1. 6.4 Day 2
The Ambiguous Case of
The Law of Sines
John 14:6 Jesus answered, “I am the way and the truth and
the life. No one comes to the Father except through me.

2. The Abiguous Case occurs when you are given SSA
information. No unique triangle is determined.

3. The Abiguous Case occurs when you are given SSA
information. No unique triangle is determined.
We could have 0, 1 or 2 triangles.
1 Triangle
0 Triangles
2 Triangles

4. Remember these properties of triangles:

5. Remember these properties of triangles:
1. The sum of the angles is 180.

6. Remember these properties of triangles:
1. The sum of the angles is 180.
2. The longest side is opposite the largest
angle and the smallest side is opposite the
smallest angle.

7. Remember these properties of triangles:
1. The sum of the angles is 180.
2. The longest side is opposite the largest
angle and the smallest side is opposite the
smallest angle.
3. The sum of any 2 sides of any triangle
must be larger than the third side.

8. Solve VABC if ∠A = 41°, a = 40, b = 89

9. Solve VABC if ∠A = 41°, a = 40, b = 89
A
41° 89
B 40 C

10. Solve VABC if ∠A = 41°, a = 40, b = 89
A sin 41° sin B
=
41°
40 89
89
B 40 C

11. Solve VABC if ∠A = 41°, a = 40, b = 89
A sin 41° sin B
=
41°
40 89
89
89sin 41°
sin B =
40
B 40 C

12. Solve VABC if ∠A = 41°, a = 40, b = 89
A sin 41° sin B
=
41°
40 89
89
89sin 41°
sin B =
40
B 40 C
89sin 41°
−1
∠B = sin
40

13. Solve VABC if ∠A = 41°, a = 40, b = 89
A sin 41° sin B
=
41°
40 89
89
89sin 41°
sin B =
40
B 40 C
89sin 41°
−1
∠B = sin
40
−1
∠B ≈ sin 1.4597

14. Solve VABC if ∠A = 41°, a = 40, b = 89
A sin 41° sin B
=
41°
40 89
89
89sin 41°
sin B =
40
B 40 C
89sin 41°
−1
∠B = sin
40
−1
∠B ≈ sin 1.4597
But sine has a max of 1 and min of -1
Therefore, no angle is possible, and
therefore, no triangle is possible.

15. Solve VABC if ∠A = 80°, a = 100, b = 10

16. Solve VABC if ∠A = 80°, a = 100, b = 10
A
80° 10
B 100 C

17. Solve VABC if ∠A = 80°, a = 100, b = 10
A
sin 80° sin B
=
100 10
80° 10
B 100 C

18. Solve VABC if ∠A = 80°, a = 100, b = 10
A
sin 80° sin B
=
100 10
80° 10 10sin 80°
sin B =
100
B 100 C

19. Solve VABC if ∠A = 80°, a = 100, b = 10
A
sin 80° sin B
=
100 10
80° 10 10sin 80°
sin B =
100
−1
B C
∠B ≈ sin .0985
100

20. Solve VABC if ∠A = 80°, a = 100, b = 10
A
sin 80° sin B
=
100 10
80° 10 10sin 80°
sin B =
100
−1
B C
∠B ≈ sin .0985
100
∠B ≈ 5.65° and ∴∠C ≈ 94.35°

21. Solve VABC if ∠A = 80°, a = 100, b = 10
A
sin 80° sin B
=
100 10
80° 10 10sin 80°
sin B =
100
−1
B C
∠B ≈ sin .0985
100
∠B ≈ 5.65° and ∴∠C ≈ 94.35°
Now, the sine of a positive number occurs in both QI and
QII, so we need to check to see if the QII angle is
possible! 180° − 5.65° = 174.25° , but this would cause our
triangle’s angle sum to exceed 180 degrees, therefore,
only one triangle exists. Let’s finish the problem ...

22. Solve VABC if ∠A = 80°, a = 100, b = 10
A ∠B ≈ 5.65° and ∴∠C ≈ 94.35°
80° 10
B 100 C

23. Solve VABC if ∠A = 80°, a = 100, b = 10
A ∠B ≈ 5.65° and ∴∠C ≈ 94.35°
80° 10 sin 80° sin 94.35°
≈
100 c
B 100 C

24. Solve VABC if ∠A = 80°, a = 100, b = 10
A ∠B ≈ 5.65° and ∴∠C ≈ 94.35°
80° 10 sin 80° sin 94.35°
≈
100 c
B 100 C No avoiding using a
rounded number here.

25. Solve VABC if ∠A = 80°, a = 100, b = 10
A ∠B ≈ 5.65° and ∴∠C ≈ 94.35°
80° 10 sin 80° sin 94.35°
≈
100 c
B 100 C No avoiding using a
rounded number here.
100sin 94.35°
c≈
sin 80°

26. Solve VABC if ∠A = 80°, a = 100, b = 10
A ∠B ≈ 5.65° and ∴∠C ≈ 94.35°
80° 10 sin 80° sin 94.35°
≈
100 c
B 100 C No avoiding using a
rounded number here.
100sin 94.35°
c≈
sin 80°
c ≈ 101.25

27. Solve VABC if ∠A = 45.3°, a = 167.1, b = 185.2

28. Solve VABC if ∠A = 45.3°, a = 167.1, b = 185.2
A
45.3° 185.2
B 167.1 C

29. Solve VABC if ∠A = 45.3°, a = 167.1, b = 185.2
A sin 45.3° sin B
=
167.1 185.2
45.3° 185.2
B 167.1 C

30. Solve VABC if ∠A = 45.3°, a = 167.1, b = 185.2
A sin 45.3° sin B
=
167.1 185.2
45.3° 185.2
185.2sin 45.3°
sin B =
167.1
B 167.1 C

