PC Lesson 4.7 (25

4-7 The Law of Sines and the Law of Cosines
25. BROADCASTING A radio tower located 38 miles along Industrial Parkway transmits radio broadcasts over a 30mile radius. Industrial Parkway intersects the interstate at a 41º angle. How far along the interstate can vehicles pick
up the broadcasting signal?
SOLUTION: Draw a diagram of the situation where A represents the location of the radio tower, B represents the leftmost point
on the interstate that is within the 30-mile broadcasting radius, and C' represents the rightmost point on the interstate
within the broadcasting radius.
Due to the information given, use the Law of Sines to solve for B in ABC.
Because two angles are now known, A
find BC.
180
– (41 + 56.2 ) or about 82.8 . Apply the Law of Sines again to
Notice that rABC' is an isosceles triangle, so B ≈ C', B ≈ 56.2º, and thus C' ≈ 56.2º. Because two angles in rABC' are
now known, A ≈ 180º – (56.2º + 56.2º) or about 67.6º. Use the Law of Cosines to find BC'.
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4-7 The Law of Sines and the Law of Cosines
Notice that rABC' is an isosceles triangle, so B ≈ C', B ≈ 56.2º, and thus C' ≈ 56.2º. Because two angles in rABC' are
now known, A ≈ 180º – (56.2º + 56.2º) or about 67.6º. Use the Law of Cosines to find BC'.
Therefore, vehicles can pick up the broadcasting signal for about 33.4 miles along the interstate.
Solve each triangle. Round side lengths to the nearest tenth and angle measures to the nearest degree.
27. ABC, if A = 42 , b = 12, and c = 19
SOLUTION: Use the Law of Cosines to find the missing side measure.
Use the Law of Sines to find a missing angle measure.
Find the measure of the remaining angle.
Therefore, B
39 , C
99 and a
12.9.
29. PQR, if P = 73 , q = 7, and r = 15
SOLUTION: Use the Law of Cosines to find the missing side measure.
Use the Law of Sines to find a missing angle measure.
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Find the measure of the remaining angle.
4-7 Therefore,
The LawB of 39
Sines
of Cosines
, C and
and aLaw
99 the
12.9.
29. PQR, if P = 73 , q = 7, and r = 15
SOLUTION: Use the Law of Cosines to find the missing side measure.
Use the Law of Sines to find a missing angle measure.
Find the measure of the remaining angle.
Therefore, Q
27 , R
80 and p
14.6.
31. RST, if r = 35, s = 22, and t = 25
SOLUTION: Use the Law of Cosines to find the missing angle measure.
Use the Law of Sines to find a missing angle measure.
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Find the measure of the remaining angle.
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Find the measure of the remaining angle.
4-7 The Law of Sines and the Law of Cosines
Therefore, Q
27 , R
80 and p
14.6.
31. RST, if r = 35, s = 22, and t = 25
SOLUTION: Use the Law of Cosines to find the missing angle measure.
Use the Law of Sines to find a missing angle measure.
Find the measure of the remaining angle.
Therefore, R
96 , S
39 and T
45 .
33. BCD, if B = 16 , c = 27, and d = 3
SOLUTION: Use the Law of Cosines to find the missing side measure.
Use the Law of Sines to find a missing angle measure.
FindManual
the measure
ofbythe
remaining
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angle.
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Find the measure of the remaining angle.
4-7 The Law of Sines and the Law of Cosines
Therefore, R
96 , S
39 and T
45 .
33. BCD, if B = 16 , c = 27, and d = 3
SOLUTION: Use the Law of Cosines to find the missing side measure.
Use the Law of Sines to find a missing angle measure.
Find the measure of the remaining angle.
Therefore, C
162 , D
2 and b
24.1.
35. AIRPLANES During her shift, a pilot flies from the Columbus to Atlanta, a distance of 448 miles, and then on to the Phoenix, a distance of 1583 miles. From Phoenix, she returns home to Columbus, a distance of 1667 miles.
Determine the angles of the triangle created by her flight path.
SOLUTION: Draw a diagram to represent the situation.
Use the Law of Cosines to find an angle measure.
Use the Law of Sines to find a second angle measure.
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Find the measure of the remaining angle.
4-7 The Law of Sines and the Law of Cosines
Therefore, C
162 , D
2 and b
24.1.
