therefore, · 2019. 2. 23. · solve each triangle. round to the nearest tenth, if necessary....

Solve each triangle. Round to the nearest tenth, if necessary.

1.

SOLUTION:

Because two angles are given, A = 180 – (110 + 38 ) or 32 . Use the Law of Sines to find a and b.

Therefore, , a 11.2, and b 19.8.

2.

SOLUTION:

Because two angles are given, H = 180 – (53 + 112 ) or 15 . Use the Law of Sines to find f and g.

Therefore, , f 55.5, and g 64.5.

3.

SOLUTION:

Because two angles are given, K = 180 – (40 + 58 ) or 82 . Use the Law of Sines to find j and ℓ.

Therefore, , j 16.2, and 21.4.

4.

SOLUTION:

Because two angles are given, S = 180 – (62 + 56 ) or 62 . Use the Law of Sines to find r and s.

Therefore, , r 6.6, and s 7.

5.

SOLUTION:

Because two angles are given, T = 180 – (12 + 148 ) or 20 . Use the Law of Sines to find t and u.

Therefore, t 29.0, and u 17.7.

6.

SOLUTION:

Because two angles are given, D = 180 – (25 + 72 ) or 83º. Use the Law of Sines to find band c.

Therefore, b 12.8, and c 28.7.

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.

8. ARCHITECTURE An architect’s client wants to build a home based on the architect Jon Lautner’s Sheats-Goldstein House. The length of the patio will be 60 feet. The left side of the roof will be at a 49º angle of elevation, and the right side will be at an 18º angle of elevation. Determine the lengths of the left and right sides of the roof and the angle at which they will meet.



Because two angles are given, θ is 180° − (49° + 18°) or 113°. Use the Law of Sines to find x and y .

Therefore, the left and right sides of the roof are about 20.1 and 49.2 feet, respectively, and the angle

at which they meet is about 113 .

9. TRAVEL For the initial 90 miles of a flight, the pilotheads 8º off course in order to avoid a storm. The pilot then changes direction to head toward the destination for the remainder of the flight, making a

157 angle to the first flight course. a. Determine the total distance of the flight. b. Determine the distance of a direct flight to the destination

SOLUTION: a. Draw a diagram to model the situation.

(Not drawn to scale) Because two angles are given, the missing angle is


Therefore, the distance of the flight is 90 + 48.4 or about 138.4 miles. b. Find the length of the side opposite the 157 angle.

Therefore, the distance of a direct flight is about 135.9 miles.

Find all solutions for the given triangle, if possible. If no solution exists, write no solution. Round side lengths to the nearest tenth and angle measures to the nearest degree.

10. a = 9, b = 7, A = 108

SOLUTION: Draw a diagram of a triangle with the given dimensions.

(Not drawn to scale) Notice that A is obtuse and a > b because 9 > 7. Therefore, one solution exists. Apply the Law of Sines to find B.

Because two angles are now known, C ≈ 180 – (108 + 48 ) or about 24 . Apply the Law of Sinesto find c.

Therefore, the remaining measures of are

B 48 , C 24 , and c 3.9.

11. a = 14, b = 15, A = 117

SOLUTION:

A is obtuse and a 12. Therefore, one solution exists. Apply the Law of Sines to find B.

Because two angles are now known, C 180º – (27º + 17.62º) or about 135.38º. Apply the Law of Sines to find c.

Therefore, the remaining measures of ABC are

B 18 , C 135 , and c 27.8.

13. a = 35, b = 24, A = 92


Notice that A is obtuse and a > b because 35 > 24. Therefore, one solution exists. Apply the Law of Sines to find B.

Because two angles are now known, C 180 – (92 + 43.26 ) or about 44.74 . Apply the Law of Sines to find c.


B 43 , C 45 , and c 24.7.

14. a = 14, b = 6, A = 145



Because two angles are now known, C ≈ 180 – (145 + 14.23 ) or about 20.77 . Apply the Law of Sines to find c.


B 14 , C 21 , and c 8.7.

15. a = 19, b = 38, A = 30


Notice that A is acute and b > a because 38 > 19. Therefore, one solution exists. Apply the Law of Sines to find B.

