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Trig Lecture 4

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5/21/2018 Trig Lecture 4

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Lecture FourApplications of Trigonometry

Section 4.1Law of Sines

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 C

a b c

or, equivalently

sin sin sin

a b c

A B C

Proof

b

hA sin sin (1)h b A

a

hB sin sin (2)h a B

From (1) & (2)

h h

BaAb sinsin

ab

Ba

ab

Ab sinsin

b

B

a

A sinsin

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AngleSide - 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 triangleABC, 30A , 70B , and cma 0.8 . Find the length of side c.

Solution

)(180 BAC

)7030(180

100180

80

AC

ac

sinsin

CA

c a

sinsin

80sin30sin

8

cm16

Example

Find the missing parts of triangleABCif 32A , 8.81C , , and cma 9.42 .

Solution

)(180 CBB )8.8132(180

2.66

sin sin

a b

A B

sin

sin

a

Ab

B

32sin

2.66sin9.42

cm.147

sin sin

c a

C A

sin

sin

a

Ac

C

42.9sin 81.8

sin32

80.1 cm

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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 aacross the river.

Solution

180B A C 180 31.10 112.90

36

sin sin

a b

A B

347.6

sin 31.1 sin 3 6

a

31.136

347.6sin

sina

305. 5 ta f

Example

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 atAand a bearing of N 15 E from the eastern station atB. How far is the fire

from the western station?

Solution

90 42 48BAC

90 15 105ABC

180 105 48 27C

sin sin

b c

B C

110

sin 105 sin 27

b

110sin105

sin 27

b

423b mi

The fire is about 234 miles from the western station.

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Example

Find distancexif a= 562ft., 7.5B and 3.85A

Solution

AB

ax

sinsin

Ax

Ba

sin

sin

3.85sin

7.5sin562

ft0.56

xA

B

a

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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 pointD. A jeep following the balloon runs out of gas at pointA. The

nearest service station is due north of the jeep at pointB. The bearing of the balloon from the jeep atA

is N 13E, while the bearing of the balloon from the service station at Bis S 19E. If the angle of

elevation of the balloon fromAis 12, how far will the people in the jeep have to walk to reach the

service station at pointB?

Solution

AC

DC12tan

12tan

DCAC

12tan

450

ft117,2

)1913(180 ACB

148

Using triangleABC

19sin148sin

2117AB

19sin

148sin2117AB

3, 400ft

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Ambiguous Case

SideAngleSide (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 angleBin triangle ABC if a= 2, b= 6, and 30A

Solution

a

A

b

B sinsin

aB

Absinsin

2

30sin6

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

314

4.35sin205

3782.0

)3782.0(1sinA

22.2A

180 22.2 157.8A

35.4 157.8C A

193.2 180

4.122)4.352.22(180B

C

Bcb

sin

sin

4.35sin

4.122sin314

ft458

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Example

Find the missing parts in triangle ABC if a= 54 cm, b= 62 cm, andA= 40.

Solution

a

A

b

B sinsin

aB

Absinsin

54

40sin62

738.0

48)738.0(sin 1B 13248180B

)4840(180 C )13240(180 C

92 8

A

Cac

sin

sin

ACa

csin

sin

40sin

92sin54

40sin8sin54

cm84

cm12

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Area of a Triangle (SAS)

In any triangleABC, the areaAis given by the following formulas:

1

2sinA bc A 1

2 sinA ac B 1

2 sinA ab C

Example

Find the area of triangleABCif 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 sin

a b

A B

27.3

sin 24 40 sin 102 40

a

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

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Example

Find the area of triangleABC.

Solution

1sin

2A ac B

1 34.0 42.0 sin 55 102

2586ft

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Number of Tr iangles Satisfying the Ambiguous Case (SSA)

Let sides aand band angleAbe given in triangleABC. (The law of sines can be used to calculate thevalue of sinB.)

1. If applying the law of sines results in an equation having sinB> 1, then no triangle satisfies the

given conditions.

2. If sinB= 1, then one triangle satisfies the given conditions andB= 90.

3. If 0 < sinB< 1, then either one or two triangles satisfy the given conditions.

a) If sinB= k, then let 11

sinB k and use1

B forBin the first triangle.

b)Let2 1

180B B . If2

180A B , then a second triangle exists. In this case, use2

B for

Bin the second triangle.

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Exercises Section 4.1Law of Sines

1. In triangleABC, 110B , 40C and inb 18 . Find the length of side c.

2. In triangleABC, 4.110A , 8.21C and inc 246 . Find all the missing parts.

3.

Find the missing parts of triangleABC,if 34B , 82C , and cma 6.5 .

4. Solve triangleABCifB= 5540, b= 8.94 m, and a= 25.1 m.

5. Solve triangleABC ifA= 55.3, a= 22.8ft., and b= 24.9 ft.

6. Solve triangleABC givenA= 43.5, a= 10.7 in., and c= 7.2 in.

7. If 26 , 22, 19,A s and r findx

8. 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 friends car in theparking lot is 35. A minute and ahalf later, after flying directly over this friends 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?

9. A satellite is circling above the earth. When the satellite is directly above pointB, angleAis

75.4. If the distance between pointsBandDon 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?

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10. 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?

11. The dimensions of a land are given in the figure. Find the area of the property in square feet.

12. 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?

13. 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

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14. A hill has an angle of inclination of 36. A study completed by a states 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?

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