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.