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that his angle of elevation to the plane is 81° and that his
angle of elevation to the top of Cartalk Tower is 8.6°. As-
suming that Tom and Ray and the airplane are in a plane
perpendicular to the ground, how high is the airplane?
46. A plane flies in a direction of 105° from airport A. After a
time, it turns and proceeds in a direction of 267°. Finally, it
lands at airport B, 120 miles directly south of airport A.
How far has the plane traveled? [Note: Aerial navigation di-
rections are explained in Exercise 41 of Section 8.2.]
47. Charlie is afraid of water; he can’t swim and refuses to get
in a boat. However, he must measure the width of a river for
his geography class. He has a long tape measure but no way
to measure angles. While pondering what to do, he paces
along the side of the river using the five paths joining points
A, B, C, and D. If he can’t determine the width of the river,
he will flunk the course.
E
A
C
B
D
81°
67°
8.6°
800 ft
1 mile
616 CHAPTER 8 Triangle Trigonometry
(a) Save Charlie from disaster by explaining how he can de-
termine the width AE simply by measuring the lengths
AB, AC, AD, BC, and BD and using trigonometry.
(b) Charlie determines that AB 75 feet, AC 25 feet,
AD 90 feet, BC 80 feet, and BD 22 feet. How
wide is the river between A and E?
48. A plane flies in a direction of 85° from Chicago. It then
turns and flies in the direction of 200° for 150 miles. It is
then 195 miles from its starting point. How far did the plane
fly in the direction of 85°? (See the note in Exercise 46.)
49. A hinged crane makes an angle of 50° with the ground. A
malfunction causes the lock on the hinge to fail and the top
part of the crane swings down. How far from the base of the
crane does the top hit the ground?
50. When 0° u 90°, the figure and parts (a)–(d) below pro-
vide a proof of the supplementary angle identity:
sin u sin(180° u).
(a) What is the second coordinate of D?
(b) Use the point-in-the-plane description to find
sin(180° u).
(c) Use right triangle DEO to find sin u.
(d) What do you conclude from parts (b) and (c)?
51. Given triangle ABC, with B 60°, a 7, and c 15, solve
the triangle as follows.
(a) Show that b 13. [Hint: Example 1 of Section 8.3.]
(b) Use the Law of Sines to find angle C.
(c) Use the fact that the sum of the angles is 180° to find
angle A.
x
y
OE
D
r
h
180° −
θ
θ
14.6 m
19 m
50°
■ Find the area of a triangle when an angle and its sides are
known.
■ Use Heron’s Formula to find the area of a triangle when its sides
are known.
The proof of the Law of Sines enables us to prove a useful fact.
SPECIAL TOPICS 8.4.A The Area of a Triangle 617
8.4.A SPECIAL TOPICS The Area of a Triangle
Section Objectives
Area of
a Triangle
The area of a triangle containing an angle C with sides of lengths a and b is
1
2
ab sin C.
Proof Place the vertex of angle C at the origin, with side b on the positive
x-axis (Figure 8–56).* Then b is the base and h is the altitude of the triangle so that
area of triangle ABC
1
2
base altitude
1
2
b
h.
The proof of the Law of Sines on page 606 shows that h a sin C. Therefore,
area of triangle ABC
1
2
b
h
1
2
b
a sin C
1
2
ab sin C. ■
EXAMPLE 1
Find the area of the triangle shown in Figure 8–57.
SOLUTION
1
2
8
13 sin 130° 39.83 square centimeters. ■
x
y
C
b
DA
B
c
h
a
Figure 8–56
*Figure 8–56 is the case when C is larger than 90°; the argument when C is less than 90° is similar.
8 cm 13 cm
130°
Figure 8–57
Here is a useful formula for the area of a triangle in terms of its sides.
Heron’s
Formula
The area of a triangle with sides a, b, c is
s(s a
)(s b
)(s c
)
,
where s
1
2
(a b c).
Proof The preceding area formula and the Pythagorean identity,
sin
2
C 1 cos
2
C (1 cos C)(1 cos C),
show that the area of triangle ABC (standard notation) is
1
2
ab sin C
1
2
ab
sin C
2
1
4
a
2
b
2
sin
2
C
1
4
a
2
b
2
(1 c
os
2
C )
1
2
ab(1
cos
C)
1
2
ab(1
cos C
)
.
