1. 9.2 - 9.3 The Law of Sines and The Law of Cosines
MAT 204 F08
The Law of Sines and The Law of Cosines
In these sections, we will study the following topics:
Solving oblique triangles (4 cases)
Solving a triangle using Law of Sines
Solving a triangle using Law of Cosines

3. An oblique triangle has either:
MAT 204 F08
In this chapter, we will work with oblique triangles  triangles that do NOT contain a right angle.
An oblique triangle has either:
three acute angles
two acute angles and one obtuse angle or

4. Every triangle has 3 sides and 3 angles.
MAT 204 F08
Every triangle has 3 sides and 3 angles.
To solve a triangle means to find the lengths of its sides and the measures of its angles.
To do this, we need to know at least three of these parts, and at least one of them must be a side.

5. Here are the four possible combinations of parts:
MAT 204 F08
Here are the four possible combinations of parts:
Two angles and one side (ASA or SAA)
Two sides and the angle opposite one of them (SSA)
Two sides and the included angle (SAS)
Three sides (SSS)

6. Two angles and one side (ASA or SAA)
MAT 204 F08
Case 1:
Two angles and one side (ASA or SAA)

7. Two sides and the angle opposite one of them (SSA)
MAT 204 F08
Case 2:
Two sides and the angle opposite one of them (SSA)

8. Two sides and the included angle (SAS)
MAT 204 F08
Case 3:
Two sides and the included angle (SAS)

9. MAT 204 F08
Case 4:
Three sides (SSS)

10. The Law of Sines Three equations for the price of one! MAT 204 F08 C b

11. Solving Case 1: ASA or SAA

12. Solving Case 1: ASA or SAA

13. Example using Law of Sines
MAT 204 F08
Example using Law of Sines
A ship takes a sighting on two buoys. At a certain instant, the bearing of buoy A is N 44.23° W, and that of buoy B is N 62.17° E. The distance between the buoys is 3.60 km, and the bearing of B from A is N 87.87° E. Find the distance of the ship from each buoy.

14. Continued from above

15. MAT 204 F08
Solving Case 2: SSA
In this case, we are given two sides and an angle opposite.
This is called the AMBIGUOUS CASE. That is because it may yield no solution, one solution, or two solutions, depending on the given information.

16. SSA --- The Ambiguous Case

17. No Triangle If , then side is not sufficiently long enough to form a triangle.

18. One Right Triangle If , then side is just long enough to form a right triangle.

19. Two Triangles If and , two distinct triangles can be formed from the given information.

20. One Triangle If , only one triangle can be formed.

22. Continued from above

24. Continued from above

26. Making fairly accurate sketches can help you to determine the number of solutions.

27. Example: Solve ABC where A = 27.6, a =112, and c = 165.

28. Continued from above

29. To deal with Case 3 (SAS) and Case 4 (SSS), we do not have enough information to use the Law of Sines.
So, it is time to call in the Law of Cosines.

30. B A C c b a
The Law of Cosines

31. Using Law of cosines to Find the Measure of an Angle
*To find the angle using Law of Cosines, you will need to solve the Law of Cosines formula for CosA, CosB, or CosC.
For example, if you want to find the measure of angle C, you would solve the following equation for CosC:
To solve for C, you would take the cos-1 of both sides.

32. Guidelines for Solving Case 3: SAS
When given two sides and the included angle, follow these steps:
Use the Law of Cosines to find the third side.
Use the Law of Cosines to find one of the remaining angles. {You could use the Law of Sines here, but you must be careful due to the ambiguous situation. To keep out of trouble, find the SMALLER of the two remaining angles (It is the one opposite the shorter side.)}
Find the third angle by subtracting the two known angles from 180.

33. Solving Case 3: SAS
Example: Solve ABC where a = 184, b = 125, and C = 27.2.

34. Continued from above

35. Solving Case 3: SAS
Example: Solve ABC where b = 16.4, c = 10.6, and A = 128.5.

36. Continued from above

37. Guidelines for Solving Case 4: SSS
When given three sides, follow these steps:
Use the Law of Cosines to find the LARGEST ANGLE *(opposite the largest side).
Use the Law of Sines to find either of the two remaining angles.
Find the third angle by subtracting the two known angles from 180.

39. Solving Case 4: SSS
Example: Solve ABC where a = 128, b = 146, and c = 222.

40. Continued from above

41. (Let bold red represent the given info)
When to use what……
(Let bold red represent the given info)
SAS
AAS
ASA
Be careful!! May have 0, 1, or 2 solutions.
SSS
SSA
Use Law of Sines
Use Law of Cosines

44. Continued from above

45. MAT 204 F08
End of Sections 9.2 – 9.3