section 2.4 law of sines and cosines - mrsk.camrsk.ca/ap/lawofsinescosines.pdf · section 2.4 –...
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28
Section 2.4 – Law of Sines and Cosines
Oblique Triangle
A triangle that is not a right triangle, either acute or obtuse.
The measures of the three sides and the three angles of a triangle can be found if at least one side and
any other two measures are known.
The Law of Sines
There are many relationships that exist between the sides and angles in a triangle.
One such relation is called the law of sines.
Given triangle ABC
sin sin sinA B Ca b c
or, equivalently
sin sin sin
a b cA B C
Proof
bhA sin sin (1)h b A
ahB sin sin (2)h a B
From (1) & (2)
h h
BaAb sinsin
ab
Baab
Ab sinsin
b
Ba
A sinsin
29
Angle – Side - Angle (ASA or AAS)
If two angles and the included side of one triangle are equal, respectively, to two angles and the
included side of a second triangle, then the triangles are congruent.
Example
In triangle ABC, 30A , 70B , and cma 0.8 . Find the length of side c.
Solution
)(180 BAC
)7030(180
100180
80
AC
acsinsin
C
Ac a sin
sin
80sin
30sin8
cm 16
Example
Find the missing parts of triangle ABC if 32A , 8.81C , , and cma 9.42 .
Solution
)(180 CBB )8.8132(180
2.66
sin sina b
A B
sinsin
aA
b B
32sin2.66sin9.42
cm .147
sin sinc a
C A
sinsin
aA
c C
42.9sin81.8sin32
80.1 cm
30
Example
You wish to measure the distance across a River. You determine that C = 112.90°, A = 31.10°, and
347.6 tb f . Find the distance a across the river.
Solution
180B A C
180 31.10 112.90
36
sin sina b
A B
347.6sin31.1 sin36
a
31.136
347.6 sinsin
a
305. 5 ta f
Example
Find distance x if a = 562 ft., 7.5B and 3.85A
Solution
AB
axsinsin
Ax Ba
sinsin
3.85sin7.5sin562
56.0 ft
31
Example
A hot-air balloon is flying over a dry lake when the wind stops blowing. The balloon comes to a stop
450 feet above the ground at point D. A jeep following the balloon runs out of gas at point A. The
nearest service station is due north of the jeep at point B. The bearing of the balloon from the jeep at A
is N 13 E, while the bearing of the balloon from the service station at B is S 19 E. If the angle of
elevation of the balloon from A is 12, how far will the people in the jeep have to walk to reach the
service station at point B?
Solution
ACDC12tan
12tan
DCAC
12tan
450
ft 117,2
)1913(180 ACB
148
Using triangle ABC
19sin148sin
2117AB
19sin148sin2117AB
3, 400 ft
32
Ambiguous Case
Side – Angle – Side (SAS)
If two sides and the included angle of one triangle are equal, respectively, to two sides and the included
angle of a second triangle, then the triangles are congruent.
Example
Find angle B in triangle ABC if a = 2, b = 6, and 30A
Solution
a
Ab
B sinsin
aB Absinsin
230sin6
5.1
1 sin 1
Since 1sinB
is impossible, no such triangle exists.
Example
Find the missing parts in triangle ABC if C = 35.4, a = 205 ft., and c = 314 ft.
Solution
c
CaA sinsin
3144.35sin205
3782.0
)3782.0(1sinA
22.2A
180 22.2 157.8A
35.4 157.8C A
193.2 18 0
4.122)4.352.22(180B
CBcb
sinsin
4.35sin4.122sin314
ft 458
33
Example
Find the missing parts in triangle ABC if a = 54 cm, b = 62 cm, and A = 40.
Solution
a
Ab
B sinsin
aB Absinsin
5440sin62
738.0
48)738.0(sin 1B 13248180B
)4840(180 C )13240(180 C
92 8
ACac
sinsin
ACac
sinsin
40sin92sin54
40sin8sin54
cm 84
cm 12
34
Area of a Triangle (SAS)
In any triangle ABC, the area A is given by the following formulas:
12
sinA bc A 12
sinA ac B 12
sinA ab C
Example
Find the area of triangle ABC if 24 40 , 27.3 , 52 40A b cm and C
Solution
180 24 40 52 40B
40 4060 60
180 24 52
102.667
sin sina b
A B
27.3
sin 24 40 sin 102 40a
27.3sin 24 40
sin 102 40a
11.7 cm
1 sin2
A ac B
1 (11.7)(27.3)sin 52 402
2127 cm
Example
Find the area of triangle ABC.
