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Advanced Math

Trigonometric 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 [pic]. What is the length of the two legs?

Law of Sines

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.

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. [pic]

14. [pic]

15. [pic]

16. [pic]

17. [pic]

18. [pic]

19. [pic]

20. [pic]

21. [pic]

22. [pic]

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.

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. [pic]

30. [pic]

31. [pic]

32. [pic]

Area of a Triangle Given SSS (aka Heron’s Formula)

Find the area of the triangle whose sides have the given lengths:

33. [pic]

34. [pic]

35. [pic]

36. [pic]

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.

42.

43.

44.

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

[pic] and [pic]. 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 22(

to the horizontal and the angle of elevation of the sun is 52(,

find the height of the tree.

[pic]

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.

[pic]

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.

-----------------------

A

B

C

376

x

98.4(

24.6(

A

B

C

26.7

x

52(

70(

A

B

C

45

(

120(

36

A

B

C

6.5

100(

46(

20(

30(

A

2

C

B

A

B

C

12

68(

12

C

A

B

6.5

80(

3.4

x

39(

42

C

B

A

21

60.1

122.5

(

154.6

C

B

A

15

x

108(

18

C

B

A

20

10

(

12

C

B

A

10

120(

18

C

B

A

B

40

12

44

C

Round ALL answers in entire packet to nearest hundredth.

A

85(

35(

x

3

C

B

A

10

x

40(

18

C

B

A

50

x

100(

10

C

A

B

30(

(

11

4

C

B

A

110

(

38(

138

C

B

A

40(

100

(

8

C

A

B

38

48

x

30(

C

A

B

25(

1000

x

98(

C

A

B

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