and 84, sketch a graph of the function and
compare the graph of g with the graph of f "x# ! arcsin x.
5
arctan 12
3
arcsin 4
arctan 58
23 73. tan arccos
x2
arctan,
2
4
5
13
65. csc cos1
82. arccos
23
1 cos1
2
In Exercises 97 and 98, write the function in terms of the sine
function by using the identity
A cos $t 1 B sin $t ! A2 1 B2 sin $t 1 arctan
!
A
.
B
Use a graphing utility to graph both forms of the function.
What does the graph imply?
97. f t 3 cos 2t 3 sin 2t
98. f t 4 cos t 3 sin t
In Exercises 79–82, fill in the blank.
79. arctan
80. arcsin
9
arcsin,
x
36 x 2
6
x0
arccos,
0
3
arcsin
81. arccos
x 2 2x 10
x
6
In Exercises 99–104, fill in the blank. If not possible,
state the reason. (Note: The notation x → c# indicates that
x approaches c from the right and x → c " indicates that x
approaches c from the left.)
99. As x → 1, the value of arcsin x → .
100. As x → 1, the value of arccos x → .
Section 4.7
101. As x → , the value of arctan x → .
102. As x → 1, the value of arcsin x → .
103. As x → 1, the value of arccos x → .
349
3 ft
104. As x → , the value of arctan x → .
105. DOCKING A BOAT A boat is pulled in by means of
a winch located on a dock 5 feet above the deck of the
boat (see figure). Let " be the angle of elevation from
the boat to the winch and let s be the length of the rope
from the winch to the boat.
β θ
α
1 ft
x
Not drawn to scale
(a) Use a graphing utility to graph # as a function of x.
(b) Move the cursor along the graph to approximate
the distance from the picture when # is maximum.
(c) Identify the asymptote of the graph and discuss its
meaning in the context of the problem.
s
5 ft
Inverse Trigonometric Functions
θ
(a) Write " as a function of s.
(b) Find " when s 40 feet and s 20 feet.
106. PHOTOGRAPHY A television camera at ground
level is filming the lift-off of a space shuttle at a point
750 meters from the launch pad (see figure). Let " be
the angle of elevation to the shuttle and let s be the
height of the shuttle.
108. GRANULAR ANGLE OF REPOSE Different types of
granular substances naturally settle at different angles
when stored in cone-shaped piles. This angle " is
called the angle of repose (see figure). When rock
salt is stored in a cone-shaped pile 11 feet high, the
diameter of the pile’s base is about 34 feet. (Source:
Bulk-Store Structures, Inc.)
11 ft
θ
17 ft
s
θ
750 m
Not drawn to scale
(a) Write " as a function of s.
(b) Find " when s 300 meters and s 1200
meters.
107. PHOTOGRAPHY A photographer is taking a picture
of a three-foot-tall painting hung in an art gallery.
The camera lens is 1 foot below the lower edge of the
painting (see figure). The angle # subtended by the
camera lens x feet from the painting is
3x
, x > 0.
# arctan 2
x 4
(a) Find the angle of repose for rock salt.
(b) How tall is a pile of rock salt that has a base
diameter of 40 feet?
109. GRANULAR ANGLE OF REPOSE When whole
corn is stored in a cone-shaped pile 20 feet high, the
diameter of the pile’s base is about 82 feet.
(a) Find the angle of repose for whole corn.
(b) How tall is a pile of corn that has a base diameter
of 100 feet?
110. ANGLE OF ELEVATION An airplane flies at an
altitude of 6 miles toward a point directly over an
observer. Consider " and x as shown in the figure.
6 mi
θ
x
Not drawn to scale
(a) Write " as a function of x.
(b) Find " when x 7 miles and x 1 mile.
350
Chapter 4
Trigonometry
111. SECURITY PATROL A security car with its spotlight
on is parked 20 meters from a warehouse. Consider
and as shown in the figure.
In Exercises 127–134, use the results of Exercises 115–117
and a calculator to approximate the value of the expression.
Round your result to two decimal places.
128. arcsec1.52
130. arccot10
16
132. arccot 7 127. arcsec 2.54
129. arccot 5.25
5
131. arccot 3
25
133. arccsc 3 135. AREA In calculus, it is shown that the area of
the region bounded by the graphs of y 0,
y 1x 2 1, x a, and x b is given by
x
(a) Write " as a function of x.
(b) Find " when x 5 meters and x 12 meters.
EXPLORATION
TRUE OR FALSE? In Exercises 112–114, determine
whether the statement is true or false. Justify your answer.
5 1
112. sin
6
2
5
1
113. tan
4
y
1
−2
In Exercises 119–126, use the results of Exercises 115–117 to
evaluate each expression without using a calculator.
2 3 3 (see figure). Find the area for the following values of a
and b.
(a) a 0, b 1
(b) a 1, b 1
(c) a 0, b 3
(d) a 1, b 3
y=
115. Define the inverse cotangent function by restricting the
domain of the cotangent function to the interval 0, ,
and sketch its graph.
116. Define the inverse secant function by restricting the
domain of the secant function to the intervals 0, 2
and 2, , and sketch its graph.
117. Define the inverse cosecant function by restricting the
domain of the cosecant function to the intervals
2, 0 and 0, 2, and sketch its graph.
125. arccsc
Area arctan b arctan a
1 5
arcsin 2
6
5
arctan 1 4
arcsin x
114. arctan x arccos x
119. arcsec 2
121. arccot1
123. arccsc 2
134. arccsc12
120. arcsec 1
122. arccot 3 124. arccsc1
126. arcsec 23
3
136. THINK ABOUT IT
the functions
a
b 2
x
Use a graphing utility to graph
f x x and gx 6 arctan x.
For x > 0, it appears that g > f. Explain why you
know that there exists a positive real number a such
that g < f for x > a. Approximate the number a.
137. THINK ABOUT IT Consider the functions given by
f x sin x and f 1x arcsin x.
(a) Use a graphing utility to graph the composite
functions f $ f 1 and f 1 $ f.
(b) Explain why the graphs in part (a) are not the
graph of the line y x. Why do the graphs of
f $ f 1 and f 1 $ f differ?
138. PROOF Prove each identity.
(a) arcsinx arcsin x
(b) arctanx arctan x
1 (c) arctan x arctan ,
x
2
2
x
(e) arcsin x arctan
1 x 2
(d) arcsin x arccos x 1
x2 + 1
x > 0
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