Because two angles are now known, C =180 – (30 + 90 ) or 60 . Apply the Law of Sines to findc.


B = 90 , C = 30 , and c 32.9.

16. a = 5, b = 6, A = 63

SOLUTION: Notice that A is acute and a < b because 5 < 6. So, this problem has no solution, one solution, or two

solutions. Find h.

Because a < h, no triangle can be formed with sides

a = 5, b = 6, and A = 63 . Therefore, this problem has no solution.

17. a = 10, b = , A = 45


Notice that A is acute and b > a because > 10. Therefore, one solution exists. Apply the Law of Sines to find B.

Because two angles are now known, C ≈ 180 – (45 + 90 ) or 45 . Apply the Law of Sines to findc.


B =90 , C = 45 , and c =10.

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 – (28 + 22.06 ) or about 129.94 . Apply the Law of Sines to find x.

Therefore, the length of cable needed for the initial rise is about 41 feet.

Find two triangles with the given angle measure and side lengths. Round side lengths to the nearest tenth and angle measures to the nearest degree.

19. A = 39 , a = 12, b = 17

SOLUTION:

A is acute, and h = 17 sin 39 or about 10.7. Notice that a h because 12 > 10.7. Therefore, two different triangles are possible with the given angle and side measures. Angle B willbe acute, and angle B' will be obtuse, as shown below.

Make a reasonable sketch of each triangle and applythe Law of Sines to find each solution. Start with thecase in which B is acute. Solution 1 B is acute.

Find B.

Find C.

Apply the Law of Sines to find c.

Solution 2 B is obtuse.

Note that . To find B', find an obtuse angle with a sine that is also 0.8915. To do this, subtract the measure given by your calculator tonearest degree, 63º, from 180º. Therefore, B' is approximately 117º. Find C.


Therefore, the missing measures for acute

are

B 63 , C 78 , and c 18.7, while the missingmeasures for obtuse are B' 117 , C 24 , and c 7.8.

20. A = 26 , a = 5, b = 9

SOLUTION:

A is acute, and h =9 sin 26 or about 3.9. Notice that a h because 5 > 3.9.Therefore, two different triangles are possible with the given angle and side measures. Angle B will be acute, and angle B' will be obtuse, as shown below.


Find B.

Find C.



Note that . To find B', find an obtuse angle with a sine that is also 0.2630. To do this, subtract the measure given by your calculator to

nearest degree, 52 , from 180 . Therefore, B' is

approximately 128 . Find C.



are B 52 , C 102 , and c 11.2, while the missing measures for obtuse are B'

128 , C 26 , and c 5.0.

21. A = 61 , a = 14, b = 15

SOLUTION:

A is acute, and h = 14sin 61 or about 12.2. Notice that a h because 14 > 12.2. Therefore, two different triangles are possible with the given angle and side measures. Angle B willbe acute, and angle B' will be obtuse, as shown below.


Find B.

Find C.








are

B 70 , C 49 , and c 12.2, while the missing measures for obtuse are B' 110 , C 9 , and c 2.5.

22. A = 47 , a = 25, b = 34

SOLUTION:



Find B.

Find C.



Note that . To find B', find an obtuse angle with a sine that is also 0.9946. To do this, subtract the measure given by your calculator tonearest degree, 84º, from 180º. Therefore, B' is approximately 96 . Find C.




96 , C 37 , and c 20.6.

23. A = 54 , a = 31, b = 36

SOLUTION:

A is acute, and h = 36 sin 54º or about 29.1. Notice that a h because 31 > 29.1. Therefore, two different triangles are possible with the given angle and side measures. Angle B willbe acute, and angle B' will be obtuse, as shown below.


Find B.

Find C.



Note that . To find B', find an obtuse angle with a sine that is also 0.9395. To do this, subtract the measure given by your calculator tonearest degree, 70º, from 180º. Therefore, B' is approximately 180º – 70º or 110º. Find C.




110 , C 16 , and c 10.6

24. A = 18 , a = 8, b = 13

SOLUTION:



Find B.

Find C.







150 , C 12 , and c 5.4.

