8-6 The Law of Cosines Objective To apply the Law of Cosines Essential Understanding If you know the measures of two side lengths and the measure of the included angle (SAS), or all three side lengths (SSS), then you can find all the other measures of the triangle.

A farmer needs to put a pipe through a hill for irrigation. The farmer attaches a 14.5 meter rope and an 11.2 meter rope at each entry point of the pipe and makes a triangle. The ends meet at a 58 0 angle. What is the length of the pipe the farmer needs? 14.5 m 11.2 m 58 0 Can you use Law of Sines? No, you don’t know the angles that are opposite the sides Pythagorean Theorem? Not a right triangle x

Law of Cosines For any triangle ABC, the Law of Cosines relates the cosine of each angle to the side lengths of the triangle. c 2 = a 2 + b 2 − 2abcosC b 2 = a 2 + c 2 − 2accosB a 2 = b 2 + c 2 − 2bccosA b c a B A C

Using the Law of Cosines (SAS) Find b to the nearest tenth. b 2 = a 2 + c 2 − 2accosB Law of Cosines b 2 = − 2(22)(10)cos44 Substitute b b b CB A

b 2 = a 2 + c 2 − 2accosB Law of Cosines = − 2(6.7)(7.1)cosV Substitute Find V to the nearest tenth. Using the Law of Cosines (SSS) Solve for angle V = − 95.14cosV Substitute V U T

Examples 14.5 m 11.2 m 58 0 x Law of Cosines c 2 = a 2 + b 2 − 2abcosC x 2 = − 2(11.2)(14.5)cos58 0 x 2 = x = 12.8 m xoxo c 2 = a 2 + b 2 − 2abcosC 4 2 = − 2(5)(7)cosx o 16 = − 70cosx o -58 = − 70cosx o.829 = cosx o 34 o = x p.529: 1-4, 7-15 odd