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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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