Hungerford T.W., Shaw D.J. Contemporary Precalculus: A Graphing Approach
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EXAMPLE 1
(a) Find the length of an arc of a circle of radius 8 ft that is intercepted by a
central angle of 2.3 radians.
(b) Find the length of an arc of a circle of radius 12 inches that is intercepted by
a central angle of 60°.
SOLUTION
(a) We apply the arc length formula with u 2.3 and r 8:
s ur (2.3)8 18.4 ft.
(b) The arc length formula requires that the angle be in radians. Since 60° is p/3
radians, we have
s ur
p
3
12 4p 12.57 inches. ■
EXAMPLE 2
A central angle intercepts an arc of length 14.64 m on a circle of radius 6 m. Find
the radian measure of the angle.
SOLUTION We solve the arc length formula s ur for u:
u
s
r
14
6
.64
2.44 radians. ■
EXAMPLE 3
A nautical mile is defined to be the length of an arc on the surface of the earth that
is intercepted by an angle u of measure 1/60 of 1° whose vertex is at the center of
the earth (see Figure 6–12, which is not to scale). Using 3960 miles as the radius
of the earth, find the length of a nautical mile in terms of ordinary miles.
SOLUTION First we express the angle u in radians by using the formula in the
box on page 433:
6
1
0
°
6
1
0
1
p
80
radians
10,
p
800
radians .0002908882 radians.
Now we use the arc length formula in the preceding box and find that the length
of a nautical mile is
s ur .0002908882(3960) 1.15 miles. ■
EXAMPLE 4
The second hand on a large clock is 6 inches long. How far does the tip of the sec-
ond hand move in 15 seconds?
436 CHAPTER 6 Trigonometric Functions
θ
Figure 6–12
SOLUTION The second hand makes a full revolution every 60 seconds, that is,
it moves through an angle of 2p radians. During a 15-second interval it will
make
1
6
5
0
1
4
of a revolution, moving through an angle of p/2 radians (Figure 6–13).
If we think of the second hand as the radius of a circle, then during a 15-second
interval, its tip travels along the arc intercepted by an angle of p/2 radians. There-
fore, the distance (arc length) traveled by the tip of the second hand is
s ur
p
2
6 3p 9.425 inches. ■
AREA OF A CIRCULAR SECTOR
Look at the shaded sector with a central angle of u radians in Figure 6–14. Since
a full circle angle is 2p, the area A of the shaded sector is
2
u
p
of the area of the
entire circular disc (which is, as usual, pr
2
). Hence,
A
2
u
p
(pr
2
)
u
2
r
2
1
2
r
2
u.
SPECIAL TOPICS 6.1.A Arc Length and Angular Speed 437
π
2
12
6
57
48
39
210
111
Figure 6–13
Area of
a Sector
The area A of a sector with central angle u radians in a circle of radius r is
A
1
2
r
2
u.
r
θ
Figure 6–14
EXAMPLE 5
In a circle of radius 5 ft, find the area of a sector with central angle
(a) 2.4 radians (b) 45°.
SOLUTION
(a) Using the area formula in the preceding box, we have
A
1
2
r
2
u
1
2
5
2
2.4
6
2
0
30 square feet.
(b) The formula is valid only when the angle is expressed in radians. Since 45 is
p4, the area is
A
1
2
r
2
u
1
2
5
2
p
4
25
8
p
9.8175 square feet. ■
ANGULAR SPEED
Suppose a point P moves along a circle of radius r at a constant rate, tracing out
an arc of length s, as in Figure 6–15. The linear speed of P is the rate at which the
distance traveled by P along the circle changes with respect to time, namely,
s
t
.
The angular speed of P is the rate at which the angle u changes with respect to
time:
u
t
.
The linear speed of P is denoted by v and the angular speed by v (the lower case
Greek letter omega). Using the arc length formula s ur, we have this summary:
EXAMPLE 6
A merry-go-round makes eight revolutions per minute.
(a) What is the angular speed of the merry-go-round in radians per minute?
(b) How fast is a horse 12 feet from the center traveling?
(c) How fast is a horse 4 feet from the center traveling?
SOLUTION
(a) Each revolution of the merry-go-round corresponds to a central angle of 2p
radians, so it travels through an angle of 8
2p 16p radians in one minute.
Therefore, its angular speed is
v
u
t
16
1
p
16p radians per minute.
Angle traced out by OP
Time elapsed
Distance traveled by P
Time elapsed
438 CHAPTER 6 Trigonometric Functions
Linear and
Angular Speed
Suppose a point P moves along a circle of radius r and center O for a period
of time t. If the arc traveled by P has length s and the angle traced out by OP
measures u radians, then
v
s
t
u
t
r
and v
u
t
.
