Notes for the Law of Sines, including the Ambiguous case. Mr. Fjelstrom's BM 13.1.

1. 6.1 - Law of Sines

2. An oblique triangle is a triangle that has no right angles.
C
a
b
A B
c
To solve an oblique triangle, you need to know the
measure of at least one side and the measures of any
other two parts of the triangle – two sides, two angles,
or one angle and one side.
2

3. Four Cases for solving a triangle…
3

4. Four Cases for solving a triangle…
1. Two angles and any side (AAS or ASA)
A
A
c c
B
C
3

5. Four Cases for solving a triangle…
1. Two angles and any side (AAS or ASA)
A
A
c c
B
C
2. Two sides and an angle opposite one of them (SSA)
c
C a
3

6. Four Cases for solving a triangle…
1. Two angles and any side (AAS or ASA)
A
A
c c
B
C
2. Two sides and an angle opposite one of them (SSA)
c
C a
3. Three sides (SSS)
c
b
a
3

7. Four Cases for solving a triangle…
1. Two angles and any side (AAS or ASA)
A
A
c c
B
C
2. Two sides and an angle opposite one of them (SSA)
c
C a
3. Three sides (SSS)
c
b
c
a
4. Two sides and their included angle (SAS) B
a
3

8. Case 1 and Case 2:
4

9. Law of Sines
Case 1 and Case 2:
If ABC is an oblique triangle with sides a, b, and c, then
C C
a
a
b
h
h b
A B B
c c
A
Acute Triangle Obtuse Triangle
4

10. Example 1 (ASA):
Find the remaining angle and sides of the triangle.
5

11. Example 1 (ASA):
Find the remaining angle and sides of the triangle.
C
10°
a = 4.5 ft
b
60°
B
c
A
5

12. Example 1 (ASA):
Find the remaining angle and sides of the triangle.
C
10°
The third angle in the triangle is
A = 180° – A – B a = 4.5 ft
= 180° – 10° – 60° b
60°
= 110°
110° B
c
A
5

13. Example 1 (ASA):
Find the remaining angle and sides of the triangle.
C
10°
The third angle in the triangle is
A = 180° – A – B a = 4.5 ft
= 180° – 10° – 60° b
60°
= 110°
110° B
c
A
Use the Law of Sines to find side b and c.
5

14. Example 1 (ASA):
Find the remaining angle and sides of the triangle.
C
10°
The third angle in the triangle is
A = 180° – A – B a = 4.5 ft
= 180° – 10° – 60° 4.15 ft b
60°
= 110°
110° B
c
A
Use the Law of Sines to find side b and c.
5

15. Example 1 (ASA):
Find the remaining angle and sides of the triangle.
C
10°
The third angle in the triangle is
A = 180° – A – B a = 4.5 ft
= 180° – 10° – 60° 4.15 ft b
60°
= 110°
110° B
c
A
0.83 ft
Use the Law of Sines to find side b and c.
5

16. Watch out for SSA!
6

17. Watch out for SSA!
SSA is not a congruency property.
SO, if we are given t wo sides and the NON-included angle in a
triangle, there are three possible scenarios…
6

18. Watch out for SSA!
SSA is not a congruency property.
SO, if we are given t wo sides and the NON-included angle in a
triangle, there are three possible scenarios…
➡ Option 1: No triangle is formed
6

19. Watch out for SSA!
SSA is not a congruency property.
SO, if we are given t wo sides and the NON-included angle in a
triangle, there are three possible scenarios…
➡ Option 1: No triangle is formed
➡ Option 2: One triangle is formed
6

20. Watch out for SSA!
SSA is not a congruency property.
SO, if we are given t wo sides and the NON-included angle in a
triangle, there are three possible scenarios…
➡ Option 1: No triangle is formed
➡ Option 2: One triangle is formed
➡ Option 3: Two triangles are formed
6

21. Watch out for SSA!
SSA is not a congruency property.
SO, if we are given t wo sides and the NON-included angle in a
triangle, there are three possible scenarios…
➡ Option 1: No triangle is formed
➡ Option 2: One triangle is formed
➡ Option 3: Two triangles are formed
How is it determined? If …
6

22. Watch out for SSA!
SSA is not a congruency property.
SO, if we are given t wo sides and the NON-included angle in a
triangle, there are three possible scenarios…
➡ Option 1: No triangle is formed
➡ Option 2: One triangle is formed
➡ Option 3: Two triangles are formed
How is it determined? If …
opp ≥ adj
0 or 1 triangle
6

