1. 5.6 Law of Cosines

2. I. Law of Cosines In any triangle with opposite sides a, b, and c: The Law of Cosines is used to solve any triangle where you are given, in order, SAS and SSS. AB C b c a

3. Derivation: 3 situations C (x, y) B (c, 0)Ac a b C (x, y) B (c, 0)A c a b C (x, y) B (c, 0) Ac a b

5. II. Examples Solve the following triangles: A.)

6. Note: General rule is to use the Law of Cosines for finding the first, (or first and second) missing angle of a triangle because the Law of Cosines will return an obtuse angle. For example, if we use the Law of Sines to solve the last example for angle C, we get This value of angle B contradicts the Law of Sines for b=10.

7. B.)

9. III. Area of a Triangle A.) In any triangle with sides a, b, and c opposite angles A, B, and C:

10. B.) Ex.- Find the area of a regular decagon with sides 5 inches in length. A regular decagon can be divided into 10 congruent isosceles triangles each with a vertex angle of 36º and two base angles of 72º. 36 5 72 xx

11. IV. Heron’s Formula In any triangle with sides a, b, and c and semi- perimeter s: