Law of Cosines Section 5-6

Law of Cosines Law of Cosines: Let ∆ABC be any triangle with sides a, b, c opposite angles A, B, C. Then:

Notes about Law of Cosines The Law of Cosines is used for triangles which fit the SSS or SAS situations. Why must we use law of cosines here instead of the law of sines? Once another angle is found it’s usually easier to use the law of sines to finish solving the triangle. However, realize that using the Law of Cosines is better to find missing angles, because the arccosine function will distinguish obtuse angles from acute angles. (Think back to the unit circle here!)

You try! Ex 1: Solve ∆ABC given that a=11, b=5, and C=20°.

You try! Ex 2: Solve ∆DEF given that d=9, e=7, and f=5.

Heron’s Formula Let a, b, and c be the sides of ∆ABC, and let s denote the semiperimeter (a+b+c)/2. Then the area of ∆ABC is given by:

Heron’s Formula Ex: Ex 4: Find the area of a triangle with sides 13, 15, and 18.

Triangle Area Area of a Triangle: ∆Area =

Find the area Ex 3: Find the area of a regular octagon inscribed inside a circle of radius 9 inches.