23. Watch out for SSA!
SSA is not a congruency property.
SO, if we are given t wo sides and the NON-included angle in a
triangle, there are three possible scenarios…
➡ Option 1: No triangle is formed
➡ Option 2: One triangle is formed
➡ Option 3: Two triangles are formed
How is it determined? If …
opp ≥ adj opp ≤ adj
0 or 1 triangle 0 or 2 triangles
6

24. Example 2 (SSA):
Use the Law of Sines to solve the triangle.
A = 110°, a = 125 inches, b = 100 inches
7

25. Example 2 (SSA):
Use the Law of Sines to solve the triangle.
A = 110°, a = 125 inches, b = 100 inches
C
a = 125 in
b = 100 in
110°
B
c
A
7

26. Example 2 (SSA):
Use the Law of Sines to solve the triangle.
A = 110°, a = 125 inches, b = 100 inches
C
There is either 0 or 1 triangle
satisfying the given conditions
a = 125 in
because opp ≥ adj
b = 100 in
110°
B
c
A
7

27. Example 2 (SSA):
Use the Law of Sines to solve the triangle.
A = 110°, a = 125 inches, b = 100 inches
C
There is either 0 or 1 triangle
satisfying the given conditions
a = 125 in
because opp ≥ adj
b = 100 in
110° 48.74°
B
c
A
7

28. Example 2 (SSA):
Use the Law of Sines to solve the triangle.
A = 110°, a = 125 inches, b = 100 inches
C 21.26°
There is either 0 or 1 triangle
satisfying the given conditions
a = 125 in
because opp ≥ adj
b = 100 in
110° 48.74°
B
c
A
C ≈ 180° – 110° – 48.74°
= 21.26°
7

29. Example 2 (SSA):
Use the Law of Sines to solve the triangle.
A = 110°, a = 125 inches, b = 100 inches
C 21.26°
There is either 0 or 1 triangle
satisfying the given conditions
a = 125 in
because opp ≥ adj
b = 100 in
110° 48.74°
B
c
A 48.23 in
C ≈ 180° – 110° – 48.74°
= 21.26°
7

30. Example 3 (SSA):
Use the Law of Sines to solve the triangle.
A = 76°, a = 18 inches, b = 20 inches
8

31. Example 3 (SSA):
Use the Law of Sines to solve the triangle.
A = 76°, a = 18 inches, b = 20 inches
There is either 0 or 2 triangles
satisfying the given conditions
because opp ≤ adj
8

32. Example 3 (SSA):
Use the Law of Sines to solve the triangle.
A = 76°, a = 18 inches, b = 20 inches
There is either 0 or 2 triangles
C
satisfying the given conditions
because opp ≤ adj
b = 20 in
a = 18 in
76°
A
B
There is no angle whose sine is 1.078.
8

33. Example 4 (SSA):
Use the Law of Sines to solve the triangle.
A = 58°, a = 11.4 cm, b = 12.8 cm
9

34. Example 4 (SSA): C
Use the Law of Sines to solve the triangle.
A = 58°, a = 11.4 cm, b = 12.8 cm
b = 12.8 cm
a = 11.4 cm
58°
A
B1 c
9

35. Example 4 (SSA): C
Use the Law of Sines to solve the triangle.
A = 58°, a = 11.4 cm, b = 12.8 cm
There is either 0 or 2 b = 12.8 cm
a = 11.4 cm
triangles that can be
formed (opp ≤ adj) 58°
A
B1 c
9

36. Example 4 (SSA): C
Use the Law of Sines to solve the triangle.
A = 58°, a = 11.4 cm, b = 12.8 cm
There is either 0 or 2 b = 12.8 cm
a = 11.4 cm
triangles that can be
formed (opp ≤ adj) 72.2° 58°
A
B1 c
9

37. Example 4 (SSA): C
49.8°
Use the Law of Sines to solve the triangle.
A = 58°, a = 11.4 cm, b = 12.8 cm
There is either 0 or 2 b = 12.8 cm
a = 11.4 cm
triangles that can be
formed (opp ≤ adj) 72.2° 58°
A
B1 c
C ≈ 180° – 58° – 72.2° = 49.8°
9