Solution
1 sin2
A ac B
1 34.0 42.0 sin 55 102
2586 ft
35
Number of Triangles Satisfying the Ambiguous Case (SSA)
Let sides a and b and angle A be given in triangle ABC. (The law of sines can be used to calculate the
value of sin B.)
1. If applying the law of sines results in an equation having sin B > 1, then no triangle satisfies the
given conditions.
2. If sin B = 1, then one triangle satisfies the given conditions and B = 90°.
3. If 0 < sin B < 1, then either one or two triangles satisfy the given conditions.
a) If sin B = k, then let 11
sinB k and use 1
B for B in the first triangle.
b) Let 2 1
180B B . If 2
180A B , then a second triangle exists. In this case, use 2
B for
B in the second triangle.
36
Law of Cosines (SAS)
2 2 2 2 cosa b c bc A
2 2 2 2 cosb a c ac B
2 2 2 2 cosc a b ab C
Derivation
222 )( hxca
222 2 hxcxc (1)
222 hxb (2)
From (2):
(1) 222 2 bcxca
cxbca 2222 (3)
bxA cos
xAb cos
(3) Acbbca cos2222
37
Example
Find the missing parts in triangle ABC if A = 60, b = 20 in, and c = 30 in.
Solution
Abccba cos2222
60cos)30)(20(23020 22
700
26a
a
AbB sinsin
26
60sin20
6662.0
)6662.0(sin 1B
42
BAC 180
4260180
78
Example
A surveyor wishes to find the distance between two inaccessible points A
and B on opposite sides of a lake. While standing at point C, she finds that
AC = 259 m, BC = 423 m, and angle ACB = 132°40′. Find the distance
AB.
Solution
2 2 2 2 cosAB AC BC AC BC C
2 2259 423 2 259 423 cos 132 40
394510
628AB
38
Law of Cosines (SSS) - Three Sides
2 2 2
cos2
b c aAbc
2 2 2
cos2
a c bBac
2 2 2
cos2
a b cCab
Example
Solve triangle ABC if a = 34 km, b = 20 km, and c = 18 km
Solution
bc
acbA2
cos222
)18)(20(2341820 222
6.0
)6.0(cos 1 A
127
OR
abcbaC
2cos
222
)20)(34(2182034 222
91.0
1 cos (0.91)C
25
aAcC sinsin
34
127sin18
4228.0
)4228.0(sin 1C
25
180 A CB
25127180
28
39
Example
A plane is flying with an airspeed of 185 miles per hour with heading 120. The wind currents are running
at a constant 32 miles per hour at 165 clockwise from due north. Find the true course and ground speed
of the plane.
Solution
120180
60
165360
60165360
135
cos2222
WVWVWV
135cos)32)(185(232185 22
621,43
mphWV 210
210sin
32
sin
210
135sin32sin
1077.0
)1077.0(sin 1 6
The true course is:
120 120 6 126 .
The speed of the plane with respect to the
ground is 210 mph.
Example
Find the measure of angle B in the figure of a roof truss.
Solution
2 2 2cos
2a c bB
ac
2 2 211 9 62(11)(9)
2 2 21 11 9 6cos2(11)(9)
B
33
40
Heron’s Area Formula (SSS)
If a triangle has sides of lengths a, b, and c, with semi-perimeter
12
s a b c
Then the area of the triangle is:
A s s a s b s c
Example
The distance “as the crow flies” from Los Angeles to New York is 2451 miles, from New York to
Montreal is 331 miles, and from Montreal to Los Angeles is 2427 miles. What is the area of the triangular
region having these three cities as vertices? (Ignore the curvature of Earth.)
