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7.1e: Exercises - Law of Sines

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    68256
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    For all exercises below, assume  \(\alpha\) or \(A\) is the angle opposite side \(a\),  \(\beta\) or \(B\) is the angle  opposite side \(b\), and \(\gamma\) or \(C\) is the angle  opposite side \(c\).

    A: Concepts.

    Exercise \(\PageIndex{A}\)

    1) Describe the altitude of a triangle.

    2) Compare right triangles and oblique triangles.

    3) When can you use the Law of Sines to find a missing angle?

    4) In the Law of Sines, what is the relationship between the angle in the numerator and the side in the denominator?

    5) What type of triangle results in an ambiguous case?

    Answers to Odd Exercises:
    1. The altitude extends from any vertex to the opposite side or to the line containing the opposite side at a \(90^{\circ}\) angle.
    3. When the known values are the side opposite the missing angle and another side and its opposite angle.
    5. A triangle with two given sides and a non-included angle.

    B: Find a side or angle

    Exercise \(\PageIndex{B}\)

    \( \bigstar \) Find the length of the specified side or the length of side \(x\). Round each answer to the nearest tenth.

    11) \(A=37^{\circ}, B=49^{\circ}, c=5\)
           Find side \(b\).
    12) \(A=132^{\circ}, C=23^{\circ}, b=10\)
          Find side \(a\).
    13) Find side \(c\) when \(B=37^{\circ}, C=21^{\circ}, b=23\)

    15)CNX_Precalc_Figure_08_01_201.jpg

    16)

    CNX_Precalc_Figure_08_01_202.jpg

    17)

    CNX_Precalc_Figure_08_01_203.jpg

    18)

    CNX_Precalc_Figure_08_01_204.jpg

    19)

    CNX_Precalc_Figure_08_01_205.jpg

    20)

    CNX_Precalc_Figure_08_01_206.jpg

    Answers to Odd Exercises:

    11. \(b\approx 3.78\)
    13. \(c\approx 13.70\)
    15. \( x \approx 12.3 \qquad\) 
    17. \( x \approx 12.2 \qquad\) 
    19. \( x \approx 16.0\).

    \( \bigstar \) State the number of possible triangles that can be formed from the given information. Find all possible values for angle \(x\) f. Round each answer to the nearest tenth.

    22) \(a=24, b=5, B=22^{\circ}\)
          Find angle \(A\). 
    23) \(a=13, b=6, B=20^{\circ}\)
          Find angle \(A\). 
    24) \(A=12^{\circ}, a=2, b=9\)
          Find angle \(B\). 

    25)

    CNX_Precalc_Figure_08_01_207.jpg

    26)

    CNX_Precalc_Figure_08_01_208.jpg

    27)

    CNX_Precalc_Figure_08_01_209.jpg

    28)

    CNX_Precalc_Figure_08_01_210.jpg

    29) 

    CNX_Precalc_Figure_08_01_211.jpg

    30)

    CNX_Precalc_Figure_08_01_212.jpg 

    Answers to Odd Exercises:

    23. Two triangles. \(A\approx 47.8^{\circ}\)  or  \(A'\approx 132.2^{\circ}\)
    25. One triangle. \(29.7^{\circ} \qquad\) 
    27. Two triangles. \(x=76.9^{\circ} \)  or  \(x=103.1^{\circ} \qquad\) 
    29.Two triangles. \(69.4^{\circ}\)  or  \(110.6^{\circ}\)

    C: Solve a Triangle

    Exercise \(\PageIndex{C}\)

    \( \bigstar \) Determine whether there is no triangle, one triangle, or two triangles. Then solve each triangle, if possible. Round each answer to the nearest tenth.

    32) \(\alpha =43^{\circ}, \gamma =69^{\circ}, a=20\)

    33) \(\alpha =35^{\circ}, \gamma =73^{\circ}, c=20\)

    34) \(\alpha =60^{\circ}, \beta =20^{\circ}, c =60 \)

    35) \(a=4, \alpha =60^{\circ}, \beta =100^{\circ}\)

    36) \(b=10, \beta =95^{\circ}, \gamma =30^{\circ}\)

    37) \(\gamma =113^{\circ}, b=10, c=32\)

    38) \(\alpha =119^{\circ}, a=14, b=26\)

    39) \(a=12, c=17, \alpha =35^{\circ}\)

    40) \(b=3.5, c=5.3, \gamma =80^{\circ}\)

    41) \(a=7, c=9, \alpha =43^{\circ}\)

    42) \(a=20.5, b=35.0, \beta =25^{\circ}\)

    43) \(b=13, c=5, \gamma =10^{\circ}\)

    44) \(a=7, b=3, \beta =24^{\circ}\)

