Law of Cosines

Law of Cosines is your last result for solving a triangle. You use it when Law of Sines doesn't work.

Formula for Law of Cosines: Opposite leg^2=adjacent leg^2+other adjacent leg^2=-2(adjacent leg)(other adjacent leg)cos(angle between the two adjacent legs

Ex.1 In triangle ABC, angle B=130, c=5, and a=8. Find angle A, angle C, and leg b.

• First, you need to draw a picture of the triangle:
• Second, you want to find leg b since it is the opposite of angle B.
• You plug the numbers into the formula: b^2=5^2+8^2-2(5)(8)cos130. Now you take the square root of both sides. When you plug that into your calculator, you get 11.850. So b=11.850.
• Now you can find angle A by using the formula. 8^2=5^2+11.850^2-2(5)(11.850)cosA. Then you subtract 5^2+11.850^2 to the other side, giving you 8^2-5^2-11.850^2=-2(5)(11.850)cosA. Now take the inverse of cosA. A=cos-1((8^2-5^2-11.850^2)/(-2(5)(11.850))=31.142
• To find angle C, just subtract the angles from 180. 180-130-31.142=18.858.

-Amber :)