38. Example 4 (SSA): C
49.8°
Use the Law of Sines to solve the triangle.
A = 58°, a = 11.4 cm, b = 12.8 cm
There is either 0 or 2 b = 12.8 cm
a = 11.4 cm
triangles that can be
formed (opp ≤ adj) 72.2° 58°
A
B1 c
10.3 cm
C ≈ 180° – 58° – 72.2° = 49.8°
Example continues …
9

39. Example 4 (SSA) continued:
Use the Law of Sines to solve the second triangle. C
A = 58°, a = 11.4 cm, b = 12.8 cm 49.8°
b = 12.8 cm
a = 11.4 cm
a=11.4 cm
72.2° 58°
A
B1 c
10.3 cm
10

40. Example 4 (SSA) continued:
Use the Law of Sines to solve the second triangle. C
A = 58°, a = 11.4 cm, b = 12.8 cm 49.8°
b = 12.8 cm
a = 11.4 cm
a=11.4 cm
72.2° 58°
A
B1 c
10.3 cm
C
b = 12.8 cm
a = 11.4 cm
58°
A
B2 c
10

41. Example 4 (SSA) continued:
Use the Law of Sines to solve the second triangle. C
A = 58°, a = 11.4 cm, b = 12.8 cm 49.8°
B2 ≈ 180° – 72.2° = 107.8 ° b = 12.8 cm
a = 11.4 cm
a=11.4 cm
72.2° 58°
A
B1 c
10.3 cm
C
b = 12.8 cm
a = 11.4 cm
58°
107.8°
A
B2 c
10

42. Example 4 (SSA) continued:
Use the Law of Sines to solve the second triangle. C
A = 58°, a = 11.4 cm, b = 12.8 cm 49.8°
B2 ≈ 180° – 72.2° = 107.8 ° b = 12.8 cm
a = 11.4 cm
a=11.4 cm
72.2° 58°
C ≈ 180° – 58° – 107.8° = 14.2° A
B1 c
10.3 cm
C
14.2°
b = 12.8 cm
a = 11.4 cm
58°
107.8°
A
B2 c
10

43. Example 4 (SSA) continued:
Use the Law of Sines to solve the second triangle. C
A = 58°, a = 11.4 cm, b = 12.8 cm 49.8°
B2 ≈ 180° – 72.2° = 107.8 ° b = 12.8 cm
a = 11.4 cm
a=11.4 cm
72.2° 58°
C ≈ 180° – 58° – 107.8° = 14.2° A
B1 c
10.3 cm
C
14.2°
b = 12.8 cm
a = 11.4 cm
58°
107.8°
A
B2 c
3.3 cm
10

44. Area of an Oblique Triangle
11

45. Area of an Oblique Triangle
11

46. Area of an Oblique Triangle
Example 5:
Find the area of the triangle.
A = 74°, b = 103 inches, c = 58 inches
11

47. Area of an Oblique Triangle
C
Example 5:
103 in
Find the area of the triangle. a
b
A = 74°, b = 103 inches, c = 58 inches
74°
A B
c
58 in
11

48. Area of an Oblique Triangle
C
Example 5:
103 in
Find the area of the triangle. a
b
A = 74°, b = 103 inches, c = 58 inches
74°
A B
c
58 in
11

49. Area of an Oblique Triangle
C
Example 5:
103 in
Find the area of the triangle. a
b
A = 74°, b = 103 inches, c = 58 inches
74°
A B
c
58 in
11

50. Area of an Oblique Triangle
C
Example 5:
103 in
Find the area of the triangle. a
b
A = 74°, b = 103 inches, c = 58 inches
74°
A B
c
58 in
11

51. Application:
A flagpole at a right angle to the horizontal is located on a slope
that makes an angle of 14° with the horizontal. The flagpole
casts a 16-meter shadow up the slope when the angle of
elevation from the tip of the shadow to the sun is 20°. How tall
is the flagpole?
12

52. Application:
A flagpole at a right angle to the horizontal is located on a slope
that makes an angle of 14° with the horizontal. The flagpole
casts a 16-meter shadow up the slope when the angle of
elevation from the tip of the shadow to the sun is 20°. How tall
is the flagpole?
12

53. Application:
A flagpole at a right angle to the horizontal is located on a slope
that makes an angle of 14° with the horizontal. The flagpole
casts a 16-meter shadow up the slope when the angle of
elevation from the tip of the shadow to the sun is 20°. How tall
is the flagpole?
14°
12