Exercise 22 uses the Law of Cosines to show that
1
2
ab(1 cos C)
(a b
4
)
2
c
2
(a
2
b) c
(a
2
b) c
s(s c)
and
1
2
ab(1 cos C)
c
2
(a
4
b)
2
c (a
2
b)
c (a
2
b)
(s a)(s b).
Combining these facts completes the proof.
Area
1
2
ab sin C
1
2
ab(1
cos
C)
1
2
a
b(1
cos C)
s(s a
)(s b
)(s c
)
. ■
EXAMPLE 2
Find the area of the triangle whose sides have lengths 7, 9, and 12.
SOLUTION Apply Heron’s Formula with a 7, b 9, c 12, and
s
1
2
(a b c)
1
2
(7 9 12) 14.
The area is
s(s a
)(s b
)(s c
)
14(14
7)(1
4 9)
(14
12)
980
31.3 square units. ■
618 CHAPTER 8 Triangle Trigonometry
EXERCISES 8.4.A
In Exercises 1–8, find the area of triangle ABC (standard nota-
tion) under the given conditions.
1. a 4, b 8, C 37°
2. b 10, c 14, A 36°
3. c 7, a 12, B 68°
4. a 9, b 13, C 75°
5. a 11, b 15, c 18
6. a 4, b 12, c 14
7. a 7, b 9, c 11
8. a 17, b 27, c 40
In Exercises 9 and 10, find the area of the triangle with the
given vertices.
9. (0, 0), (2, 5), (3, 1) 10. (4, 2), (5, 7), (3, 0)
In Exercises 11 and 12, find the area of the polygonal region.
[Hint: Divide the region into triangles.]
11.
12.
13. A triangular banner is hung from a window along the side of
a building. The edges that touch the window are 20 and 24
feet long respectively. The third side is parallel to the
ground. The angle between the 20-foot side and the third
side is 44°. What is the area of the banner?
14. A triangular lot has sides of length 120 feet and 160 feet.
The angle between these sides is 42°. Adjacent to this lot is
a rectangular lot whose longest side has length 200 feet and
whose shortest side is the same length as the shortest side of
the triangular lot. What is the total area of both lots?
15. If a gallon of paint covers 400 square feet, how many gal-
lons are needed to paint a triangular deck with sides of
lengths 65 feet, 72 feet, and 88 feet?
16. Find the volume of the prism in the figure. The volume is
given by the formula V
1
3
Bh, where B is the area of the
base and h is the height.
h
10
46°34°
36°
20
30
40
30
23
1
3
80°
120°
135°
130°
75°
120
55
135
68.4
103°
72°
89°
96°
SPECIAL TOPICS 8.4.A The Area of a Triangle 619
17. A rigid plastic triangle ABC rests on three vertical rods, as
shown in the figure. What is its area?
18. Prove that the area of triangle ABC (standard notation) is
given by
a
2
s
2
in
si
B
n
s
A
in C
.
19. A regular polygon has n equal sides and n equal angles
formed by the sides. For example, a regular polygon of
three sides is an equilateral triangle and a regular polygon of
4 sides is a square. If a regular polygon of n sides is
inscribed in a circle of radius r, then Exercise 20 shows that
its area is
1
2
nr
2
sin
36
n
0
°
.
Find the area of
(a) A square inscribed in a circle of radius 5.
(b) A regular hexagon (6 sides) inscribed in a circle of
radius 2.
20. Prove the area formula in Exercise 19 as follows.
(a) Draw lines from each vertex of the polygon to the
center of the circle, as shown in the figure for n 6.
(b) Explain why all the triangles are congruent and hence,
all the angles at the center of the circle have equal
measure. [Hint: SSS.]
(c) Explain why the measure of each center angle is
360°/n.
(d) Find the area of each triangle. [Hint: See the first box
on page 617.]
(e) The area of the polygon is the sum of the areas of these
n triangles. Use this fact and part (d) to obtain the area
formula in Exercise 19.