Linear Speed Angular Speed
O
r
P
P
s
θ
Figure 6–15
(b) The horse that is 12 feet from the center travels along a circle of radius 12 and
travels through an angle of 16p radians in 1 minute. Therefore, its linear
speed is
v
u
t
r
192p feet per minute,
which is approximataely 6.9 miles per hour.
(c) For the horse that is 4 feet from the center, r 4, u 16p, and t 1. Hence, the
linear speed is
v
u
t
r
64p feet per minute (about 2.28 mph). ■
By rewriting the linear speed equation, we obtain the relationship between
linear and angular speed.
v
s
t
u
t
r
r
u
t
rv.
EXAMPLE 7
A bicycle, each of whose wheels is 26 inches in diameter, travels at a constant 14 mph.
(a) What is the angular speed of one of its wheels (in radians per hour)?
(b) How many revolutions per minute does each wheel make?
SOLUTION Within each computation, we must be careful to use comparable
units.
(a) Since the wheel size is given in inches, we also express the speed in terms of
inches. Recall that there are 5280 feet in a mile and 12 inches in a foot, which
means that
1 mile 5280
12 63,360 inches.
Thus, the bike’s linear speed is
v 14 miles per hour 14(63,360) inches per hour.
The radius of the wheel is
2
2
6
13 inches. As we saw above, v rv, so the
angular speed is
v
v
r
14(6
1
3
3
,360)
68,233.85 radians per hour.
(b) Each revolution corresponds to an angle of 2p radians, so the number of
revolutions per hour is
68,2
2
3
p
3.85
. Dividing this number by 60 gives the revo-
lutions per minute.
6
1
0
68,2
2
3
p
3.85
181 revolutions per minute. ■
16p
4
1
16p
12
1
SPECIAL TOPICS 6.1.A Arc Length and Angular Speed 439
Linear and
Angular Speed
If a point P moves along a circle of radius r with linear speed v and angu-
lar speed v, then
v rv.
440 CHAPTER 6 Trigonometric Functions
EXERCISES 6.1.A
In Exercises 1–4, find the length of the circular arc subtended
by the central angle whose radian measure is given. Assume
the circle has diameter 10.
1. 1 radian 2. 2 radians
3. 1.75 radians 4. 2.2 radians
5. The second hand on a clock is 6 centimeters long. How
far does its tip travel in 40 seconds? (See Exercise 51 in
section 6.1.)
6. The second hand on a clock is 5 centimeters long. How far
does its tip travel in 2 minutes and 15 seconds? (See Exer-
cise 54 in section 6.1.)
In Exercises 7–10, find the radian measure of the angle u.
7. 8.
9. 10.
In Exercises 11–14, an arc length and angle (in radian meas-
ure) are given. Find the radius r of the circle.
11. 12.
13. 14.
45.6
r
3.8
25
3
r
16
2
r
9
1.5
r
50.83
10
θ
30
15
θ
85
20
θ
175
70
θ
In Exercises 15–18, assume that a wheel on a car has radius
36 centimeters. Find the angle (in radians) that the wheel turns
while the car travels the given distance.
15. 2 meters ( 200 centimeters) 16. 5 meters
17. 720 meters
18. 1 kilometer ( 1000 meters)
In Exercises 19–22, the latitudes of a pair of cities are given.
Assume that one city is directly south of the other and that the
earth is a perfect sphere of radius 3960 miles. Find the dis-
tance between the two cities. [The latitude of a point P on the
earth is the degree measure of the angle u between the point
and the plane of the equator (with the center of the earth being
the vertex), as shown in the figure. Remember that angles are
measured in radians in the arc length formula.]
19. The North Pole (latitude 90° north) and Springfield, Illinois
(latitude 40° north).
20. San Antonio, Texas (latitude 29.5° north), and Mexico City,
Mexico (latitude 20° north).
21. Cleveland, Ohio (latitude 41.5° north), and Tampa, Florida
(latitude 28° north).
22. Copenhagen, Denmark (latitude 54.3° north), and Rome,
Italy (latitude 42° north).
In Exercises 23–26, find the area of the sector with the given
central angle and the given radius.
23. Angle 1 radian; radius 5 ft
24. Angle 1.8 radians; radius 10 in
25. Angle 2.4 radians; radius 8 m
26. Angle 4 radians; radius 9 cm
27. If a sector of a circle of radius 2 ft has an area of 4 sq ft, find
the radian measure of the central angle.
28. If a sector of a circle of radius 5 m has an area of 37.5 sq m,
find the radian measure of the central angle.
29. If the area of a sector of a circle with a central angle of
1.5 radians is 4.32 sq ft, find the radius of the circle.