    45) \(\beta =119^{\circ}, b=8.2, a=11.3\)

    46) \(a=2.3, c=1.8, \gamma =28^{\circ}\)

     

    47) Solve the triangle 

    CNX_Precalc_Figure_08_01_214.jpg

    Answers to Odd Exercises:
    33. one triangle, \(\beta =72^{\circ}, a\approx 12.0, b\approx 19.9\)
    35. one triangle, \(\gamma =20^{\circ}, b\approx 4.5, c\approx 1.6\)
    37. one triangle, \(\alpha \approx 50.3^{\circ}, \beta \approx 16.7^{\circ}, a\approx 26.7\)
    39. two triangles, \(\gamma \approx 54.3^{\circ}, \beta \approx 90.7^{\circ}, b\approx 20.9\) or \(\gamma '\approx 125.7^{\circ}, \beta '\approx 19.3^{\circ}, b'\approx 6.9\)
    41. two triangles, \(\beta \approx 75.7^{\circ}, \gamma \approx 61.3^{\circ}, b\approx 9.9\) or \(\beta '\approx 18.3^{\circ}, \gamma '\approx 118.7^{\circ}, b'\approx 3.2\)
    43. two triangles, \(\alpha \approx 143.2^{\circ}, \beta \approx 26.8^{\circ}, a\approx 17.3\) or \(\alpha '\approx 16.8^{\circ}, \beta '\approx 153.2^{\circ}, a'\approx 8.3\)
    45. no triangle possible
    47. one triangle, \(A\approx 39.4^{\circ}, C\approx 47.6^{\circ}, BC\approx 20.7\).
     

    D: Extensions

    Exercise \(\PageIndex{D}\)

    50) Find the radius of the circle in the Figure below. Round to the nearest tenth.

    CNX_Precalc_Figure_08_01_221.jpg

    51) Find the diameter of the circle in the Figure below. Round to the nearest tenth.

    CNX_Precalc_Figure_08_01_222.jpg

    52) Find \(m\angle ADC\) in the Figure below. Round to the nearest tenth.

    CNX_Precalc_Figure_08_01_223.jpg

    53) Find side \( AD\) in the Figure below. Round to the nearest tenth.

    CNX_Precalc_Figure_08_01_224.jpg

    54) Solve both triangles in the Figure below. Round each answer to the nearest tenth.

    CNX_Precalc_Figure_08_01_225.jpg

    55) Find side \( AB\) in the parallelogram shown below. Round to the nearest tenth.

    CNX_Precalc_Figure_08_01_226.jpg

    56) Solve the triangle in the Figure below. (Hint: Draw a perpendicular from \(H\) to \(JK\). Round each answer to the nearest tenth.

     

    CNX_Precalc_Figure_08_01_227.jpg

    57) Solve the triangle in the Figure below. (Hint: Draw a perpendicular from \(N\) to \(LM\). Round each answer to the nearest tenth.

    CNX_Precalc_Figure_08_01_228.jpg

    58) In the Figure to the right, \(ABCD\) is not a parallelogram. \(\angle m\) is obtuse. Solve both triangles. Round each answer to the nearest tenth.

    CNX_Precalc_Figure_08_01_229.jpg
    Answers to Odd Exercises:

    51. \(10.1 \qquad\) 53. \(AD\approx 13.8 \qquad\) 55. \(AB\approx 2.8 \qquad\)  57. \(L\approx 49.7^{\circ}, \; N\approx 56.3^{\circ},  \; LN\approx 5.8\).

    E: Real-World Applications

    Exercise \(\PageIndex{E}\)

    59) A pole leans away from the sun at an angle of \(7^{\circ}\) to the vertical, as shown in the Figure below. When the elevation of the sun is \(55^{\circ}\), the pole casts a shadow \(42\) feet long on the level ground. How long is the pole? Round the answer to the nearest tenth.

    CNX_Precalc_Figure_08_01_231.jpg

    60) To determine how far a boat is from shore, two radar stations \(500\) feet apart find the angles out to the boat, as shown in the Figure below. Determine the distance of the boat from station \(A\) and the distance of the boat from shore. Round your answers to the nearest whole foot.

    CNX_Precalc_Figure_08_01_232.jpg

    61) The Figure below shows a satellite orbiting Earth. The satellite passes directly over two tracking stations \(A\) and \(B\), which are \(69\) miles apart. When the satellite is on one side of the two stations, the angles of elevation at \(A\) and \(B\) are measured to be \(86.2^{\circ}\) and \(83.9^{\circ}\) respectively. How far is the satellite from station \(A\) and how high is the satellite above the ground? Round answers to the nearest whole mile.

