Hungerford T.W., Shaw D.J. Contemporary Precalculus: A Graphing Approach
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An angle in the coordinate plane is said to be in standard position if its ver-
tex is at the origin and one of its sides (which we call the initial side) is on the
positive x-axis, as shown in Figure 8–21. The other side of the angle will be called
its terminal side.
586 CHAPTER 8 Triangle Trigonometry
Initial Initial
Terminal
Terminal
Figure 8–21
Let u be an acute angle in standard position. Choose any point (x, y) on the
terminal side of u (except the origin) and consider the right triangle with vertices
(0, 0), (x, 0), and (x, y) shown in Figure 8–22.
The legs of this triangle have lengths x and y, respectively. The distance for-
mula shows that the hypotenuse has length
(x 0
)
2
(
y 0)
2
x
2
y
2
,
which we denote by r. The triangle shows that
sin u
y
r
cos u
x
r
tan u
y
x
.
We now have a description of the trigonometric functions in terms of the coordi-
nate plane rather than right triangles. Furthermore, this description makes sense
for any angle. Consequently, we make the following definition.
(0, 0)
(x, y)
(x, 0)
r
θ
x
y
Figure 8–22
Point-in-the-Plane
Description
Let u be an angle in standard position and (x, y) any point (except the ori-
gin) on its terminal side. Then the trigonometric functions are defined by
these rules:
sin u
y
r
, cos u
x
r
, tan u
y
x
(x 0),
where r
x
2
y
2
is the distance from (x, y) to the origin.
The discussion preceding the box shows that when u is an acute angle, these
definitions produce the same numbers for sin u, cos u, and tan u as does the right
triangle definition on page 585. It can be shown that the values of the trigono-
metric functions of u are independent of the point that is chosen on the terminal
side ( just as the definition for acute angles is independent of the size of the right
triangle).
EXAMPLE 10
Find sin u, cos u, and tan u for the angle u shown in Figure 8–23.
SOLUTION Since (5, 7) is on the terminal side of u, we apply the definitions
in the preceding box with (x, y) (5, 7) and
r
x
2
y
2
(5)
2
7
2
74:
sin u
y
r
7
74
cos u
x
r
7
5
4
tan u
y
x
7
5
7
5
.
■
Figure 8–23
For most angles, we use a calculator in degree mode to approximate the val-
ues of the trigonometric functions, as in Figure 8–24. But there are a few angles
for which we can compute the exact values of the trigonometric functions.
EXAMPLE 11
Find the exact values of the trigonometric functions at u 90°.
SOLUTION We use the point (0, 1) on the terminal side of an angle of 90° in
standard position (Figure 8–25). In this case, r
x
2
y
2
0
2
1
2
1, so
sin 90°
y
r
1
1
1
cos 90°
x
r
0
1
0
tan 90°
y
x
1
0
(undefined). ■
Figure 8–25
The last part of Example 11 shows that the domain of the tangent function
excludes 90°, whereas the domains of sine and cosine include all angles.
EXAMPLE 12
Find the exact values of sin 135°, cos 135°, and tan 135°.
SOLUTION Construct an angle of 135° in standard position and let P be the
point on the terminal side that is 1 unit from the origin (Figure 8–26 on the next
page). Draw a vertical line from P to the x-axis, forming a right triangle with hy-
potenuse 1 and two angles of 45°. Each side of this triangle has length 2
/2, as
(0, 1)
90°
x
y
(−5, 7)
74
θ
x
y
ALTERNATE 8.1 Trigonometric Functions of Angles 587
Figure 8–24
explained in Example 2 of the Geometry Review Appendix. Therefore, the coor-
dinates of P are (2
/2, 2
/2), and we have
sin 135°
y
r
2
1
/2
2
2
cos 135°
x
r
1
2
/2
2
2
tan 135°
y
x
1.
Figure 8–26 ■
2
/2
2
/2
135°
45°
45°
1
x
y
P
2
2
2
2
2
22
,
()
2
−
588 CHAPTER 8 Triangle Trigonometry
EXERCISES ALTERNATE 8.1
Use the exercises for Section 8.1 on page 582.
8.2 Applications of Right Triangle Trigonometry
■ Use right triangle trigonometry to solve applied problems.
The following examples illustrate a variety of practical applications of triangle
trigonometry.
EXAMPLE 1
Lola and her sister Harper see a tree on the river’s edge directly opposite them. They
walk along the riverbank for 120 feet and note that the angle formed by their path and
a line to the tree measures 70°, as indicated in Figure 8–27. How wide is the river?