P
Equator
θ
SPECIAL TOPICS 6.1.A Arc Length and Angular Speed 441
30. If the area of a sector of a circle with a central angle of
2.2 radians is 23.276 sq in, find the radius of the circle.
31. One day in Syene (modern day Aswan), the Greek mathe-
matician Eratosthenes (c. 276–194
B.C.E.) noted that the
sun’s rays shown directly down a deep well. One year later
at the same time in Alexandria (which is 500 miles due
north of Syene), he determined that the sun’s rays were at an
angle of approximately 7.2, as shown in the figure. From
this information Eratosthenes was able to find the approxi-
mate radius and circumference of the earth. You should do
the same.
32. A person is standing on the equator. The earth is making one
full revolution each 24 hours. Assuming that the radius of
the earth is 3960 miles, find the person’s approximate
(a) linear velocity (in mph)
(b) angular velocity (in radians per hour)
33. One end of a rope is attached to a winch (circular drum) of
radius 2 feet and the other to a steel beam on the ground.
When the winch is rotated, the rope wraps around the drum
and pulls the object upward (see the figure). Through what
angle must the winch be rotated to raise the beam 6 feet
above the ground?
Syene
7.2
Alexandria
500 miles
34. A circular saw blade has an angular speed of 15,000 radians
per minute.
(a) How many revolutions per minute does the saw
make?
(b) How long will it take the saw to make 6000 revolu-
tions?
35. A circular gear rotates at the rate of 200 revolutions per
minute (rpm).
(a) What is the angular speed of the gear in radians per
minute?
(b) What is the linear speed of a point on the gear 2 inches
from the center in inches per minute and in feet per
minute?
36. A wheel in a large machine is 2.8 feet in diameter and
rotates at 1200 rpm.
(a) What is the angular speed of the wheel?
(b) How fast is a point on the circumference of the wheel
traveling in feet per minute? In miles per hour?
37. A riding lawn mower has wheels that are 15 inches in
diameter. If the wheels are making 2.5 revolutions per
second.
(a) What is the angular speed of a wheel?
(b) How fast is the lawn mower traveling in miles per hour?
38. A weight is attached to the end of a 4-foot-long leather cord.
A boy swings the cord in a circular motion over his head. If
the weight makes 12 revolutions every 10 seconds, what is
its linear speed?
39. A merry-go-round horse is traveling at 10 feet per second,
and the merry-go-round is making 6 revolutions per
minute. How far is the horse from the center of the
merry-go-round?
40. The pedal sprocket of a bicycle has radius 4.5 inches, and
the rear wheel sprocket has radius 1.5 inches (see figure).
If the rear wheel has a radius of 13.5 inches and the cyclist
is pedaling at the rate of 80 rpm, how fast is the bicycle
traveling in feet per minute? In miles per hour?
442 CHAPTER 6 Trigonometric Functions
6.2 The Sine, Cosine, and Tangent Functions
■ Use the unit circle to define the sine, cosine and tangent
functions.
■ Find the exact values of sine, cosine, and tangent at p3, p4,
p6, and integer multiples of these numbers.
■ Use the point-in-the-plane description to evaluate the
trigonometric functions.
Unlike most of the functions seen thus far, the definitions of the sine and cosine
functions do not involve algebraic formulas. Instead, these functions are defined
geometrically, using the unit circle.* Recall that the unit circle is the circle of
radius 1 with center at the origin, whose equation is x
2
y
2
1.
Both the sine and cosine functions have the set of all real numbers as domain.
Their rules are given by the following three-step geometric process:
1. Given a real number t, construct an
angle of t radians in standard posi-
tion.
2. Find the coordinates of the point P
where the terminal side of this angle
meets the unit circle x
2
y
2
1,
say P (a, b).
3. The value of the cosine function at
t (denoted cos t) is the x-coordinate
of P:
cos t a.
The value of the sine function at t
(denoted sin t) is the y-coordinate
of P:
sin t b.
Section Objectives
(a, b)
−1
−1
1
1
x
y
P
x
y
t radians
NOTE
If you have read Chapter 8, use Alternate Section 6.2 on page 452 in place of this section.
Instructors who wish to cover all
six trigonometric functions
simultaneously should
incorporate Section 6.6 into
Sections 6.2–6.4, as follows.
Instructors who prefer to
introduce triangle trigonometry
early should consult the chart on
page xiv.
Subsection of Cover at
Section 6.6 the end of
Part I Section 6.2
Part II Section 6.3
Part III Section 6.4
*If you have previously studied the trigonometry of triangles, the definition given here may not look
familiar. If this is the case, just concentrate on this definition and don’t worry about relating it to any
definition you remember from the past. The connection between this definition and the trigonometry
of triangles will be explained in Chapter 8.
Sine and
Cosine
If P is the point where the terminal side of an angle of t radians in standard
position meets the unit circle, then
P has coordinates (cos t, sin t).
