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Unformatted text preview: If a triangle has sides of lengths a , b , and c opposite the angles A , B , and C , respectively, then a 2 = b 2 + c 2 2 bc cos A , (2.9) b 2 = c 2 + a 2 2 ca cos B , (2.10) c 2 = a 2 + b 2 2 ab cos C . (2.11) To prove the Law of Cosines, let ABC be an oblique triangle. Then ABC can be acute, as in Figure 2.2.1(a), or it can be obtuse, as in Figure 2.2.1(b). In each case, draw the altitude from the vertex at C to the side AB . In Figure 2.2.1(a) the altitude divides AB into two line segments with lengths x and c x , while in Figure 2.2.1(b) the altitude extends the side AB by a distance x . Let h be the height of the altitude. h b a c A B C x c x (a) Acute triangle h b a c A B C 180 B x (b) Obtuse triangle Figure 2.2.1 Proof of the Law of Cosines for an oblique triangle ABC...
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 Fall '11
 Dr.Cheun
 Calculus, Angles, Law Of Cosines

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