Section 8 – The Law of Cosines Chapter 5 Section 8 – The Law of Cosines

The Law of Cosines Just like the Law of Sines, the Law of Cosines incorporates the sides and angles of any triangle. Law of Cosines a2 = b2 + c2 – 2bc(cos A) b2 = a2 + c2 – 2ac(cos B) c2 = a2 + b2 – 2ab(cos C)

Derivation C2 = (a-x)2 + h2 C2 = a2 - 2ax+x2+h2 C2 = a2 - 2ax+b2 C2 = a2 - 2a(bcosC)+b2 C2 = a2+b2 - 2abcosC)

Examples Solve each triangle below. EX 1: A = 120 degrees, b = 9, and c = 5 EX 2: B = 65 degrees, a = 8, and c = 3

Heron’s Formula Heron’s Formula is a formula used to figure out the area of any given triangle when you have the perimeter of the triangle where s is the semiperimeter

Example EX 3: Use Heron’s Formula to figure out the area of a triangle with sides 6, 8, and 13. EX 4: Use Heron’s Formula to figure out the area of a triangle with sides 5, 8, and 19.

Assignment Chapter 5, Section 8 pgs 331-332 #12-24E,30