54. Application:
A flagpole at a right angle to the horizontal is located on a slope
that makes an angle of 14° with the horizontal. The flagpole
casts a 16-meter shadow up the slope when the angle of
elevation from the tip of the shadow to the sun is 20°. How tall
is the flagpole?
Flagpole height: b
14°
12

55. Application:
A flagpole at a right angle to the horizontal is located on a slope
that makes an angle of 14° with the horizontal. The flagpole
casts a 16-meter shadow up the slope when the angle of
elevation from the tip of the shadow to the sun is 20°. How tall
is the flagpole?
Flagpole height: b
14°
12

56. Application:
A flagpole at a right angle to the horizontal is located on a slope
that makes an angle of 14° with the horizontal. The flagpole
casts a 16-meter shadow up the slope when the angle of
elevation from the tip of the shadow to the sun is 20°. How tall
is the flagpole?
Flagpole height: b
14°
12

57. Application:
A flagpole at a right angle to the horizontal is located on a slope
that makes an angle of 14° with the horizontal. The flagpole
casts a 16-meter shadow up the slope when the angle of
elevation from the tip of the shadow to the sun is 20°. How tall
is the flagpole?
Flagpole height: b
16 m
14°
12

58. Application:
A flagpole at a right angle to the horizontal is located on a slope
that makes an angle of 14° with the horizontal. The flagpole
casts a 16-meter shadow up the slope when the angle of
elevation from the tip of the shadow to the sun is 20°. How tall
is the flagpole?
Flagpole height: b
16 m
14°
12

59. Application:
A flagpole at a right angle to the horizontal is located on a slope
that makes an angle of 14° with the horizontal. The flagpole
casts a 16-meter shadow up the slope when the angle of
elevation from the tip of the shadow to the sun is 20°. How tall
is the flagpole?
20°
Flagpole height: b
16 m
14°
12

60. Application:
A flagpole at a right angle to the horizontal is located on a slope
that makes an angle of 14° with the horizontal. The flagpole
casts a 16-meter shadow up the slope when the angle of
elevation from the tip of the shadow to the sun is 20°. How tall
is the flagpole?
20°
70°
Flagpole height: b
16 m
14°
12

61. Application:
A flagpole at a right angle to the horizontal is located on a slope
that makes an angle of 14° with the horizontal. The flagpole
casts a 16-meter shadow up the slope when the angle of
elevation from the tip of the shadow to the sun is 20°. How tall
is the flagpole?
20°
70°
34°
Flagpole height: b
16 m
14°
12

62. Application:
A flagpole at a right angle to the horizontal is located on a slope
that makes an angle of 14° with the horizontal. The flagpole
casts a 16-meter shadow up the slope when the angle of
elevation from the tip of the shadow to the sun is 20°. How tall
is the flagpole?
A
20°
70°
34° B
Flagpole height: b
16 m
C
14°
12

63. Application:
A flagpole at a right angle to the horizontal is located on a slope
that makes an angle of 14° with the horizontal. The flagpole
casts a 16-meter shadow up the slope when the angle of
elevation from the tip of the shadow to the sun is 20°. How tall
is the flagpole?
A
20°
70°
34° B
Flagpole height: b
16 m
C
14°
12

64. Application:
A flagpole at a right angle to the horizontal is located on a slope
that makes an angle of 14° with the horizontal. The flagpole
casts a 16-meter shadow up the slope when the angle of
elevation from the tip of the shadow to the sun is 20°. How tall
is the flagpole?
A
20°
70°
34° B
Flagpole height: b
16 m
C
14°
12

65. Application:
A flagpole at a right angle to the horizontal is located on a slope
that makes an angle of 14° with the horizontal. The flagpole
casts a 16-meter shadow up the slope when the angle of
elevation from the tip of the shadow to the sun is 20°. How tall
is the flagpole?
A
20°
70°
34° B
Flagpole height: b
16 m
C
14°
12

66. Application:
A flagpole at a right angle to the horizontal is located on a slope
that makes an angle of 14° with the horizontal. The flagpole
casts a 16-meter shadow up the slope when the angle of
elevation from the tip of the shadow to the sun is 20°. How tall
is the flagpole?
A
20°
70°
34° B
Flagpole height: b
16 m
C
14°
The flagpole is approximately 9.5 meters tall.
12

67. Homework:
Assignment #2 p. 436
#7, 8, 13, 19-22, 29-31, 35,
36, 39, 40, 42