1. 6.4 The Law of Sines
Matthew 16: 26 What good will it be for a man if he gains
the whole world, yet forfeits his soul? Or what can a man give
in exchange for his soul?

2. The Law of Sines

3. The Law of Sines
- used for oblique (non-right) triangles

4. The Law of Sines
- used for oblique (non-right) triangles
- proof is in the book

5. The Law of Sines
- used for oblique (non-right) triangles
- proof is in the book
sin A sin B sinC
= =
a b c

6. The Law of Sines
- used for oblique (non-right) triangles
- proof is in the book
sin A sin B sinC
= =
a b c
Use when given: ASA or SAA (only 1 triangle possible)

7. The Law of Sines
- used for oblique (non-right) triangles
- proof is in the book
sin A sin B sinC
= =
a b c
Use when given: ASA or SAA (only 1 triangle possible)
Use Law of Cosines when given: SAS or SSS

8. The Law of Sines
- used for oblique (non-right) triangles
- proof is in the book
sin A sin B sinC
= =
a b c
Use when given: ASA or SAA (only 1 triangle possible)
Use Law of Cosines when given: SAS or SSS
SSA is called the Ambiguous Case as we could have
0, 1 or 2 triangles. We will deal with this case later.

9. Solve VABC if ∠A = 38°, ∠B = 64°, c = 81

10. Solve VABC if ∠A = 38°, ∠B = 64°, c = 81
Draw a diagram!!

11. Solve VABC if ∠A = 38°, ∠B = 64°, c = 81
A
38°
81
Draw a diagram!!
64°
B C

12. Solve VABC if ∠A = 38°, ∠B = 64°, c = 81
A
∠C = 78°
38°
81
Draw a diagram!!
64°
B C

13. Solve VABC if ∠A = 38°, ∠B = 64°, c = 81
A
∠C = 78°
38°
81
Draw a diagram!!
64°
B C
sin 78° sin 38°
=
81 a

14. Solve VABC if ∠A = 38°, ∠B = 64°, c = 81
A
∠C = 78°
38°
81
Draw a diagram!!
64°
B C
sin 78° sin 38°
=
81 a
81sin 38°
a=
sin 78°

15. Solve VABC if ∠A = 38°, ∠B = 64°, c = 81
A
∠C = 78°
38°
81
Draw a diagram!!
64°
B C
sin 78° sin 38°
=
81 a
81sin 38°
a=
sin 78°
a ≈ 51.0

16. Solve VABC if ∠A = 38°, ∠B = 64°, c = 81
A
∠C = 78° a ≈ 51.0
38°
81
Draw a diagram!!
64°
B C
sin 78° sin 38°
=
81 a
81sin 38°
a=
sin 78°
a ≈ 51.0

17. Solve VABC if ∠A = 38°, ∠B = 64°, c = 81
A
∠C = 78° a ≈ 51.0
38°
81
Draw a diagram!!
64°
B C
sin 78° sin 38° sin 78° sin 64°
= =
81 a 81 b
81sin 38°
a=
sin 78°
a ≈ 51.0

18. Solve VABC if ∠A = 38°, ∠B = 64°, c = 81
A
∠C = 78° a ≈ 51.0
38°
81
Draw a diagram!!
64°
B C
sin 78° sin 38° sin 78° sin 64°
= =
81 a 81 b
81sin 38° 81sin 64°
a= b=
sin 78° sin 78°
a ≈ 51.0

19. Solve VABC if ∠A = 38°, ∠B = 64°, c = 81
A
∠C = 78° a ≈ 51.0
38°
81
Draw a diagram!!
64°
B C
sin 78° sin 38° sin 78° sin 64°
= =
81 a 81 b
81sin 38° 81sin 64°
a= b=
sin 78° sin 78°
a ≈ 51.0 b ≈ 74.4

20. Solve VABC if ∠A = 38°, ∠B = 64°, c = 81
A
∠C = 78° a ≈ 51.0
38°
81
Draw a diagram!! b ≈ 74.4
64°
B C
sin 78° sin 38° sin 78° sin 64°
= =
81 a 81 b
81sin 38° 81sin 64°
a= b=
sin 78° sin 78°
a ≈ 51.0 b ≈ 74.4

21. Solve VABC if ∠A = 56°, ∠B = 74°, b = 78.2

22. Solve VABC if ∠A = 56°, ∠B = 74°, b = 78.2
A
56° 78.2
74°
B C

23. Solve VABC if ∠A = 56°, ∠B = 74°, b = 78.2
A
∠C = 50°
56° 78.2
74°
B C

24. Solve VABC if ∠A = 56°, ∠B = 74°, b = 78.2
A
∠C = 50°
56° 78.2
74°
B C
sin 74° sin 56°
=
78.2 a

25. Solve VABC if ∠A = 56°, ∠B = 74°, b = 78.2
A
∠C = 50°
56° 78.2
74°
B C
sin 74° sin 56°
=
78.2 a
78.2sin 56°
a=
sin 74°