Roadmap
SECTION 6.2 The Sine, Cosine, and Tangent Functions 443
With your calculator in radian mode and parametric graphing mode, set the range
values as follows:
0 t 2p, 1.8 x 1.8, 1.2 y 1.2*
Then graph the curve given by the parametric equations
x cos t and y sin t.
The graph is the unit circle. Use the trace to move around the circle. At each point,
the screen will display three numbers: the values of t, x, and y. For each t, the cur-
sor is on the point where the terminal side of an angle of t radians meets the unit
circle, so the corresponding x is the number cos t and the corresponding y is the
number sin t.
GRAPHING EXPLORATION
The tangent function is defined as the quotient of the sine and cosine functions.
Its value at the number t, denoted tan t, is given by
tan t
c
s
o
in
s
t
t
.
EXAMPLE 1
Evaluate the three trigonometric functions at
(a) t p (b) t p/2.
SOLUTION
(a) Construct an angle of p radians, as in Figure 6–16. Its terminal side lies on
the negative x-axis and intersects the unit circle at P (1, 0). Hence,
sin p y-coordinate of P 0
cos p x-coordinate of P 1
tan p
c
s
o
in
s
p
p
0
1
0
*Parametric graphing is explained in Special Topics 3.3.A. These settings give a square viewing
window on calculators with a screen measuring approximately 95 by 63 pixels (such as TI-84+), and
hence the unit circle will look like a circle. For wider screens, adjust the x range settings to obtain a
square window.
π
1
1−1
−1
x
P
y
Figure 6–16
(b) An angle of p/2 radians (Figure 6–17) has its terminal side on the positive
y-axis and intersects the unit circle at P (0, 1).
cos
p
2
x-coordinate of P 0
sin
p
2
y-coordinate of P 1
tan
p
2
c
s
o
in
s
(
(
p
p
/
/
2
2
)
)
1
0
undefined
The definitions of sine and cosine show that sin t and cos t are defined for
every real number t. Example 1(b), however, shows that tan t is not defined when
the x-coordinate of the point P is 0. This occurs when P has coordinates (0, 1) or
(0, 1), that is, when t p2, 3p2, 5p2, etc. Consequently, the domain
(set of inputs) of each trigonometric function is as follows.
444 CHAPTER 6 Trigonometric Functions
π
1
2
−1
x
P
y
Figure 6–17
Function Domain
f (t) sin t All real numbers
g(t) cos t All real numbers
h(t) tan t All real numbers except
p/2, 3p/2, 5p/2, . . .
In most cases, evaluating trigonometric functions is not as simple as in
Example 1. Usually, you must use the SIN, COS, and TAN keys on a calculator (in
radian mode) to approximate sin t, cos t, and tan t, as illustrated in Figure 6–18.
EXAMPLE 2
When a baseball is hit by a bat, the horizontal distance d traveled by the ball is
approximated by
d
v
2
sin
16
t cos t
,
where the ball leaves the bat at an angle of t radians and has initial velocity v feet
per second, as shown in Figure 6–19.
Figure 6–18
TECHNOLOGY TIP
Throughout this chapter, be sure your
calculator is set for RADIAN mode. Use
the MODE(S) menu on TI and HP and
the SETUP menu on Casio.
■
(a) How far does the ball travel when the initial velocity is 90 feet per second and
t .7?
(b) If the initial velocity is 105 feet per second and t 1, how far does the ball
travel?
Figure 6–19
SOLUTION
(a) Let v 90 and t .7 in the formula for d. Then a calculator (in radian mode)
shows that
d
v
2
sin
16
t cos t
90
2
sin
1
.
6
7 cos .7
249.44 feet.
(b) Now let v 105 and t 1. Then
d
v
2
sin
16
t cos t
105
2
si
1
n
6
1 cos 1
313.28. ■
SPECIAL VALUES
The trigonometric functions can be evaluated exactly at t p/3, t p/4,
t p/6, and any integer multiples of these numbers by using the following facts
(which are explained in Examples 2–4 of the Geometry Review Appendix):*
A right triangle with hypotenuse A right triangle with hypotenuse
1 and angles of p/6 and p/3 1 and two angles of p/4 radians
radians has sides of lengths 1/2 has two sides of length 2
/2.
(opposite the angle of p/6) and
3
/2 (opposite the angle of p/3).
Figure 6–20
2
1
2
π
2
π
4
π
4
2
2
1
2
3
1
2
π
2
π
3
π
6
d
t
SECTION 6.2 The Sine, Cosine, and Tangent Functions 445
*Angles in the Geometry Review are given in degree measure: 60°, 45°, 30° instead of radian measure
p/3, p/4, p/6, as is done here.