SOLUTION Using the right triangle whose one leg is the width w of the river
and whose other leg is the 120-foot path on this side of the river, we see that
tan 70°
o
ad
p
j
p
a
o
c
s
e
i
n
te
t
1
w
20
.
Solving this equation for w, we have
w 120 tan 70° 329.7.
So the river is about 330 feet wide. ■
EXAMPLE 2
A plane takes off, making an angle of 18° with the ground. After the plane travels
three miles along this flight path, how high (in feet) is it above the ground?
Section Objective
w
70°
120
Figure 8–27
SECTION 8.2 Applications of Right Triangle Trigonometry 589
h
18°
3 miles
Figure 8–28
SOLUTION Figure 8–28 shows that
sin 18°
hy
o
p
p
o
p
t
o
e
s
n
i
u
te
se
h
3
Multiply both sides by 3: h 3 sin 18° .92705 miles.
Since there are 5280 feet in a mile, the height of the plane in feet is
h .92705
5280 4894.8 feet. ■
EXAMPLE 3
According to the safety sticker on an extension ladder, the distance from the foot
of the ladder to the base of the wall on which it leans should be one-fourth of the
length of the ladder. If the ladder is in this position, what angle does it make with
the ground?
SOLUTION Let c be the distance from the foot of the ladder to the base of the
wall. Then the ladder’s length is 4c, as shown in Figure 8–29. If u is the angle the
ladder makes with the ground, then
cos u
hy
a
p
d
o
ja
t
c
e
e
n
n
u
t
se
4
c
c
1
4
.
Using the COS
1
key, we find that u 75.52°, as shown in Figure 8–30. ■
c
θ
4c
Figure 8–29
Figure 8–30
In many practical applications, one uses the angle between the horizontal and
some other line (for instance, the line of sight from an observer to a distant object).
This angle is called the angle of elevation or the angle of depression, depending
on whether the line is above or below the horizontal, as shown in Figure 8–31.
590 CHAPTER 8 Triangle Trigonometry
Horizontal
Angle of elevation
Angle of depression
Figure 8–31
EXAMPLE 4
A surveyor stands on one edge of a ravine. By using the method in Example 1, she
determines that the ravine is 125 feet wide. She then determines that the angle of
depression from the edge where she is standing to a point on the bottom of the ravine
is 57.5°, as shown in Figure 8–32 (which is not to scale). How deep is the ravine?
125 ft
Angle of
depression
d
57.5°
SOLUTION From Figure 8–32, we see that
tan 57.5°
o
ad
p
j
p
a
o
c
s
e
i
n
te
t
1
d
25
Multiply both sides by 125: d 125 tan 57.5°
d 196.21 feet ■
EXAMPLE 5
A wire is to be stretched from the top of a 10-meter-high building to a point on the
ground. From the top of the building, the angle of depression to the ground point
is 22°. How long must the wire be?
Figure 8–32
SOLUTION Figure 8–33 shows that the sum of the angle of depression and the
angle a is 90°. Hence, a measures 90° 22° 68°. We know the length of
the side of the triangle adjacent to the angle a and must find the hypotenuse w (the
length of the wire). Using the cosine function, we see that
cos 68°
hy
a
p
d
o
ja
t
c
e
e
n
n
u
t
se
1
w
0
w
cos
10
68°
26.7 meters. ■
EXAMPLE 6
A large American flag flies from a pole on top of the Terminal Tower in Cleveland
(Figure 8–34). At a point 203 feet from the base of the tower, the angle of eleva-
tion to the bottom of the flag pole is 74°, and the angle of elevation to the top of
the pole is 75.285°. To the nearest foot, how long is the flagpole?
SECTION 8.2 Applications of Right Triangle Trigonometry 591
Wire
α
Angle of depression
22°
10
Figure 8–33
Figure 8–36
75.285°
74°
c
d
203
Figure 8–34
Figure 8–35
SOLUTION By abstracting the given information, we see that there are two
right triangles, as shown in Figure 8–35 (which is not to scale). The length of the
flagpole is c d. We can use the two triangles to find c and d.
Larger Triangle Smaller Triangle
20
c
3
o
ad
p
j
p
a
o
c
s
e
i
n
te
t
tan 75.285°
2
d
03
o
ad
p
j
p
a
o
c
s
e
i
n
te
t
tan 74°
c 203 tan 75.285° 773 d 203 tan 74° 708
As shown in Figure 8–36, the length of the flagpole is
c d 773 708 65 feet. ■
EXAMPLE 7
Phil Embree stands on the edge of one bank of a canal and observes a lamp post on the
edge of the other bank of the canal. His eye level is 152 centimeters above the ground
(approximately 5 feet). The angle of elevation from eye level to the top of the lamp
post is 12°, and the angle of depression from eye level to the bottom of the lamp post
is 7°, as shown in Figure 8–37. How wide is the canal? How high is the lamp post?