21. What is the area of a triangle whose sides have lengths 12,
20, and 36? (If your answer turns out strangely, try drawing
a picture.)
r
A
C
B
Horizontal plane
4
4
3
5
5
6
22. Complete the proof of Heron’s Formula as follows. Let
s
1
2
(a b c).
(a) Show that
1
2
ab(1 cos C )
(a b
4
)
2
c
2
(a
2
b) c
(a
2
b) c
s(s c).
620 CHAPTER 8 Triangle Trigonometry
[Hint: Use the Law of Cosines to express cos C in terms
of a, b, c; then simplify.]
(b) Show that
1
2
ab(1 cos C)
c
2
(a
4
b)
2
c (a
2
b)
c (a
2
b)
(s a)(s b).
Chapter 8 Review
IMPORTANT CONCEPTS
Section 8.1
[Alternate Section 8.1]
Trigonometric functions of
angles 574 [585]
Point-in-the-plane description 575
[586]
Right triangle description 576 [585]
Solving right triangles 579
Section 8.2
Applications of right triangle
trigonometry 588
Angles of elevation and
depression 590
Section 8.3
Standard notation for
triangles 597
Law of Cosines 597–598
Solution of oblique triangles 598–602
Section 8.4
Law of Sines 606
Ambiguous case 608
Special Topics 8.4.A
Area formulas for triangles 617
IMPORTANT FACTS & FORMULAS
■
Right Triangle Description: In a right triangle containing an angle u,
sin u
hy
o
p
p
o
p
t
o
e
s
n
i
u
te
se
cos u
hy
a
p
d
o
ja
t
c
e
e
n
n
u
t
se
tan u
o
ad
p
j
p
a
o
c
s
e
i
n
te
t
cot u
o
ad
p
j
p
a
o
c
s
e
i
n
te
t
sec u
hy
a
p
d
o
ja
t
c
e
e
n
n
u
t
se
csc u
hy
o
p
p
o
p
t
o
e
s
n
i
u
te
se
■
Law of Cosines: a
2
b
2
c
2
2bc cos A
■
Law of Cosines—Alternate Form: cos A
b
2
2
c
b
2
c
a
2
■
Law of Sines:
sin
a
A
sin
b
B
sin
c
C
■
sin D sin(180° D)
■
Area of triangle ABC
ab s
2
in C
■
Heron’s Formula:
Area of triangle ABC s(s a
)(s b
)(s c
)
where s
1
2
(a b c).
CHAPTER 8 Review 621
Note: Standard notation is used for triangles.
In Questions 1 and 2, find sin u , cos u , and tan u .
1. 2.
3. Which of the following statements about the angle u is
true?
(a) sin u 3/4
(b) cos u 5/4
(c) tan u 3/5
(d) sin u 4/5
(e) sin u 4/3
4. Suppose u is a real number. Consider the right triangle with
sides as shown in the figure. Then
(a) x 1
(b) x 2
(c) x 4
(d) x 2 (cos u sin u)
(e) none of the above
5. Use the right triangle in the figure to find sec u.
(a) sec u 7/4
(b) sec u 4/
65
(c) sec u 7/
65
(d) sec u
65
/7
(e) sec u
65
/4
6. Find the length of side h in the triangle, where angle A mea-
sures 40° and the distance from C to A is 25.
B
C
h
A
θ
8
3
θ
5
3
In Questions 7–10, angle B is a right angle. Solve triangle
ABC.
7. a 10, c 14 8. A 40°, b 10
9. C 35°, a 12 10. A 56°, a 11
11. From a point on level ground 145 feet from the base of a
tower, the angle of elevation to the top of the tower is 57.3°.
How high is the tower?
12. A pilot in a plane at an altitude of 22,000 feet observes that
the angle of depression to a nearby airport is 26°. How
many miles is the airport from a point on the ground directly
below the plane?
13. A road rises 140 feet per horizontal mile. What angle does
the road make with the horizontal?
14. A lighthouse keeper 100 feet above the water sees a boat
sailing in a straight line directly toward her. As she watches,
the angle of depression to the boat changes from 25° to 40°.
How far has the boat traveled during this time?
In Questions 15–18, use the Law of Cosines to solve triangle
ABC.
