Essential Question: What is the law of cosines, and when do we use it?

10-1: The Law of Cosines In any triangle ABC, with side lengths a, b, c – which are opposite their respective angle, the Law of Cosines states: a 2 = b 2 + c 2 – 2bc cos A b 2 = a 2 + c 2 – 2ac cos B c 2 = a 2 + b 2 – 2ab cos C Basically: It’s just like the Pythagorean Theorem, then subtracting two times those two sides times the cosine of the angle. Proof on board

10-1: The Law of Cosines The law of Cosines can be used to solve triangles in the following cases: Given two sides and an angle between them (SAS) Given three sides (SSS) The law of cosines helps us solve the situations where the law of sines cannot.

10-1: The Law of Cosines Example 1: Solve a Triangle with SAS Information. Solve triangle ABC below C A B 110 ° 1016 c c 2 = a 2 + b 2 – 2ab cos C c 2 = – 2(16)(10) cos 110 c 2 = – 320( ) c 2 = c 2 = c  (you can give 21.6 as an answer, but use 4 digits to continue solving) 1) Use law of cosines to find c 2) Use law of sines to find A (or B) 3) Find the last angle B = 180 – 110 – 44.2 = 25.8

10-1: The Law of Cosines Example 2: Solve a Triangle with SSS Information Solve a triangle where a = 20, b = 15 and c = 8.3 1) Use the law of cosines to find any angle 2) Use the law of sines to find another angle 3) Use common sense to find the third angle c 2 = a 2 + b 2 – 2ab cos C = -600 cos C = – 2(20)(15) cos C = cos C = – 600 cos Ccos -1 (.92685) = C = 625 – 600 cos C22.1 ° = C A = 180 – 22.1 – 42.7 A = 115.2

10-1: The Law of Cosines Example 3: The distance between two vehicles Two trains leave a station on different tracks. The tracks make an angle of 125 ° with the station as the vertex. The first train travels at an average speed of 100 km/h, and the second train travels at an average speed of 65 km/h. How far apart are the trains after 2 hours? These questions are helped if you draw a diagram. Station 1st 130 2nd ° x

10-1: The Law of Cosines Example 3: The distance between two vehicles Use the law of cosines x 2 = – 2(130)(200) cos 125 x 2 = – cos 125 x 2 = x = Station 1st 130 2nd ° x

10-1: The Law of Cosines Assignment Page 622 Problems 1 – 25, odds Show work