592 CHAPTER 8 Triangle Trigonometry
12°
7°
152
Figure 8–37
SOLUTION Abstracting the essential information, we obtain the diagram in
Figure 8–38.
12°
A
ED
C
B
7°
152 152
Figure 8–38
We must find the height of the lamp post BD and the width of the canal AC (or
ED). The eye level height AE of the observer is 152 centimeters. Since AC and ED
are parallel, CD also has length 152 centimeters. In right triangle ACD, we know
the angle of 7° and the side CD opposite it. We must find the adjacent side AC.
The tangent function is needed.
tan 7°
o
ad
p
j
p
a
o
c
s
e
i
n
te
t
1
A
5
C
2
or, equivalently, AC
ta
1
n
52
7°
AC
ta
1
n
52
7°
1237.94 centimeters.
So the canal is approximately 12.3794 meters* wide (about 40.6 feet). Now using
right triangle ACB, we see that
tan 12°
o
ad
p
j
p
a
o
c
s
e
i
n
te
t
B
A
C
C
12
B
37
C
.94
or, equivalently,
BC 1237.94(tan 12°) 263.13 centimeters.
Therefore, the height of the lamp post BD is BC CD 263.13 152 415.13
centimeters or, equivalently, 4.1513 meters. ■
*Remember, 100 centimeters 1 meter.
SECTION 8.2 Applications of Right Triangle Trigonometry 593
EXERCISES 8.2
In Exercises 1–4, solve the right triangle.
1.
2.
3.
4.
5. What is the width of the river in Example 1 if the angle to the
tree is 40° (and all the other information is the same)?
6. Suppose the plane in Example 2 takes off at an angle of 5°
and travels along this path for one mile. How high (in feet)
is the plane above the ground?
7. Suppose you have a 24-foot-long ladder and you ignore the
safety advice in Example 3 by placing the foot of the ladder
9 feet from the base of the wall. What angle does the ladder
make with the ground?
8. The surveyor in Example 4 stands at the edge of another
ravine, which is known to be 115 feet wide. She notes that the
angle of depression from the edge she is standing on to the
bottom of the oposite side is 64.3°. How deep is this ravine?
9. How long a wire is needed in Example 5 if the angle of de-
pression is 25.8°?
10. Suppose that the flagpole on the Terminal Tower (Example 6)
has been replaced. Now from a point 240 feet from the base
of the tower, the angle of elevation to the bottom of the flag-
pole is 71.3°, and the angle of elevation to the top of the pole
is 72.9°. To the nearest foot, how long is the new flagpole?
11. A 20-foot-long ladder leans on a wall of a building. The
foot of the ladder makes an angle of 50° with the ground.
AB
C
b
a
14.5
63°
6
4.5
B
A
C
7.5
12
13
5
B
C
A
B
C
A
c
a
8
16°
How far above the ground does the top of the ladder touch
the wall?
12. A pilot flying at an altitude of 14,500 feet notes that his
angle of depression to the control tower of a nearby airport
is 15°. If the plane continues flying at this altitude toward
the control tower, how far must it travel before it is directly
over the tower?
13. A straight road leads from an ocean beach into the nearby hills.
The road has a constant upward grade of 3°. After taking this
road for one mile, how high above sea level (in feet) are you?
14. If you travel the road in Exercise 13 for a mile and a half,
how high above sea level are you?
15. A powerful searchlight projects a beam of light vertically up-
ward so that it shines on the bottom of a cloud. A clinometer,
600 feet from the searchlight, measures the angle u, as shown
in the figure. If u measures 80°, how high is the cloud?
16. A wire from the top of a TV tower to the ground makes an
angle of 49.5° with the ground and touches ground 225 feet
from the base of the tower. How high is the tower?
600 ft
5280 ft
Ocean
h = ?
Sea level
Road
3°
15°
14,500
17. Find the distance across the pond (from B to C) if AC is
110 feet and angle A measures 38°.
18. The Seattle Space Needle casts a 225-foot-long shadow.
If the angle of elevation from the tip of the shadow to the
top of the Space Needle is 69.6°, how high is the Space
Needle?
19. Batman is on the edge of a 200-foot-deep chasm and wants
to jump to the other side. A tree on the edge of the chasm is
directly across from him. He walks 20 feet to his right and
notes that the angle to the tree is 54°. His jet belt enables
him to jump a maximum of 24 feet. How wide is the chasm,
and is it safe for Batman to jump?