25. BROADCASTING A radio tower located 38 miles along Industrial Parkway transmits radio broadcasts over a 30-mile radius. Industrial Parkwayintersects 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 180 – (41 + 56.2 ) or about 82.8 . Apply the Law of Sines again to find BC.

Notice that ABC' is an isosceles triangle, so B ≈ C', B ≈ 56.2º, and thus C' ≈ 56.2º. Because two angles in ABC' are now known, A ≈ 180º – (56.2º + 56.2º) ≈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.

26. BOATING The light from a lighthouse can be seen from an 18-mile radius. A boat is anchored so that it can just see the light from the lighthouse. A second boat is located 25 miles away from the lighthouse and is headed straight toward it, making a 44º angle with the lighthouse and the first boat. Find the distance between the two boats when the secondboat enters the radius of the lighthouse light.

SOLUTION: Draw a diagram to represent the situation.



So, A ≈ 180º – (44º + 74.75º) or about 61.25º. Notice that ABC is an isosceles triangle. Draw an altitude from vertex A to BC'.

Use the sine function to find x.

Therefore, the distance between the two boats whenthe second boat enters the radius of the lighthouse light is 2(9.17) or about 18.3 miles.

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.

28. XYZ, if x = 5, y = 18, and z = 14

SOLUTION: Use the Law of Cosines to find a missing angle measure.



Therefore, X 11 , Y 137 and Z 32 .

29. PQR, if P = 73 , q = 7, and r = 15




Therefore, Q 27 , R 80 and p 14.6.

30. JKL, if J = 125 , k = 24, and l = 33




Therefore, K 23 , L 32 and j 50.7.

31. RST, if r = 35, s = 22, and t = 25

SOLUTION: Use the Law of Cosines to find the missing angle measure.



Therefore, R 96 , S 39 and T 45 .

32. FGH, if f = 39, g = 50, and h = 64




Therefore, F 38 , G 51 and H 91 .

33. BCD, if B = 16 , c = 27, and d = 3




Therefore, C 162 , D 2 and b 24.1.

34. LMN, if l = 12, m = 4, and n = 9




Therefore, L 131 , M 15 and N 34 .

35. AIRPLANES During her shift, a pilot flies from theColumbus 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.


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 .

36. CATCH Lola rolls a ball on the ground at an angle of 23° to the right of her dog Buttons. If the ball rollsa total distance of 48 feet, and she is standing 30 feetaway, how far will Buttons have to run to retrieve the ball?

SOLUTION: Draw a diagram to model the situation.

Use the Law of Cosines to find x.

Therefore, Buttons will have to run 23.5 feet to retrieve that ball.

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 is about 47.6 cm2.

38. x = 29 in., y = 25 in., z = 27 in.




39. x = 58 ft, y = 40 ft, z = 63 ft



Therefore, the area of is about 1133.0 ft2.

40. x = 37 mm, y = 10 mm, z = 34mm




41. x = 8 yd, y = 15 yd, z = 8 yd




42. x = 133 mi, y = 82 mi, z = 77 mi




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.


Therefore, the total area is 963.1 + 3548.4 or about 4511.5 square steps. b. Use dimensional analysis to convert the area fromsquare steps to square feet.

Therefore, the area is about 14,617 square feet.

44. DANCE During a performance, a dancer remainedwithin a triangular area of the stage.

a. Find the area of stage used in the performance. b. If the stage is 250 square feet, determine the percentage of the stage used in the performance.

SOLUTION: a. Use Heron’s Formula to find the area. First find s.

Therefore, the area of stage used in the performanceis 163.9 square feet. b. The stage is 250 square feet. So, 163.9 ÷ 250 = 0.656 or about 66% of the stage is used in the performance

Find the area of each triangle to the nearest tenth.

45. ABC, if A = 98 , b = 13 mm, and c = 8 mm

SOLUTION:

Therefore, the area of ABC is about 51.5 mm2.