31. Solve VABC if ∠A = 45.3°, a = 167.1, b = 185.2
A sin 45.3° sin B
=
167.1 185.2
45.3° 185.2
185.2sin 45.3°
sin B =
167.1
B 167.1 C −1
∠B ≈ sin .7878

32. Solve VABC if ∠A = 45.3°, a = 167.1, b = 185.2
A sin 45.3° sin B
=
167.1 185.2
45.3° 185.2
185.2sin 45.3°
sin B =
167.1
B 167.1 C −1
∠B ≈ sin .7878
∠B ≈ 52° or ∠B ≈ 180 − 52° ≈ 128°

33. Solve VABC if ∠A = 45.3°, a = 167.1, b = 185.2
A sin 45.3° sin B
=
167.1 185.2
45.3° 185.2
185.2sin 45.3°
sin B =
167.1
B 167.1 C −1
∠B ≈ sin .7878
∠B ≈ 52° or ∠B ≈ 180 − 52° ≈ 128°
We have 2 triangles!

34. Solve VABC if ∠A = 45.3°, a = 167.1, b = 185.2
A sin 45.3° sin B
=
167.1 185.2
45.3° 185.2
185.2sin 45.3°
sin B =
167.1
B 167.1 C −1
∠B ≈ sin .7878
∠B ≈ 52° or ∠B ≈ 180 − 52° ≈ 128°
We have 2 triangles!
∠C ≈ 82.7° ∠C ≈ 6.7°
(continued on next slide)

35. (continued from previous slide)
A
V:
1 V2 :
45.3° 185.2 ∠B ≈ 52° ∠B ≈ 128°
∠C ≈ 82.7° ∠C ≈ 6.7°
B 167.1 C

36. (continued from previous slide)
A
V:
1 V2 :
45.3° 185.2 ∠B ≈ 52° ∠B ≈ 128°
∠C ≈ 82.7° ∠C ≈ 6.7°
B 167.1 C
V:
1 sin 45.3° sin 82.7°
≈
167.1 c

37. (continued from previous slide)
A
V:
1 V2 :
45.3° 185.2 ∠B ≈ 52° ∠B ≈ 128°
∠C ≈ 82.7° ∠C ≈ 6.7°
B 167.1 C
V:
1 sin 45.3° sin 82.7°
≈
167.1 c
167.1sin 82.7°
c≈
sin 45.3°

38. (continued from previous slide)
A
V:
1 V2 :
45.3° 185.2 ∠B ≈ 52° ∠B ≈ 128°
∠C ≈ 82.7° ∠C ≈ 6.7°
B 167.1 C
V:
1 sin 45.3° sin 82.7°
≈
167.1 c
167.1sin 82.7°
c≈
sin 45.3°
c ≈ 233.2

39. (continued from previous slide)
A
V:
1 V2 :
45.3° 185.2 ∠B ≈ 52° ∠B ≈ 128°
∠C ≈ 82.7° ∠C ≈ 6.7°
B C c ≈ 233.2
167.1
V:
1 sin 45.3° sin 82.7°
≈
167.1 c
167.1sin 82.7°
c≈
sin 45.3°

40. (continued from previous slide)
A
V:
1 V2 :
45.3° 185.2 ∠B ≈ 52° ∠B ≈ 128°
∠C ≈ 82.7° ∠C ≈ 6.7°
B C c ≈ 233.2
167.1
V:
1 sin 45.3° sin 82.7° V2 : sin 45.3° sin 6.7°
≈ ≈
167.1 c 167.1 c
167.1sin 82.7°
c≈
sin 45.3°

41. (continued from previous slide)
A
V:
1 V2 :
45.3° 185.2 ∠B ≈ 52° ∠B ≈ 128°
∠C ≈ 82.7° ∠C ≈ 6.7°
B C c ≈ 233.2
167.1
V:
1 sin 45.3° sin 82.7° V2 : sin 45.3° sin 6.7°
≈ ≈
167.1 c 167.1 c
167.1sin 82.7° 167.1sin 6.7°
c≈ c≈
sin 45.3° sin 45.3°

42. (continued from previous slide)
A
V:
1 V2 :
45.3° 185.2 ∠B ≈ 52° ∠B ≈ 128°
∠C ≈ 82.7° ∠C ≈ 6.7°
B C c ≈ 233.2
167.1
V:
1 sin 45.3° sin 82.7° V2 : sin 45.3° sin 6.7°
≈ ≈
167.1 c 167.1 c
167.1sin 82.7° 167.1sin 6.7°
c≈ c≈
sin 45.3° sin 45.3°
c ≈ 27.4

43. (continued from previous slide)
A
V:
1 V2 :
45.3° 185.2 ∠B ≈ 52° ∠B ≈ 128°
∠C ≈ 82.7° ∠C ≈ 6.7°
B C c ≈ 233.2 c ≈ 27.4
167.1
V:
1 sin 45.3° sin 82.7° V2 : sin 45.3° sin 6.7°
≈ ≈
167.1 c 167.1 c
167.1sin 82.7° 167.1sin 6.7°
c≈ c≈
sin 45.3° sin 45.3°

44. (continued from previous slide)
A
V:
1 V2 :
45.3° 185.2 ∠B ≈ 52° ∠B ≈ 128°
∠C ≈ 82.7° ∠C ≈ 6.7°
B C c ≈ 233.2 c ≈ 27.4
167.1
V:
1 sin 45.3° sin 82.7° V2 : sin 45.3° sin 6.7°
≈ ≈
167.1 c 167.1 c
167.1sin 82.7° 167.1sin 6.7°
c≈ c≈
sin 45.3° sin 45.3°
Organize your work and your answers

45. HW #7
Opportunity dances with those who are already on the
dance floor.
Jackson Brown