Solution
The semiperimeter s is:
12
s a b c
12
2451 331 2427
2604.5
A s s a s b s c
2604.5 2604.5 262451 3304.5 2604.51 2427
2401,700 mi
41
Exercises Section 2.4 – Law of Sines and Cosines
1. In triangle ABC, 110B , 40C and inb 18 . Find the length of side c.
2. In triangle ABC, 4.110A , 8.21C and inc 246 . Find all the missing parts.
3. Find the missing parts of triangle ABC, if 34B , 82C , and cma 6.5 .
4. Solve triangle ABC if B = 55°40′, b = 8.94 m, and a = 25.1 m.
5. Solve triangle ABC if A = 55.3°, a = 22.8 ft., and b = 24.9 ft.
6. Solve triangle ABC given A = 43.5°, a = 10.7 in., and c = 7.2 in.
7. If 26 , 22, 19,A s and r find x
8. If a = 13 yd., b = 14 yd., and c = 15 yd., find the largest angle.
9. Solve triangle ABC if b = 63.4 km, and c = 75.2 km, A = 124 40
10. Solve triangle ABC if 42.3 , 12.9 , 15.4A b m and c m
11. Solve triangle ABC if a = 832 ft., b = 623 ft., and c = 345 ft.
12. Solve triangle ABC if 9.47 , 15.9 , 21.1a ft b ft and c ft
13. The diagonals of a parallelogram are 24.2 cm and 35.4 cm and intersect at an angle of 65.5. Find
the length of the shorter side of the parallelogram
14. A man flying in a hot-air balloon in a straight line at a constant rate of 5 feet per second, while
keeping it at a constant altitude. As he approaches the parking lot of a market, he notices that the
angle of depression from his balloon to a friend’s car in the parking lot is 35. A minute and a half
later, after flying directly over this friend’s car, he looks back to see his friend getting into the car
and observes the angle of depression to be 36. At that time, what is the distance between him and
his friend?
15. A satellite is circling above the earth. When the satellite is
directly above point B, angle A is 75.4. If the distance between
points B and D on the circumference of the earth is 910 miles and
the radius of the earth is 3,960 miles, how far above the earth is
the satellite?
42
16. A pilot left Fairbanks in a light plane and flew 100 miles toward Fort in still air on a course with
bearing of 18. She then flew due east (bearing 90) for some time drop supplies to a snowbound
family. After the drop, her course to return to Fairbanks had bearing of 225. What was her
maximum distance from Fairbanks?
17. The dimensions of a land are given in the figure. Find the area of the property in square feet.
18. The angle of elevation of the top of a water tower from point A on the ground is 19.9. From point
B, 50.0 feet closer to the tower, the angle of elevation is 21.8. What is the height of the tower?
19. A 40-ft wide house has a roof with a 6-12 pitch (the roof rises 6 ft for a run of 12 ft). The owner
plans a 14-ft wide addition that will have a 3-12 pitch to its roof. Find the lengths of AB and BC
20. A hill has an angle of inclination of 36. A study completed by a state’s highway commission
showed that the placement of a highway requires that 400 ft of the hill, measured horizontally, be
removed. The engineers plan to leave a slope alongside the highway with an angle of inclination of
62. Located 750 ft up the hill measured from the base is a tree containing the nest of an endangered
hawk. Will this tree be removed in the excavation?
43
21. A cruise missile is traveling straight across the desert at 548 mph at an altitude of 1 mile. A gunner
spots the missile coming in his direction and fires a projectile at the missile when the angle of
elevation of the missile is 35. If the speed of the projectile is 688 mph, then for what angle of
elevation of the gun will the projectile hit the missile?
22. When the ball is snapped, Smith starts running at a 50 angle to the line of scrimmage. At the
moment when Smith is at a 60 angle from Jones, Smith is running at 17 ft/sec and Jones passes the
ball at 60 ft/sec to Smith. However, to complete the pass, Jones must lead Smith by the angle .
Find (find in his head. Note that can be found without knowing any distances.)
44
23. A rabbit starts running from point A in a straight line in the direction 30 from the north at 3.5 ft/sec.
At the same time a fox starts running in a straight line from a position 30 ft to the west of the rabbit
6.5 ft/sec. The fox chooses his path so that he will catch the rabbit at point C. In how many seconds
will the fox catch the rabbit?
24. An engineer wants to position three pipes at the vertices of a triangle. If the pipes A, B, and C have
radii 2 in, 3 in, and 4 in, respectively, then what are the measures of the angles of the triangle ABC?
25. A solar panel with a width of 1.2 m is positioned on a flat roof. What is the angle of elevation of
the solar panel?
26. Andrea and Steve left the airport at the same time. Andrea flew at 180 mph on a course with bearing
80, and Steve flew at 240 mph on a course with bearing 210. How far apart were they after 3 hr.?