    CNX_Precalc_Figure_08_01_233.jpg

    62) A communications tower is located at the top of a steep hill, as shown in the Figure below. The angle of inclination of the hill is \(67^{\circ}\). A guy wire is to be attached to the top of the tower and to the ground, \(165\) meters downhill from the base of the tower. The angle formed by the guy wire and the hill is \(16^{\circ}\). Find the length of the cable required for the guy wire to the nearest whole meter.

     

    CNX_Precalc_Figure_08_01_234.jpg

    63) The roof of a house is at a \(20^{\circ}\) angle. An \(8\)-foot solar panel is to be mounted on the roof and should be angled \(38^{\circ}\) relative to the horizontal for optimal results. (See the Figure below). How long does the vertical support holding up the back of the panel need to be? Round to the nearest tenth.

    CNX_Precalc_Figure_08_01_235.jpg

    64) Similar to an angle of elevation, an angle of depression is the acute angle formed by a horizontal line and an observer’s line of sight to an object below the horizontal. A pilot is flying over a straight highway. He determines the angles of depression to two mileposts, \(6.6\) km apart, to be \(37^{\circ}\) and \(44^{\circ}\) as shown in the Figure below. Find the distance of the plane from point \(A\) to the nearest tenth of a kilometer.

    CNX_Precalc_Figure_08_01_236.jpg

    65) A pilot is flying over a straight highway. He determines the angles of depression to two mileposts, \(4.3\) km apart, to be \(32^{\circ}\) and \(56^{\circ}\), as shown in the Figure below. Find the distance of the plane from point \(A\) to the nearest tenth of a kilometer.

    CNX_Precalc_Figure_08_01_237.jpg

    66) In order to estimate the height of a building, two students stand at a certain distance from the building at street level. From this point, they find the angle of elevation from the street to the top of the building to be \(39^{\circ}\). They then move \(300\) feet closer to the building and find the angle of elevation to be \(50^{\circ}\). Assuming that the street is level, estimate the height of the building to the nearest foot.

    67) In order to estimate the height of a building, two students stand at a certain distance from the building at street level. From this point, they find the angle of elevation from the street to the top of the building to be \(35^{\circ}\). They then move \(250\) feet closer to the building and find the angle of elevation to be \(53^{\circ}\). Assuming that the street is level, estimate the height of the building to the nearest foot.

    68) Points \(A\) and \(B\) are on opposite sides of a lake. Point \(C\) is \(97\) meters from \(A\). The measure of \(\angle BAC\) is determined to be \(101^{\circ}\), and the measure of \(\angle ACB\) is determined to be \(53^{\circ}\). What is the distance from \(A\) to \(B\),rounded to the nearest whole meter?

    69) A man and a woman standing \(3.5\) miles apart spot a hot air balloon between them at the same time. If the angle of elevation from the man to the balloon is \(27^{\circ}\), and the angle of elevation from the woman to the balloon is \(41^{\circ}\), find the altitude of the balloon to the nearest tenth of a mile.

    70) Two search teams spot a stranded climber between them on a mountain. The first search team is \(0.5\) miles from the second search team, and both teams are at an altitude of \(1\) mile. The angle of elevation from the first search team to the stranded climber is \(15^{\circ}\). The angle of elevation from the second search team to the climber is \(22^{\circ}\). What is the altitude of the climber? Round to the nearest hundredth of a mile.

    71) A street light is mounted on a pole. A \(6\)-foot-tall man is standing on the street a short distance from the pole, casting a shadow. The angle of elevation from the tip of the man’s shadow to the top of his head of \(28^{\circ}\). A \(6\)-foot-tall woman is standing on the same street on the opposite side of the pole from the man. The angle of elevation from the tip of her shadow to the top of her head is \(28^{\circ}\). If the man and woman are \(20\) feet apart, how far is the street light from the tip of the shadow of each person? Round the distance to the nearest tenth of a foot.

    72) Three cities, \(A\), \(B\), and \(C\), are located so that city \(A\) is due east of city \(B\). If city \(C\) is located \(35^{\circ}\) west of north from city \(B\) and is \(100\) miles from city \(A\) and \(70\) miles from city \(B\), how far is city \(A\) from city \(B\)? Round the distance to the nearest tenth of a mile.

    Answers to Odd Exercises:

    59. \(51.4\) ft
    61. The distance from the satellite to station \(A\) is approximately \(1716\) miles. The satellite is approximately \(1706\) miles above the ground.
    63. \(2.6\) ft \( \qquad \) 65. \(5.6\) km  \( \qquad \) 67. \(371\) ft \( \qquad \)  69. \(1.1\) miles \( \qquad \)  71. \(21.3\) ft.

    \( \bigstar \)


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