15. a 12, b 8, c 14
16. a 7.5, b 3.2, c 6.4
17. a 10, c 14, B 115°
18. a 7, b 8.6, C 72.4°
19. Two trains depart simultaneously from the same station.
The angle between the two tracks on which they leave is
120°. One train travels at an average speed of 45 mph and
the other at 70 mph. How far apart are the trains after 3
hours?
20. A 40-foot-high flagpole sits on the side of a hill. The hillside
makes a 17° angle with the horizontal. How long is a wire
that runs from the top of the pole to a point 72 feet downhill
from the base of the pole?
In Questions 21–26, use the Law of Sines to solve triangle ABC.
21. B 124°, C 40°, c 3.5
22. A 96°, B 44°, b 12
23. a 75, c 95, C 62°
24. a 5, c 2.5, C 30°
25. a 3.5, b 4, A 60°
26. a 3.8, c 2.8, C 41°
27. Find the area of triangle ABC if b 24, c 15, and A 55°.
28. Find the area of triangle ABC if a 10, c 14, and B 75°.
29. A boat travels for 8 kilometers in a straight line from the
dock. It is then sighted from a lighthouse that is 6.5 kilome-
ters from the dock. The angle determined by the dock, the
lighthouse (vertex), and the boat is 25°. How far is the boat
from the lighthouse?
REVIEW QUESTIONS
5
4
3
θ
x
2 cos
θ
2 sin
θ
7
4
θ
30. A pole tilts 12° from the vertical, away from the sun, and
casts a 34-foot-long shadow on level ground. The angle of
elevation from the end of the shadow to the top of the pole
is 64°. How long is the pole?
In Questions 31–34, solve triangle ABC.
31. A 48°, B 75°, b 50
32. A 67°, c 125, a 100
33. a 12, c 6, B 76°
34. a 90, b 70, c 40
35. Two surveyors, Joe and Alice, are 240 meters apart on a
riverbank. Each sights a flagpole on the opposite bank. The
angle from the pole to Joe (vertex) to Alice is 63°. The angle
from the pole to Alice (vertex) to Joe is 54°. How far are Joe
and Alice from the pole?
36. A surveyor stakes out points A and B on opposite sides of a
building. Point C on the side of the building is 300 feet from
A and 440 feet from B. Angle ACB measures 38°. What is
the distance from A to B?
37. A woman on the top of a 448-foot-high building spots a
small plane. As she views the plane, its angle of elevation is
62°. At the same instant, a man at the ground-level entrance
to the building sees the plane and notes that its angle of ele-
vation is 65°.
(a) How far is the woman from the plane?
(b) How far is the man from the plane?
(c) How high is the plane?
38. A straight road slopes at an angle of 10° with the horizontal.
When the angle of elevation of the sun (from horizontal) is
62.5°, a telephone pole at the side of the road casts a 15-foot
shadow downhill, parallel to the road. How high is the tele-
phone pole?
622 CHAPTER 8 Triangle Trigonometry
39. Find angle ABC.
40. Use the Law of Sines to prove Engelsohn’s equations: For
any triangle ABC (standard notation),
a
c
b
sin A
si
n C
sin B
and
a
c
b
sin A
si
n C
sin B
.
In Questions 41–44, find the area of triangle ABC under the
given conditions.
41. There is an angle of 30°, the sides of which have lengths 5
and 8.
42. There is an angle of 40°, the sides of which have lengths 3
and 12.
43. The sides have lengths 7, 11, and 14.
44. The sides have lengths 4, 8, and 10.
12
10
18
A
B
C
Chapter
8
Test
Sections 8.1 and 8.2
1. In the right triangle shown below, find
(a) sin u (b) cos u (c) tan u
2. A 160-ft long ramp connects a ground-level parking lot with
the entrance of a building. If the entrance is 7 feet above the
ground, what angle does the ramp make with the ground?
2b
1
θ
1 + 4b
2
3. Find side c of the right triangle shown below, if csc C 3.5
and b 31.5.
4. Ruth is flying a kite. Her hand is 3 feet above ground level
and is holding the end of a 310-ft long kite string, which
makes an angle of 55° with the horizontal. How high is the
kite above the ground?