45
27. A submarine sights a moving target at a distance of 820 m. A torpedo is fired 9 ahead of the target,
and travels 924 m in a straight line to hit the target. How far has the target moved from the time the
torpedo is fired to the time of the hit?
28. A tunnel is planned through a mountain to connect points A and B on two existing roads. If the
angle between the roads at point C is 28, what is the distance from point A to B? Find CBA and
CAB to the nearest tenth of a degree.
29. A 6-ft antenna is installed at the top of a roof. A guy wire is to be attached to the top of the antenna
and to a point 10 ft down the roof. If the angle of elevation of the roof is 28, then what length guy
wire is needed?
30. On June 30, 1861, Comet Tebutt, one of the
greatest comets, was visible even before sunset.
One of the factors that causes a comet to be extra
bright is a small scattering angle . When Comet
Tebutt was at its brightest, it was 0.133 a.u. from
the earth, 0.894 a.u. from the sun, and the earth was
1.017 a.u. from the sun. Find the phase angle and
the scattering angle for Comet Tebutt on June 30,
1861. (One astronomical unit (a.u) is the average
distance between the earth and the sub.)
46
31. A human arm consists of an upper arm of 30 cm and a lower arm of 30 cm. To move the hand to the
point (36, 8), the human brain chooses angle 1 2
and to the nearest tenth of a degree.
32. A forest ranger is 150 ft above the ground in a fire tower when she spots an angry grizzly bear east
of the tower with an angle of depression of 10. Southeast of the tower she spots a hiker with an
angle of depression of 15. Find the distance between the hiker and the angry bear.
33. Two ranger stations are on an east-west line 110 mi apart. A forest fire is located on a bearing N 42°
E from the western station at A and a bearing of N 15° E from the eastern station at B. How far is
the fire from the western station?
43
Solution Section 2.4 – Law of Sines and Cosines
Exercise
In triangle ABC, 110B , 40C and inb 18 . Find the length of side c.
Solution
)(180 CBA 180 110 40
30
sin sina b
A B sin sin
c bC B
18sin30 sin110
a
18sin 40 sin110
c
18sin30sin110
a
18sin 40sin110
c
9.6 in 12.3 in
Exercise
In triangle ABC, 4.110A , 8.21C and inc 246 . Find all the missing parts.
Solution
CAB 180
180 110.4 21.8
47.8
sin sina c
A C sin sin
b cB C
246sin110.4 sin 21.8
a
24647.8 sin 21.8
b
246sin110.4sin 21.8
a
246sin 47.8sin 21.8
b
621 in 491 in
44
Exercise
Find the missing parts of triangle ABC if 34B , 82C , and cma 6.5 .
Solution
)(180 CBA )8234(180
116180
64
ABab
sinsin
ABb a
sinsin
64sin34sin6.5
cm 5.3
ACac
sinsin
ACc a
sinsin
64sin82sin6.5
cm 2.6
Exercise
Solve triangle ABC if B = 55°40′, b = 8.94 m, and a = 25.1 m.
Solution
sin sinA Ba b
4060
sin 55sin25.1 8.94
A
25.1sin 55.667sin
8.942.31 4 18A
Since sin A > 1 is impossible, no such triangle exists.
45
Exercise
Solve triangle ABC if A = 55.3°, a = 22.8 ft, and b = 24.9 ft
Solution
sin sinB Ab a
24.9sin55.3sin 0.8978722.8
B
1sin 0.89787B
63.9 180 63.9 116.1B and B
180C A B
180 55.3 63.9C
60.8C
sinsin
a CcA
22.8sin60.8sin55.3
24.2 ft
180 55.3 116.1C
8.6C
sinsin
a CcA
22.8sin8.6sin55.3
4.15 ft
46
Exercise
Solve triangle ABC given A = 43.5°, a = 10.7 in., and c = 7.2 in
Solution
sin sinC Ac a
7.2sin 43.5sin 0.463210.7
C
1sin 0.4632C
27.6 180 27.6 152.4C and C
180B A C
180 43.5 27.6B
108.9B
sinsin
a BbA
10.7sin108.9sin 43.5
14.7 in
180 43.5 152.4B
15.9B
Is not possible
Exercise
If 19 ,22 ,26 randsA find x
Solution
1802219
66sC adr
r
8 180 180 26 66 8 A CD
sin sinr x r
D A
19sin8819sin 26
x
19sin88 19 sin 26
24x
Exercise
If a = 13 yd, b = 14 yd, and c = 15 yd, find the largest angle.
