Law of Cosines

Draw a right triangle on a rectangular coordinate system.  Label the origin A and the other vertex of the base B.  Label the upper vertex C(x,y).  Draw the altitude from C to AB.  Label the altitude y. Label the base of the altitude D.  Label AD with x.

(c - x)2 + y2 = a2

c2 - 2cx + x2 + y2 = a2

c2 - 2cx + b2 = a2

Since cos A = x/b, then b cos A = x, and

c2 - 2bc cos A + b2 = a2  or

a2 = b2 + c2 - 2bc cos A

Likewise,

b2 =a2 + c2 - 2ac cos B

c2 = a2 + b2 - 2ab cos C


Problems

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