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Advanced MathTrigonometric Relationships Packet
Area of a Triangle Given SAS
1. Find the area of a triangle with sides of length 7 and 9 _________________________and included angle 72.
2. Find the area of a triangle with sides of length 10 and 22 _________________________and included angle 10.
3. Find the area of an equilateral triangle with side of length 10. _________________________
4. A triangle has an area of 16 in2, and two of the sides of the _________________________triangle have lengths 5 in. and 7 in. Find the angle included between these two sides.
5. An isosceles triangle has an area of 24 cm2, and the vertex _________________________angle is . What is the length of the two legs?
Law of Sines
Round ALL answers in entire packet to nearest hundredth.
6. Use the Law of Sines to find the indicated side x. _________________________
7. Use the Law of Sines to find the indicated side x. _________________________
8. Use the Law of Sines to find the indicated angle . _________________________
9. Use the Law of Sines to solve the triangle. _________________________
_________________________
_________________________
10. Use the Law of Sines to solve the triangle. _________________________
A
B
C376
x98.4
24.6
A B
C
26.7
x
52 70
AB
C45
120
36
A B
C
6.546 20
_________________________
_________________________
11. Use the Law of Sines to solve the triangle. _________________________
_________________________
_________________________
12. Use the Law of Sines to solve the triangle. _________________________
_________________________
_________________________
Use the Law of Sines to solve for all possible triangles that satisfy the given conditions:
13. ___________________________________________________________________________
14. ___________________________________________________________________________
15. __________________________________________________
A
BC
2
30
100
A
B
C
12
6812
C
AB
6.5
80
3.4
_________________________
16. ___________________________________________________________________________
17. ___________________________________________________________________________
18. ___________________________________________________________________________
19. ___________________________________________________________________________
20. ___________________________________________________________________________
21. ___________________________________________________________________________
22. ___________________________________________________________________________
Law of Cosines
23. Use the Law of Cosines to find the indicated side x. _________________________
24. Use the Law of Cosines to find the indicated angle . _________________________
25. Use the Law of Cosines to find the indicated side x. _________________________
26. Use the Law of Cosines to find the indicated angle . _________________________
27. Use the Law of Cosines to solve the triangle. _________________________
A B
C
4239
x21
A B
C
154.6
122.560.1
A B
C
18108
x15
A
B
C
12
10
20
_________________________
_________________________
28. Use the Law of Cosines to solve the triangle. _________________________
_________________________
_________________________
Use the Law of Cosines to solve the triangle that satisfies the given conditions:
29. ___________________________________________________________________________
30. ___________________________________________________________________________
31. ___________________________________________________________________________
32. ___________________________________________________________________________
Area of a Triangle Given SSS (aka Heron’s Formula)
A B
C1812010
AC
44
B
12
40
Find the area of the triangle whose sides have the given lengths:
33. _________________________
34. _________________________
35. _________________________
36. _________________________
Mixed (Law of Sines or Law of Cosines)
Find the indicated side x or angle . (Use either the Law of Sines or Law of Cosines, as appropriate.)
37. _________________________
38. _________________________
39. _________________________
40. _________________________
41. _________________________
A B
C3
x35
85
A B
C
1840
x10
B A
C
100x
50
30
A B
C
4
11
10
A
B
C
13838
110
42. _________________________
43. _________________________
44. _________________________
B
A
C
8
100
40
B
A
C
30
x 48
38
BA
C
98
x
1000
25
Applications
45. Tracking a Satellite: The path of a satellite orbiting the earth causes it to pass directly over two tracking stations A and B, which are 50 mi apart. When the satellite is on one side of the two stations, the angles of elevation at A and B are measured to be 87.0 and 84.2, respectively.
a. How far is the satellite from station A? _________________________b. How high is the satellite above the ground? _________________________
46. Distance Across a River: To find the distance across a _________________________ river, a surveyor chooses points A and B, which are 200 ft apart on one side of the river. She then chooses a reference point C on the opposite side of the river and finds that
and . Approximate the distance from A to C.
47. Height of a Tree: A tree on a hillside casts a shadow 215 ft _________________________down the hill. If the angle of inclination of the hillside is 22to the horizontal and the angle of elevation of the sun is 52, find the height of the tree.
48. Length of a Guy Wire: A communications tower is located at _________________________the top of a steep hill. The angle of inclination of the hill is 58. A guy wire is to be attached to the top of the tower and to the ground, 100 m downhill from the base of the tower. The angle is determined to be 12. Find the length of the cable required for the guy wire.
49. Calculating a Distance: Observers at P and Q are located on the _________________________side of a hill that is inclined 32 to the horizontal. The observer at P determines the angle of elevation to a hot-air balloon to be 62. At the same instant, the observer at Q measures the angle of elevation to the balloon to be 71. If P is 60 m down the hill from Q, find the distance from Q to the balloon.
50. Calculating an Angle: A water tower 30 m tall is located at the _________________________top of a hill. From a distance of 120 m down the hill, it is observed that the angle formed between the top and base of the tower is 8. Find the angle of inclination of the hill.
51. Surveying: To find the distance across a small lake, a surveyor _________________________has taken the measurements shown. Find the distance across the lake using this information.
52. Towing a Barge: Two tugboats that are 120 ft apart pull a barge. _________________________If the length of one cable is 212 ft and the length of the other is 230 ft, find the angle formed by the two cables.
53. Flying Kites: A boy is flying two kites at the same time. He has _________________________380 ft of line out to one kite and 420 ft to the other. He estimates the angle between the two lines to be 30. Approximate the distance between the kites.
54. Securing a Tower: A 125-ft tower is located on the side of a _________________________mountain that is inclined 32 to the horizontal. A guy wire is to be attached to the top of the tower and anchored at a point 55 ft downhill from the base of the tower. Find the shortest length of wire needed.