You probably learned about the Law of Sines recently. The cosine function also has a rule that applies to problems related to cosine--the Law of Cosines. The Law of Cosines, like the Law of Sines, is used to solve problems involving any type of triangle, not just right triangles.
Let's try applying the Law of Cosines to the same oblique triangle you may have used to learn the Law of Sines.
The Law of Cosines can be expressed using three equations. The equation you use depends on what information you are given in the problem. (Before attempting to solve these equations, make sure to have your calculators set on degree mode.) If A, B, and C are the measures of the angles of triangle ABC and a, b, and c are the lengths of the sides opposite the angles, then the following three formulas define the Law of Cosines:
| Formula 1 | a2 = b2 + c2 - 2bc cos(A) |
| Formula 2 | b2 = a2 + c2 - 2ac cos(B) |
| Formula 3 | c2 = a2 + b2 - 2ab cos(C) |
See if you can write an equation for a specific problem. Suppose you are shown the triangle below and are asked to find c.
Could you use one of the Law of Cosine formulas to write an equation to solve for c? Think about the answer to this question--then click the Show me Button below to check your work.
Since we are given a and b and are asked to solve for c, we can use the third formula to write an equation for c. c2 = a2 + b2 - 2ab cos(C) c2 = 222 + 202 - 2(22)(20)cos(95°)