26. Solve VABC if ∠A = 56°, ∠B = 74°, b = 78.2
A
∠C = 50°
56° 78.2
74°
B C
sin 74° sin 56°
=
78.2 a
78.2sin 56°
a=
sin 74°
a ≈ 67.4

27. Solve VABC if ∠A = 56°, ∠B = 74°, b = 78.2
A a ≈ 67.4
∠C = 50°
56° 78.2
74°
B C
sin 74° sin 56°
=
78.2 a
78.2sin 56°
a=
sin 74°
a ≈ 67.4

28. Solve VABC if ∠A = 56°, ∠B = 74°, b = 78.2
A a ≈ 67.4
∠C = 50°
56° 78.2
74°
B C
sin 74° sin 56° sin 74° sin 50°
= =
78.2 a 78.2 c
78.2sin 56°
a=
sin 74°
a ≈ 67.4

29. Solve VABC if ∠A = 56°, ∠B = 74°, b = 78.2
A a ≈ 67.4
∠C = 50°
56° 78.2
74°
B C
sin 74° sin 56° sin 74° sin 50°
= =
78.2 a 78.2 c
78.2sin 56° 78.2sin 50°
a= c=
sin 74° sin 74°
a ≈ 67.4

30. Solve VABC if ∠A = 56°, ∠B = 74°, b = 78.2
A a ≈ 67.4
∠C = 50°
56° 78.2
74°
B C
sin 74° sin 56° sin 74° sin 50°
= =
78.2 a 78.2 c
78.2sin 56° 78.2sin 50°
a= c=
sin 74° sin 74°
a ≈ 67.4 c ≈ 62.3

31. Solve VABC if ∠A = 56°, ∠B = 74°, b = 78.2
A a ≈ 67.4
∠C = 50°
56° 78.2 c ≈ 62.3
74°
B C
sin 74° sin 56° sin 74° sin 50°
= =
78.2 a 78.2 c
78.2sin 56° 78.2sin 50°
a= c=
sin 74° sin 74°
a ≈ 67.4 c ≈ 62.3

32. The navigator of a hot air balloon determines the angle
of depression to the landing site is 15 degrees. Later,
when 2 miles closer but at the same altitude, it is 68
degrees. Find the altitude of the balloon.

33. The navigator of a hot air balloon determines the angle
of depression to the landing site is 15 degrees. Later,
when 2 miles closer but at the same altitude, it is 68
degrees. Find the altitude of the balloon.

34. The navigator of a hot air balloon determines the angle
of depression to the landing site is 15 degrees. Later,
when 2 miles closer but at the same altitude, it is 68
degrees. Find the altitude of the balloon.
∠ACB = 112°
∠ABC = 53°

35. The navigator of a hot air balloon determines the angle
of depression to the landing site is 15 degrees. Later,
when 2 miles closer but at the same altitude, it is 68
degrees. Find the altitude of the balloon.
sin112° sin 53°
=
x 2
∠ACB = 112°
∠ABC = 53°

36. The navigator of a hot air balloon determines the angle
of depression to the landing site is 15 degrees. Later,
when 2 miles closer but at the same altitude, it is 68
degrees. Find the altitude of the balloon.
sin112° sin 53°
=
x 2
2sin112°
x=
sin 53°
∠ACB = 112°
∠ABC = 53°

37. The navigator of a hot air balloon determines the angle
of depression to the landing site is 15 degrees. Later,
when 2 miles closer but at the same altitude, it is 68
degrees. Find the altitude of the balloon.
sin112° sin 53°
=
x 2
2sin112°
x=
sin 53°
x ≈ 2.32
∠ACB = 112°
∠ABC = 53°

38. The navigator of a hot air balloon determines the angle
of depression to the landing site is 15 degrees. Later,
when 2 miles closer but at the same altitude, it is 68
degrees. Find the altitude of the balloon.
sin112° sin 53°
=
x 2
2sin112°
x=
sin 53°
x ≈ 2.32
y
sin A =
x
∠ACB = 112°
∠ABC = 53°

39. The navigator of a hot air balloon determines the angle
of depression to the landing site is 15 degrees. Later,
when 2 miles closer but at the same altitude, it is 68
degrees. Find the altitude of the balloon.
sin112° sin 53°
=
x 2
2sin112°
x=
sin 53°
x ≈ 2.32
y
sin A =
x
∠ACB = 112°
y ≈ 2.32sin15°
∠ABC = 53°

40. The navigator of a hot air balloon determines the angle
of depression to the landing site is 15 degrees. Later,
when 2 miles closer but at the same altitude, it is 68
degrees. Find the altitude of the balloon.
sin112° sin 53°
=
x 2
2sin112°
x=
sin 53°
x ≈ 2.32
y
sin A =
x
∠ACB = 112°
y ≈ 2.32sin15°
∠ABC = 53°
y ≈ .6 miles

41. HW #6
Tomorrow: The Ambiguous Case
The greatest mistake you can make in life is to be
continually fearing you will make one.
Elbert Hubbard