Law of Cosines 7.2 JMerrill, 2009

Law of Cosines A Usually used when you have SSS or SAS In ANY triangle ABC: a2 = b2 + c2 – 2bc cosA b2 = a2 + c2 – 2ac cosB c2 = a2 + b2 – 2ab cosC BC cb a You don’t have to learn but one of these—the letters don’t matter. You start and end with the same letter: (side) 2 =(1 side) 2 +(other side) 2 – 2(1 side)(other side)Cos   

Example In the triangle below, m  A = 52 o, b = 4, & c = 7. Find a In the triangle below, m  A = 52 o, b = 4, & c = 7. Find a C AB 52 o 4 7

Example A triangular field is fenced on two sides. The third side of the field is formed by a river. If the fences measure 150 meters and 98 meters and the side along the river is 172 meters, what is the measure of the angle between the fences? A triangular field is fenced on two sides. The third side of the field is formed by a river. If the fences measure 150 meters and 98 meters and the side along the river is 172 meters, what is the measure of the angle between the fences? C 150 m A 172 m B 98 m

Example We are looking for angle C: We are looking for angle C: C 150 m A 172 m B 98 m

You Do In 1996, Allen Johnson ran the 110-meter hurdles in seconds. To help him improve his performance, Johnson used trigonometry to find the angle he made as he jumped over the top of the hurdle. What was this angle if the distance from his takeoff point to the top of the hurdle was 12.1 feet, the distance from the top of the hurdle to the landing point was 8.8 feet, and the distance from takeoff to landing was 19.7 feet.

His takeoff point to the top of the hurdle was 12.1 feet, the distance from the top of the hurdle to the landing point was 8.8 feet, and the distance from takeoff to landing was 19.7 feet. His takeoff point to the top of the hurdle was 12.1 feet, the distance from the top of the hurdle to the landing point was 8.8 feet, and the distance from takeoff to landing was 19.7 feet. top of hurdle takeoff landing A