46. JKL, if L = 67 , j = 11 yd, and k = 24 yd

SOLUTION:


47. RST, if R = 35 , s = 42 ft, and t = 26 ft

SOLUTION:


48. XYZ, if Y = 124 , x = 16 m, and z = 18 m

SOLUTION:


49. FGH, if F = 41 , g = 22 in., and h = 36 in.

SOLUTION:


50. PQR, if Q = 153 , p = 27 cm, and r = 21 cm

SOLUTION:


51. DESIGN A free-standing art project requires a triangular support piece for stability. Two sides of the triangle must measure 18 and 15 feet in length

and a nonincluded angle must measure 42 . If support purposes require the triangle to have an areaof at least 75 square feet, what is the measure of thethird side?

SOLUTION: Draw a diagram to model the situation of a triangle with side lengths of 15 and 18 feet. Because the 42º angle is a nonincluded angle, there are two possible triangles that can be formed.

Triangle 1 Use the Law of Sines to find C.

So, C = 180º – (42º + 53.41º) or about 84.59º. Use the Law of Sines again to find c.

Find the area of the triangle to make sure that it is at least 75 square feet.




Therefore, the measure of the third side is about 22.3ft or about 26.1 ft.

Use Heron’s Formula to find the area of each figure. Round answers to the nearest tenth.

52.

SOLUTION:

Find the area of PQR. Because three side lengths


1.

SOLUTION:



2.

SOLUTION:



3.

SOLUTION:



4.

SOLUTION:



5.

SOLUTION:



6.

SOLUTION:























10. a = 9, b = 7, A = 108





B 48 , C 24 , and c 3.9.

11. a = 14, b = 15, A = 117

SOLUTION:




B 18 , C 135 , and c 27.8.

13. a = 35, b = 24, A = 92





B 43 , C 45 , and c 24.7.

14. a = 14, b = 6, A = 145





B 14 , C 21 , and c 8.7.

15. a = 19, b = 38, A = 30





B = 90 , C = 30 , and c 32.9.

16. a = 5, b = 6, A = 63


solutions. Find h.



17. a = 10, b = , A = 45





B =90 , C = 45 , and c =10.


SOLUTION:






19. A = 39 , a = 12, b = 17

SOLUTION:



Find B.

Find C.






are


20. A = 26 , a = 5, b = 9

SOLUTION:



Find B.

Find C.









128 , C 26 , and c 5.0.

21. A = 61 , a = 14, b = 15

SOLUTION:



Find B.

Find C.








are


22. A = 47 , a = 25, b = 34

SOLUTION:



Find B.

Find C.







96 , C 37 , and c 20.6.

23. A = 54 , a = 31, b = 36

SOLUTION:



Find B.

Find C.







110 , C 16 , and c 10.6

24. A = 18 , a = 8, b = 13

SOLUTION:



Find B.

Find C.







150 , C 12 , and c 5.4.
















27. ABC, if A = 42 , b = 12, and c = 19





28. XYZ, if x = 5, y = 18, and z = 14





29. PQR, if P = 73 , q = 7, and r = 15





30. JKL, if J = 125 , k = 24, and l = 33





31. RST, if r = 35, s = 22, and t = 25





32. FGH, if f = 39, g = 50, and h = 64





33. BCD, if B = 16 , c = 27, and d = 3





34. LMN, if l = 12, m = 4, and n = 9









Find A.








37. x = 9 cm, y = 11 cm, z = 16 cm




38. x = 29 in., y = 25 in., z = 27 in.




39. x = 58 ft, y = 40 ft, z = 63 ft




40. x = 37 mm, y = 10 mm, z = 34mm




41. x = 8 yd, y = 15 yd, z = 8 yd




42. x = 133 mi, y = 82 mi, z = 77 mi


















SOLUTION:



SOLUTION:



SOLUTION:


48. XYZ, if Y = 124 , x = 16 m, and z = 18 m

SOLUTION:



SOLUTION:



SOLUTION:













52.

SOLUTION:


eSolutions Manual - Powered by Cognero Page 1

4-7 The Law of Sines and the Law of Cosines


1.

SOLUTION:



2.

SOLUTION:



3.

SOLUTION:



4.

SOLUTION:



5.

SOLUTION:



6.

SOLUTION:























10. a = 9, b = 7, A = 108





B 48 , C 24 , and c 3.9.