Solution
2 2 21 12 2 2 13 14 15cos2 2(13)
614)
7(
cos a b cab
C
47
Exercise
Solve triangle ABC if b = 63.4 km, and c = 75.2 km, A = 124 40
Solution
2 2 2 2 cosa b c bc A
4060
2 263.4 75.2 63.4 75.2 1242 cos
15098
122.9 ma k
sinsin b ABa
63.4sin124.67122.9
1 63.4sin124.67sin122.9
B 25.1
180 BC A
180 124.67 25.1
30.23
Exercise
Solve triangle ABC if a = 832 ft, b = 623 ft, and c = 345 ft
Solution
1 2 2 2cos
2aC b c
ab
1 8322 2 2623 345cos2(832)(623)
22
sinsin b CBc
623sin 22345
1 623sin 22sin345
B
43
180 22 4 1 1 53A
48
Exercise
Solve triangle ABC if 42.3 , 12.9 , 15.4A b m and c m
Solution
2 2 2 2 cosa b c bc A
2 212.9 15.4 2(12.9)(15.4)cos42.3
109.7
10.47 ma
12.9sin 42.310.47
sinsin b ABa
1 12.9sin 42.310.47
sin 56.0B
180 42. 81.73 56 C
Exercise
Solve triangle ABC if 9.47 , 15.9 , 21.1a ft b ft and c ft
Solution
1 2 2 2
2cos a b c
abC
2 2 21 9.47 15.9 21.1cos2(9.47)(15.9)
109.9
sinsin b CBc
15.9sin109.921.1
1 15.9sin109.921.1
sinB 25.0
180 25 109.9 45.1A
Exercise
The diagonals of a parallelogram are 24.2 cm and 35.4 cm and
intersect at an angle of 65.5. Find the length of the shorter side of the
parallelogram
Solution
2 2 217.7 12.1 2(17.7)(12.1)cos65.5 282.07x
16.8 mx c
49
Exercise
A man flying in a hot-air balloon in a straight line at a constant rate of 5 feet per second, while keeping it
at a constant altitude. As he approaches the parking lot of a market, he notices that the angle of depression
from his balloon to a friend’s car in the parking lot is 35. A minute and a half later, after flying directly
over this friend’s car, he looks back to see his friend getting into the car and observes the angle of
depression to be 36. At that time, what is the distance between him and his friend?
Solution
180 35 36 109car
450sin35 sin109
d
450sin35 sin
27109
3 ftd
Exercise
A satellite is circling above the earth. When the satellite is directly above point B, angle A is 75.4. If the
distance between points B and D on the circumference of the earth is 910 miles and the radius of the earth
is 3,960 miles, how far above the earth is the satellite?
Solution
rs
radiusdBDlengtharcC by ivides
radC3960910
180
3960910 2.13
)(180 CAD
)2.134.75(180
4.91
ADCA
sinsin3960
4.75sin4.91sin39603960x
4.75sin4.91sin39603960x
3960
4.75sin4.91sin3960
x
130x i m
35° 36°
car
5 ft/sec
d
50
Exercise
A pilot left Fairbanks in a light plane and flew 100 miles toward Fort in still air on a course with bearing
of 18. She then flew due east (bearing 90) for some time drop supplies to a snowbound family. After the
drop, her course to return to Fairbanks had bearing of 225. What was her maximum distance from
Fairbanks?
Solution
From the triangle ABC:
90 18 108ABC
360 225 90 45ACB
90 18 45 27BAC
The length AC is the maximum distance from Fairbanks:
100sin108 sin 45
b
100sin108 sin 4
134.5 5
milesb
Exercise
The dimensions of a land are given in the figure. Find the area of the
property in square feet.
Solution
211 2
148.7 93.5 sin91.5 6949.3 A ft
212 2
100.5 155.4 sin87.2 7799.5 A ft
The total area = 1 2
26949.3 7799 14,74. 85 .8 tA fA
Exercise
The angle of elevation of the top of a water tower from point A on the ground is 19.9. From point B, 50.0
feet closer to the tower, the angle of elevation is 21.8. What is the height of the tower?
