4-7 The Law of Sines and the Law of Cosines
7. GOLF A golfer misses a 12-foot putt by putting 3º off course. The hole now lies at a 129º angle between the ball
and its spot before the putt. What distance does the golfer need to putt in order to make the shot?
SOLUTION:
Draw a diagram of a triangle with two angle measures of 3 and 129 and an included side length of 12 feet.
(not drawn to scale)
Because two angles are given, the missing angle is 180° − (3° + 129°) or 48°. Use the Law of Sines to find x.
Therefore, the distance the golfer needs to putt is about 0.85 ft.
18. SKIING A ski lift rises at a 28 angle during the first 20 feet up a mountain to achieve a height of 25 feet, which is
the height maintained during the remainder of the ride up the mountain. Determine the length of cable needed for this
initial rise.
SOLUTION:
In this problem, A = 28 , a = 25 ft, and b = 20 ft. So, A is acute and a > b. Therefore, one solution exists. Apply the
Law of Sines to find B.
Because two angles are now known, the angle opposite x is 180
Law of Sines to find x.
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– (28 + 22.06 ) or about 129.94 . Apply the
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4-7 The Law of Sines and the Law of Cosines
Therefore, the distance the golfer needs to putt is about 0.85 ft.
18. SKIING A ski lift rises at a 28 angle during the first 20 feet up a mountain to achieve a height of 25 feet, which is
the height maintained during the remainder of the ride up the mountain. Determine the length of cable needed for this
initial rise.
SOLUTION:
In this problem, A = 28 , a = 25 ft, and b = 20 ft. So, A is acute and a > b. Therefore, one solution exists. Apply the
Law of Sines to find B.
Because two angles are now known, the angle opposite x is 180
Law of Sines to find x.
– (28 + 22.06 ) or about 129.94 . Apply the
Therefore, the length of cable needed for the initial rise is about 41 feet.
43. LANDSCAPING The Steele family want to expand their backyard by purchasing a vacant lot adjacent to their
property. To get a rough measurement of the area of the lot, Mr. Steele counted the steps needed to walk around
the border and diagonal of the lot.
a. Estimate the entire area in steps.
b. Mr. Steele measured his step to be 1.8 feet. Determine the area of the lot in square feet.
SOLUTION:
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a. Find the area of the Steele’s property. First, find s.
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4-7 The Law of Sines and the Law of Cosines
Therefore, the length of cable needed for the initial rise is about 41 feet.
43. LANDSCAPING The Steele family want to expand their backyard by purchasing a vacant lot adjacent to their
property. To get a rough measurement of the area of the lot, Mr. Steele counted the steps needed to walk around
the border and diagonal of the lot.
a. Estimate the entire area in steps.
b. Mr. Steele measured his step to be 1.8 feet. Determine the area of the lot in square feet.
SOLUTION:
a. Find the area of the Steele’s property. First, find s.
Use Heron's Formula find the area of the triangle.
Next, find the area of the vacant lot.
Use Heron's Formula find the area of the triangle.
Therefore, the total area is 963.1 + 3548.4 or about 4511.5 square steps.
b. Use dimensional analysis to convert the area from square steps to square feet.
Therefore, the area is about 14,617 square feet.
56. ZIP LINES A tourist attraction currently has its base connected to a tree platform 150 meters away by a zip line.
The owners now want to connect the base to a second platform located across a canyon and then connect the
platforms to each other. The bearings from the base to each platform and from platform 1 to platform 2 are given.
Find the distances from the base to platform 2 and from platform 1 to platform 2.
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Therefore, the area is about 14,617 square feet.
56. ZIP LINES A tourist attraction currently has its base connected to a tree platform 150 meters away by a zip line.
want toand
connect
base to
second platform located across a canyon and then connect the
4-7 The
Theowners
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of Sines
thetheLaw
ofa Cosines
platforms to each other. The bearings from the base to each platform and from platform 1 to platform 2 are given.
Find the distances from the base to platform 2 and from platform 1 to platform 2.
SOLUTION:
Draw a diagram to represent the situation.
Recall from Geometry that when two parallel lines are cut by a transversal, then consecutive angles are
supplementary.
∠A + 39 + 72 = 180
∠A = 69
∠C = 72° − 31° = 41°
∠B = 180° − (69° + 41°) = 70°.
Use the Law of Sines to find a.
Use the Law of Sines again to find c.
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4-7 The Law of Sines and the Law of Cosines
Use the Law of Sines again to find c.
Therefore, the distance from the base to platform 2 is about 149.02 meters, and the distance from platform 1 to
platform 2 is about 104.72 meters.
63. BUILDINGS Barbara wants to know the distance between the tops of two buildings R and S. On the top of her
building, she measures the distance between the points T and U and finds the given angle measures. Find the
distance between the two buildings.
SOLUTION:
Label the point at which
TXU is 180
and
intersect as X.
− (65 + 38 ) = 77 . Use the Law of Sines to find
SXU = 180 − 77 = 103
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= 180
+ 103
− (54
Use the Law of Sines to find
) = 23
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4-7 The Law of Sines and the Law of Cosines
SXU = 180 − 77 = 103
XSU = 180 − (54 + 103 ) = 23
Use the Law of Sines to find
Because RXT and SXU are vertical angles,
24 . Use the Law of Sines to find
RXT is 103 . Therefore,
XRT is 180
− (53 + 103 ) =
Because ∠TXU and ∠RXS are vertical angles, ∠RXS is 77°. Use the Law of Cosines to find
Therefore, the distance between the two buildings is about 40.9 meters.
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