The Law of Cosines is used while solving triangles that are not right triangles, or triangles that are; while in the form of SSS (side, side, side); or ASA (angle, side, angle)
Equation: For triangle ABC; a, b, and c are sides; as well as A, B, and C are angles. Then these equations are true:
- a2 = b2 + c2 - 2bc(cos A)
- b2 = a2 + c2 - 2ac(cos B)
- c2 = a2 + b2 - 2ab(cos C)
For the following examples, refer to the triangle at the right.
Ex 1: If b equals 10, c equals 13, and angle A equals 33 degrees, find side a.
| Step 1: | Set up the equation | a2 = 102 + 132 � 2(10)(13)(cos 33) |
|---|---|---|
| Step 2: | Simplify | a2 = 100 + 169 � (-3.452) |
| a2 = 272.452 | ||
| Step 3: | Square root both sides | a = about 16.51 |
Ex 2: If a equals 27, b equals 19, and c equals 23, find angle C.
| Step 1: | Set up the equation | 232 = 272 + 192 � 2(27)(19)(cos C)(10)(13)(cos 33) |
|---|---|---|
| Step 2: | Simplify | 529 = 729 + 361 � 1026cos C |
| 529 = 1090 � 1026cos C | ||
| Step 3: | Isolate the cosine by division | -561 = -1026cos C |
| Step 4: | Use inverse cosine | .547 = cos-1 C |
| About 56.8 degrees = C |
There are also many other useful websites on the Law of Cosines. Here are few useful ones that I have found:
