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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This note was uploaded on 01/21/2012 for the course MAC 1130 taught by Professor Dr.cheun during the Fall '11 term at FSU.
 Fall '11
 Dr.Cheun
 Calculus, Angles, Law Of Cosines

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