Trig Lecture 4

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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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