Solution
180 21.8 158.2ABC
180 19.9 158.2 1.9ACB
Apply the law of sines in triangle ABC:
50sin19.9 sin1.9
BC
50sin19.9 513.3sin1.9
BC
51
Using the right triangle: sin 21.8y
BC
513.3sin21.8 191 y ft
Exercise
A 40-ft wide house has a roof with a 6-12 pitch (the roof rises 6 ft for a run of 12 ft). The owner plans a
14-ft wide addition that will have a 3-12 pitch to its roof. Find the lengths of AB and BC .
Solution
16 612 12
tan tan 26.565
13 312 12
tan tan 14.036
180 180 26.565 153.435
180 14.036 153.435 12.529
14sin153.435 sin12.529
AB
14sin153.435 sin12.529
28.9 A fB t
14sin14.036 sin12.529
BC
14sin14.036 sin12.529
15.7C ftB
Exercise
A hill has an angle of inclination of 36. A study completed by a state’s
highway commission showed that the placement of a highway requires
that 400 ft of the hill, measured horizontally, be removed. The engineers
plan to leave a slope alongside the highway with an angle of inclination
of 62. Located 750 ft up the hill measured from the base is a tree
containing the nest of an endangered hawk. Will this tree be removed in
the excavation?
Solution
180 62 118ACB
180 118 36 26ABC
Using the law of sines:
400sin118 sin 26
x
A C
B
14 20
A C
B
36° 62°
400 ft
750 ft
x
118°
26°
52
400sin118 sin 26
805.7 ftx
Yes, the tree will have to be excavated.
Exercise
A cruise missile is traveling straight across the desert at 548 mph at an altitude of 1 mile. A gunner spots
the missile coming in his direction and fires a projectile at the missile when the angle of elevation of the
missile is 35. If the speed of the projectile is 688 mph, then for what angle of elevation of the gun will
the projectile hit the missile?
Solution
35ACB
180 35BAC
After t seconds;
The cruise missile distance: 5483600
t miles
The Projectile distance: 6883600
t miles
Using the law of sines:
548 6883600 3600
sin 35sin 145
t t
548 688sin 35 sin 1453600 3600
t t
548sin 35 688sin 145
548sin 145 sin 35688
1 548145 sin sin 35688
1 548145 sin sin 35 688
117.8
180 35 117.8 27.2BAC
The angle of elevation of the projectile must be 35 27.2 62.2
A
CB
35°
D
1 mi
548t mi
688t mi
53
Exercise
When the ball is snapped, Smith starts running at a 50
angle to the line of scrimmage. At the moment when
Smith is at a 60 angle from Jones, Smith is running at 17
ft/sec and Jones passes the ball at 60 ft/sec to Smith.
However, to complete the pass, Jones must lead Smith by
the angle . Find (find in his head. Note that can be
found without knowing any distances.)
Solution
180 60 50 70ABD
180 70 110ABC
Using the law of sines:
17 60sin sin110
t t
17 60sin sin110
17sin110sin60
1 17sin110sin 15.460
Exercise
A rabbit starts running from point A in a straight line in
the direction 30 from the north at 3.5 ft/sec. At the same
time a fox starts running in a straight line from a position
30 ft to the west of the rabbit 6.5 ft/sec. The fox chooses
his path so that he will catch the rabbit at point C. In how
many seconds will the fox catch the rabbit?
Solution
90 30 120BAC
6.5 3.5sin120 sin B
t t
6.5 3.5sin120 sin B
3.5sin120sin6.5
B
1 3.5sin120sin 286.5
B
A
B
C
D60° 50°
60t ft
17t ft
AB
C
54
180 120 28 32C
3.5 30sin 28 sin 32
t
30sin 28 3.5si
7.6 secn 32
t
It will take 7.6 sec. to catch the rabbit.
Exercise
An engineer wants to position three pipes at the vertices of a triangle. If the pipes A, B, and C have radii 2
in, 3 in, and 4 in, respectively, then what are the measures of the angles of the triangle ABC?
Solution
6 5 7AC AB BC
2 2 21 5 6 7cos 2(5)(6)
78.5A
6 7sin sin 78.5B
6sin 78.5sin7
B
1 6sin 78.5sin 7
57.1B
180 78.5 57.1 44.4C
Exercise
Andrea and Steve left the airport at the same time. Andrea flew at 180 mph on a course with bearing 80,
and Steve flew at 240 mph on a course with bearing 210. How far apart were they after 3 hr.?