20. From the top of a 130-foot-high lighthouse, the angle of de-
pression to a boat in Lake Erie is 2.5°. How far is the boat
from the lighthouse?
21. If you stand upright on a mountainside that makes a 62°
angle with the horizontal and stretch your arm straight out at
shoulder height, you may be able to touch the mountain (as
shown in the figure). Can a person with an arm reach of
27 inches, whose shoulder is five feet above the ground,
touch the mountain?
20 ft
54°
A
C
B
594 CHAPTER 8 Triangle Trigonometry
22. Alice is flying a kite. Her hand is three feet above ground
level and is holding the end of a 300-foot-long kite string,
which makes an angle of 57° with the horizontal. How high
is the kite above the ground?
23. It is claimed that the Ohio Turnpike never has an uphill grade
of more than 3°. How long must a straight uphill segment of
the road be to allow a vertical rise of 450 feet?
24. A swimming pool is three feet deep in the shallow end. The
bottom of the pool has a steady downward drop of 12°. If
the pool is 50 feet long, how deep is it at the deep end?
25. Consider a 16-foot-long drawbridge on a medieval castle, as
shown in the figure. The royal army is engaged in ignomin-
ious retreat. The king would like to raise the end of the
drawbridge 8 feet off the ground so that Sir Rodney can
jump onto the drawbridge and scramble into the castle
while the enemy’s cavalry are held at bay. Through how
much of an angle must the drawbridge be raised for the end
of it to be 8 feet off the ground?
26. Through what angle must the drawbridge in Exercise 25 be
raised in order that its end be directly above the center of the
moat?
27. A buoy in the ocean is observed from the top of a 40-meter-
high radar tower on shore. The angle of depression from the
top of the tower to the base of the buoy is 6.5°. How far is
the buoy from the base of the radar tower?
16 ft
62°
28. A 150-foot-long ramp connects a ground-level parking lot
with the entrance of a building. If the entrance is 8 feet
above the ground, what angle does the ramp make with the
ground?
29. A plane flies a straight course. On the ground directly below
the flight path, observers two miles apart spot the plane at
the same time. The plane’s angle of elevation is 46° from
one observation point and 71° from the other. How high is
the plane?
30. A man stands 20 feet from a statue. The angle of elevation
from his eye level to the top of the statue is 30°, and the
angle of depression to the base of the statue is 15°. How tall
is the statue?
31. Two boats lie on a straight line with the base of a lighthouse.
From the top of the lighthouse (21 meters above water
level), it is observed that the angle of depression of the
nearer boat is 53° and the angle of depression of the farther
boat is 27°. How far apart are the boats?
32. A rocket shoots straight up from the launchpad. Five sec-
onds after liftoff, an observer two miles away notes that the
rocket’s angle of elevation is 3.5°. Four seconds later, the
angle of elevation is 41°. How far did the rocket rise during
those four seconds?
33. From a 35-meter-high window, the angle of depression to
the top of a nearby streetlight is 55°. The angle of depres-
sion to the base of the streetlight is 57.8°. How high is the
streetlight?
53°
27°
46°
71°
2 miles
2 miles
SECTION 8.2 Applications of Right Triangle Trigonometry 595
34. A plane takes off at an angle of 6° traveling at the rate of
200 feet/second. If it continues on this flight path at the
same speed, how many minutes will it take to reach an
altitude of 8000 feet?
35. A car on a straight road passes under a bridge. Two seconds
later an observer on the bridge, 20 feet above the road, notes
that the angle of depression to the car is 7.4°. How fast (in
miles per hour) is the car traveling? [Note: 60 mph is equiv-
alent to 88 feet/second.]
36. A plane passes directly over your head at an altitude of
500 feet. Two seconds later, you observe that its angle of
elevation is 42°. How far did the plane travel during those
two seconds?
37. Laura Bernett is 5 ft-4 inches tall. She stands 10 feet from a
streetlight and casts a 4-foot-long shadow. How tall is the
streetlight? What is angle u?
38. One plane flies straight east at an altitude of 31,000 feet. A
second plane is flying west at an altitude of 14,000 feet on
a course that lies directly below that of the first plane
and directly above the straight road from Thomasville to
Johnsburg. As the first plane passes over Thomasville, the
second is passing over Johnsburg. At that instant, both
planes spot a beacon next to the road between Thomasville
to Johnsburg. The angle of depression from the first plane to
the beacon is 61°, and the angle of depression from the
second plane to the beacon is 34°. How far is Thomasville
from Johnsburg?
39. A schematic diagram of a pedestrian overpass is shown in
the figure. If you walk on the overpass from one end to the
other, how far have you walked?
15°
21°
18 ft
200 ft
θ