1. Section 1.5
Law of Sines

2. Objectives:
1. To prove the law of sines.
2. To identify the ambiguous case.
3. To solve triangles using the law
of sines.

3. Law of Sines
In ABC where a is the side length opposite A, b is opposite B, and c is opposite C, the following proportion exists:
sin C c
sin B b
sin A a =

4. To apply the law of sines, you must know either the measures of two angles and a side (ASA or SAA) or the measures of two sides and angle opposite one of the given sides (SSA).

5. Example: Solve triangle ABC
mA = 52, mB = 49, c = 16
mC = 180° - mA - mB
mC = 180 - 52 - 49
mC = 79

6. Example: Solve triangle ABC
mA = 52, mB = 49, c = 16 = 79
sin 16 52 a
a(sin 79) = (16)(sin 52)
(sin 79)
(16)(sin 52)
a =
a ≈ 12.8

7. Example: Solve triangle ABC
mA = 52, mB = 49, c = 16 = 79
sin 16 49 b
b(sin 79) = (16)(sin 49)
(sin 79)
(16)(sin 49)
b =
b ≈ 12.3

8. Example: Solve triangle ABC
mA = 52
mB = 49
mC = 79

9. SSA is called the ambiguous case because it does not always result in a unique triangle. The three possible cases follow.
1. There may be not be a solution.
2. There may be two solutions.
3. There may be a unique solution.

10. 1. SSA – no solution
No triangle exists when b  h.
sin B = h/a
a sin B = h
If b  a sin B, there is no solution to SSA; no triangle is possible.

11. C a b h B
b  a sin B

12. 2. SSA – two solutions
two triangles exist when
h  b  a
a sin B  b  a
If a sin B  b  a, the solution to SSA can be either of two triangles.

13. C a b b h B
a sin B  b  a

14. 3. SSA – one solution
one triangle exists when
b = h
b = a sin B
or, one triangle exists when
b  a. If b = a sin B or b  a, then the solution to SSA is a unique triangle.

15. C a
b = h B
b = a sin B

16. C a b h B
b  a

17. Possible Cases for SSA:
1. b < a sin B - no solution
2. b = a sin B - 1 solution (rt. triangle)
3. b > a sin B and a > b solutions
4. b > a sin B and b > a solution

18. Practice: Give the number of triangles that satisfy the given information.
c = 8, B = 35, C = 70
One

19. Practice: Give the number of triangles that satisfy the given information.
a = 12, b = 9, B = 55
None

20. Practice: Give the number of triangles that satisfy the given information.
b = 13, c = 14, B = 64
Two

21. Practice: Give the number of triangles that satisfy the given information.
a = 10, c = 12, C = 73
One

22. Practice: Solve triangle ABC
a = 6, c = 8, mC = 100 =
sin 100 8
sin A 6
6(sin 100) = 8(sin A) 8
(6)(sin 100)
sin A =
sin A ≈

23. Practice: Solve triangle ABC
a = 6, c = 8, mC = 100
mA ≈ sin
mA ≈ 47.6
mB = 180 - mA - mC
mB = 180  -100
mB ≈ 32.4

24. Practice: Solve triangle ABC
a = 6, c = 8, mC = 100 =
sin 100 8
sin 32.4 b
b(sin 100°) = 8(sin 32.4°)
sin 100°
(8)(sin 32.4°)
b =
b ≈ 4.4

25. Practice: Solve triangle ABC
mA ≈ 47.6
mB ≈ 32.4
mC = 100

26. Homework
p. 27

27. ►A. Exercises
Specify the type of pattern.
1. a = 5, B = 48°, A = 68°
1. ASA
2. SAA
3. SSA

28. ►A. Exercises
Determine the number of triangles that satisfy the given information for ∆ABC.
1. a = 5, B = 48°, A = 68°
1. 0
2. 1
3. 2

29. ►A. Exercises
Determine the number of triangles that satisfy the given information for ∆ABC.
3. A = 12°, B = 106°, c = 10
1. 0
2. 1
3. 2

30. ►A. Exercises
Determine the number of triangles that satisfy the given information for ∆ABC.
5. a = 3, b = 7, C = 40°
1. 0
2. 1
3. 2

31. ►A. Exercises
Determine the number of triangles that satisfy the given information for ∆ABC.
7. b = 18, c = 9, B = 82°
1. 0
2. 1
3. 2

32. ►A. Exercises
Determine the number of triangles that satisfy the given information for ∆ABC.
9. a = 3, b = 7, c = 8
1. 0
2. 1
3. 2

33. ►B. Exercises
Solve the triangle. Round sides to the nearest tenth. Give angle measures to the nearest tenth of a degree.
11. A = 52°, B = 49°, c = 16
Find C.

34. ►B. Exercises
Solve the triangle. Round sides to the nearest tenth. Give angle measures to the nearest tenth of a degree.
13. B = 82°, b = 12, c = 6
Find C.

35. ►B. Exercises
Solve the triangle. Round sides to the nearest tenth. Give angle measures to the nearest tenth of a degree.
15. C = 15°, a = 14, c = 12
Find A.

36. ►B. Exercises
Solve the triangle. Round sides to the nearest tenth. Give angle measures to the nearest tenth of a degree.
17. B = 80°, a = 6, b = 2

37. ■ Cumulative Review
25. If one acute angle of a right triangle has a measure of radians, what is the radian measure of the other acute angle?

38. ■ Cumulative Review
26. Find two negative angles coterminal with 5/6.

39. ■ Cumulative Review
27. In what quadrants is the cosecant negative, and why?

40. ■ Cumulative Review
In right triangle ABC, where C = 90°, suppose a = 3 and B = 40°.
28. Find b.

41. ■ Cumulative Review
In right triangle ABC, where C = 90°, suppose a = 3 and B = 40°.
29. Find cos A.