Solution
After 3 hrs., Steve flew: 3(240) = 720 mph
Andrea flew: 3(180) = 540 mph
2 2 2720 540 2 720 540 cos 210x
2 2720 540 2 720 540 cos 210 x
1144.5 miles
80°
210°
540
720
x
55
Exercise
A solar panel with a width of 1.2 m is positioned on a flat roof.
What is the angle of elevation of the solar panel?
Solution
2 2 21 1.2 1.2 0.4cos 2(1.2)(1.2)
19.2
0.41.2
o sr
r
Exercise
A submarine sights a moving target at a distance of 820 m. A torpedo is fired 9 ahead of the target, and
travels 924 m in a straight line to hit the target. How far has the target moved from the time the torpedo is
fired to the time of the hit?
Solution
2 2820 924 2 820 924 cos9 m 171.7 x
Exercise
A tunnel is planned through a mountain to connect points A and B on two existing roads. If the angle
between the roads at point C is 28, what is the distance from point A to B? Find CBA and CAB to the
nearest tenth of a degree.
Solution
By the cosine law,
2 25.3 7.6 2(5.3)(7.6)cos 28 3.8 AB
2 2 21 3.8 5.3 7.6cos 2(3.8)(5.3)
112CBA
180 112 4 028BAC
56
Exercise
A 6-ft antenna is installed at the top of a roof. A guy wire is
to be attached to the top of the antenna and to a point 10 ft
down the roof. If the angle of elevation of the roof is 28,
then what length guy wire is needed?
Solution
90 28 62
180 62 118
By the cosine law,
2 26 100 2(6)(10)cos118 13.9 ftx
Exercise
On June 30, 1861, Comet Tebutt, one of the greatest comets, was visible even before sunset. One of the
factors that causes a comet to be extra bright is a small scattering angle . When Comet Tebutt was at its
brightest, it was 0.133 a.u. from the earth, 0.894 a.u. from the sun, and the earth was 1.017 a.u. from the
sun. Find the phase angle and the scattering angle for Comet Tebutt on June 30, 1861. (One
astronomical unit (a.u) is the average distance between the earth and the sub.)
Solution
By the cosine law:
2 2 21 0.133 0.894 1.017cos 2(0.133)(0.89 )
1561
180 180 156 24
Exercise
A human arm consists of an upper arm of 30 cm and a lower
arm of 30 cm. To move the hand to the point (36, 8), the human
brain chooses angle 1 2
and to the nearest tenth of a degree.
Solution
30 8 22AC 36BC
2 2AB AC CB
2 222 36
42.19
By the cosine law:
28°
x6
10
a
A
BC
D
57
2 2 21cos2
AD DB ABADBAD DB
2 2 21 30 30 42.19cos2 30 30
ADB 89.4ADB
2180 89.4 90.6
36tan22
BCCABAC
1 36tan 58.5722
CAB
sin sin89.4 30sin89.4sin30 42.19 42.19DAB DAB
1 30sin89.4sin 45.3242.19
DAB
1CAB DAB
58.57 45.32
13.25
Exercise
A forest ranger is 150 ft above the ground in a fire tower when she spots an angry grizzly bear east of the
tower with an angle of depression of 10. Southeast of the tower she spots a hiker with an angle of
depression of 15. Find the distance between the hiker and the angry bear.
Solution
10BEC ECD
From triangle EBC: 150tan10BE
150 850.692tan10
BE
15BAC ACF
From triangle ABC:
150tan15AB
150 559.808tan15
AB
2 2 2 cos 45x AB BE AB BE
2 2559.808 850.692 2 559.808 850.692 cos 45
603 ft
A
B
CD
F
x
45°
58
Exercise
Two ranger stations are on an east-west line 110 mi apart. A forest fire is located on a bearing N 42° E
from the western station at A and a bearing of N 15° E from the eastern station at B. How far is the fire
from the western station?
Solution
90 42 48BAC
90 15 105ABC
180 105 48 27C
sin sinb c
B C
110sin105 sin 27
b
110sin105sin 27
b
4 23b mi
The fire is about 234 miles from the western station.