Law of Cosines

h a c A B C x D b - x b To derive the formula, fine the relationship between a, b, c, and A in this triangle. a 2 = (b – x) 2 + h 2 a 2 = b 2 –2bx+x 2 + h 2 a 2 = b 2 –2bx + c 2 a 2 = b 2 –2b(c cos A) + c 2 a 2 = b 2 + c 2 – 2bc cos A

Law of Cosines 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

Remember! You can only use the law of of cosines to a triangle if you know: the measure of three sides (SSS), or the measures of two sides and the included angle (SAS)

 Find c if a = 15, b = 18, and C = 34°  Since the two sides and their included angle are given, all we need to do is plug them in this equation: c 2 = a 2 + b 2 – 2ab cos C Example:

c 2 = a 2 + b 2 – 2ab cos C c 2 = – 2(15)(18)cos 34° c 2 = c = Now you know the length of side c!