A
C
B
a
b
c
CHAPTER 8 Test 623
5. In the right triangle shown here, find the exact length of side
c (no decimal approximations).
6. A wire from the top of a TV tower to the ground makes and
angle of 42° with the ground and touches the ground 200
feet from the base of the tower. How high is the tower?
7. Solve the right triangle below.
8. A plane passes directly over your head at an altitude of 5500
feet. Two seconds later, you observe that its angle of eleva-
tion is 72°. How far did the plane travel during those two
seconds?
9. Find angle u in the right triangle shown here.
10. Find angles A and C in the triangle shown here.
B
A
C
10
9
149
250
A
C
3.5
37
B
b
c
A
C
12
30
B
b
c
Sections 8.3 and 8.4; Special Topics 8.4.A
11. The side of a hill makes an angle of 17° with the horizontal.
A wire is to be run from the top of a 180-ft tower on the top
of the hill to a stake located 120 feet down the hillside from
the base of the tower. How long a wire is needed?
In Questions 12–17, solve triangle ABC (standard notation)
under the given conditions.
12. B 93.5°, C 49.5°, and b 6
13. B 26.6°, a 6.9, and c 16.5
14. a 22, c 15.3, and B 74°
15. a 5.4, b 7.2, and c 12.
16. b 15.2, c 19.3, and A 44°
17. b 12.3, c 20.2, and B 37.2°
18. Let A, B and C be the points in the plane with coordinates
(8, 5), (5, 1) and (1, 1) respectively. Find the angles of
triangle ABC.
19. A plane flies in a straight line at 425 mph for 1 hour and 42
minutes. It makes a 25° turn and flies at 375 mph for 2 hours
and 33 minutes. How far is it from its starting point?
20. A straight path makes an angle of 7° with the horizontal. A
statue at the higher end of the path casts a 9-meter long
shadow straight down the path. The angle of elevation from
the end of the shadow to the top of the statue is 33°. How tall
is the statue?
21. Each of two observers 300 feet apart measures the angle of
elevation to the top of a tree that sits on the straight line
between them. These angles are 54° and 67° respectively.
How tall is the tree?
22. If ABC is a triangle (standard notation), find its area when
(a) a 4, b 6, and c 9.
(b) a 7, b 9, and C 43°.
Russell Illig / Getty Images
624
DISCOVERY PROJECT 8 Life on a Sphere
Although we experience the surface of the earth as a level surface, it is not. The earth,
of course, is a sphere whose radius is about 6370 kilometers. When you stand on a
sphere, strange things happen, especially when the surface is particularly smooth.
Granted, the horizon is often interfered with by hills, forests, buildings, and
such. However, if you are out in an area of flat land like that found in eastern
Kansas or northern Ontario, the hills are very slight, and the ground is very much
like a sphere. The same is true of large lakes and oceans.
Generally speaking, you are not as interested in seeing the horizon as you are in
objects that the horizon might hide. On the ocean, for instance, you might want to see
another ship. On land, you might be looking for a particular building or vehicle.
1. Suppose that a person is standing on a highway and a 3.5-meter-high
truck passes by. Assume also that an average human has eyes that are
1.55 meters off the ground. If the road is flat and straight, how far away
is the truck when it disappears below the horizon?
2. Engineers building large bridges must also account for the curvature of
the earth. In the figure, a bridge with towers 200 meters tall and 900
meters apart (straight-line distance) at the base is constructed. How much
farther apart are the tops of the towers than the bases?
3. How much difference does it make if the distance between the towers is
measured along the curve of the earth?
900 m
200 m
200 m
6370 km
6370 km
6370 km
3.5 m
1.55 m
6370 km
6370 km
APPLICATIONS OF TRIGONOMETRY
Is this bridge safe?
When planning a bridge or a building, architects
and engineers must determine the stress on cables
and other parts of the structure to be sure that all
parts are adequately supported. Problems like these
can be modeled and solved by using vectors. See
Exercise 73 on page 652.
625
Chapter
© Jeff Spielman/Getty Images
(c, d)
(a, b)
(a – b, c – d)
x
u
u – v
v
–v
w
y