35. AIRPLANES During her shift, a pilot flies from the Columbus to Atlanta, a distance of 448 miles, and then on to the Phoenix, a distance of 1583 miles. From Phoenix, she returns home to Columbus, a distance of 1667 miles.
Determine the angles of the triangle created by her flight path.
SOLUTION: Draw a diagram to represent the situation.
Use the Law of Cosines to find an angle measure.
Use the Law of Sines to find a second angle measure.
Find A.
Therefore, the angles of the triangle created by the flight path are about 15.6 , 71.5 , and 92.9 .
Use Heron’s Formula to find the area of each triangle. Round to the nearest tenth.
37. x = 9 cm, y = 11 cm, z = 16 cm
SOLUTION: First, find the value of s.
Next, use Heron's Formula find the area of
.
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Therefore,
areabyofCognero
39. x = 58 ft, y = 40 ft, z = 63 ft
2
is about 47.6 cm .
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Find A.
4-7 The Law of Sines and the Law of Cosines
Therefore, the angles of the triangle created by the flight path are about 15.6 , 71.5 , and 92.9 .
Use Heron’s Formula to find the area of each triangle. Round to the nearest tenth.
37. x = 9 cm, y = 11 cm, z = 16 cm
SOLUTION: First, find the value of s.
Next, use Heron's Formula find the area of
.
Therefore, the area of
2
is about 47.6 cm .
39. x = 58 ft, y = 40 ft, z = 63 ft
SOLUTION: First, find the value of s.
Next, use Heron's Formula find the area of
Therefore, the area of
.
2
is about 1133.0 ft .
41. x = 8 yd, y = 15 yd, z = 8 yd
SOLUTION: First, find the value of s.
Next, use Heron's Formula find the area of
.
Therefore, the area of
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2
is about 20.9 ft .
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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
4-7 The Law of Sines and the Law of 2Cosines
Therefore, the area of
is about 1133.0 ft .
41. x = 8 yd, y = 15 yd, z = 8 yd
SOLUTION: First, find the value of s.
Next, use Heron's Formula find the area of
.
Therefore, the area of
2
is about 20.9 ft .
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 area of the lot 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.
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Therefore, the total area is 963.1 + 3548.4 or about 4511.5 square steps.
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4-7 The Law of Sines and the Law of
2 Cosines
Therefore, the area of
is about 20.9 ft .
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 area of the lot 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.
Find the area of each triangle to the nearest tenth.
45. ABC, if A = 98 , b = 13 mm, and c = 8 mm
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4-7 The Law of Sines and the Law of Cosines
Therefore, the area is about 14,617 square feet.
Find the area of each triangle to the nearest tenth.
45. ABC, if A = 98 , b = 13 mm, and c = 8 mm
SOLUTION: 2
Therefore, the area of ABC is about 51.5 mm .
47. RST, if R = 35 , s = 42 ft, and t = 26 ft
SOLUTION: 2
Therefore, the area of ABC is about 313.2 mm .
49. FGH, if F = 41 , g = 22 in., and h = 36 in.
SOLUTION: 2
Therefore, the area of ABC is about 259.8 mm .
Use Heron’s Formula to find the area of each figure. Round answers to the nearest tenth.
53. SOLUTION: First, find the area of
Find the area of
EFJ. Because three side lengths are given, you can use Heron’s formula. First, find s.
FGH.
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4-7 The Law of Sines and the Law of Cosines
2
Therefore, the area of ABC is about 259.8 mm .
Use Heron’s Formula to find the area of each figure. Round answers to the nearest tenth.
53. SOLUTION: First, find the area of
Find the area of
EFJ. Because three side lengths are given, you can use Heron’s formula. First, find s.
FGH.
Therefore, the total area is 110.7 + 178 or about 288.7 square millimeters.
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.
SOLUTION: Draw a diagram to represent the situation.
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4-7 SOLUTION: The Law of Sines and the Law of Cosines
Draw a diagram to represent the situation.
Recall from Geometry that when two parallel lines are cut by a transversal, then each pair of corresponding angles is
congruent. Therefore, ∠A is 180° − (72° + 39°) or 69°, as shown.
So, ∠C = 72° − 31° or 41°, and ∠B = 180° − (69° + 41°) or 70°.
Use the Law of Sines to find a.
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.
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