11. a = 14, b = 15, A = 117

SOLUTION:




B 18 , C 135 , and c 27.8.

13. a = 35, b = 24, A = 92





B 43 , C 45 , and c 24.7.

14. a = 14, b = 6, A = 145





B 14 , C 21 , and c 8.7.

15. a = 19, b = 38, A = 30





B = 90 , C = 30 , and c 32.9.

16. a = 5, b = 6, A = 63


solutions. Find h.



17. a = 10, b = , A = 45





B =90 , C = 45 , and c =10.


SOLUTION:






19. A = 39 , a = 12, b = 17

SOLUTION:



Find B.

Find C.






are


20. A = 26 , a = 5, b = 9

SOLUTION:



Find B.

Find C.









128 , C 26 , and c 5.0.

21. A = 61 , a = 14, b = 15

SOLUTION:



Find B.

Find C.








are


22. A = 47 , a = 25, b = 34

SOLUTION:



Find B.

Find C.







96 , C 37 , and c 20.6.

23. A = 54 , a = 31, b = 36

SOLUTION:



Find B.

Find C.







110 , C 16 , and c 10.6

24. A = 18 , a = 8, b = 13

SOLUTION:



Find B.

Find C.







150 , C 12 , and c 5.4.
















27. ABC, if A = 42 , b = 12, and c = 19





28. XYZ, if x = 5, y = 18, and z = 14





29. PQR, if P = 73 , q = 7, and r = 15





30. JKL, if J = 125 , k = 24, and l = 33





31. RST, if r = 35, s = 22, and t = 25





32. FGH, if f = 39, g = 50, and h = 64





33. BCD, if B = 16 , c = 27, and d = 3





34. LMN, if l = 12, m = 4, and n = 9









Find A.








37. x = 9 cm, y = 11 cm, z = 16 cm




38. x = 29 in., y = 25 in., z = 27 in.




39. x = 58 ft, y = 40 ft, z = 63 ft




40. x = 37 mm, y = 10 mm, z = 34mm




41. x = 8 yd, y = 15 yd, z = 8 yd




42. x = 133 mi, y = 82 mi, z = 77 mi


















SOLUTION:



SOLUTION:



SOLUTION:


48. XYZ, if Y = 124 , x = 16 m, and z = 18 m

SOLUTION:



SOLUTION:



SOLUTION:













52.

SOLUTION:



1.

SOLUTION:



2.

SOLUTION:



3.

SOLUTION:



4.

SOLUTION:



5.

SOLUTION:



6.

SOLUTION:























10. a = 9, b = 7, A = 108





B 48 , C 24 , and c 3.9.

11. a = 14, b = 15, A = 117

SOLUTION:




B 18 , C 135 , and c 27.8.

13. a = 35, b = 24, A = 92





B 43 , C 45 , and c 24.7.

14. a = 14, b = 6, A = 145





B 14 , C 21 , and c 8.7.

15. a = 19, b = 38, A = 30





B = 90 , C = 30 , and c 32.9.

16. a = 5, b = 6, A = 63


solutions. Find h.



17. a = 10, b = , A = 45





B =90 , C = 45 , and c =10.


SOLUTION:






19. A = 39 , a = 12, b = 17

SOLUTION:



Find B.

Find C.






are


20. A = 26 , a = 5, b = 9

SOLUTION:



Find B.

Find C.









128 , C 26 , and c 5.0.

21. A = 61 , a = 14, b = 15

SOLUTION:



Find B.

Find C.








are


22. A = 47 , a = 25, b = 34

SOLUTION:



Find B.

Find C.







96 , C 37 , and c 20.6.

23. A = 54 , a = 31, b = 36

SOLUTION:



Find B.

Find C.







110 , C 16 , and c 10.6

24. A = 18 , a = 8, b = 13

SOLUTION:



Find B.

Find C.







150 , C 12 , and c 5.4.
















27. ABC, if A = 42 , b = 12, and c = 19

SOLUTION: Use the Law of

therefore, · 2019. 2. 23. · solve each triangle. round to the nearest tenth, if necessary....

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