Pre-Calc Exam Notes 45

Pre-Calc Exam Notes 45 - B acute and obtuse. By similar...

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The Law of Cosines Section 2.2 45 For each triangle in Figure 2.2.1, we see by the Pythagorean Theorem that h 2 = a 2 x 2 (2.12) and likewise for the acute triangle in Figure 2.2.1(a) we see that b 2 = h 2 + ( c x ) 2 . (2.13) Thus, substituting the expression for h 2 in equation (2.12) into equation (2.13) gives b 2 = a 2 x 2 + ( c x ) 2 = a 2 x 2 + c 2 2 cx + x 2 = a 2 + c 2 2 cx . But we see from Figure 2.2.1(a) that x = a cos B , so b 2 = a 2 + c 2 2 ca cos B . (2.14) And for the obtuse triangle in Figure 2.2.1(b) we see that b 2 = h 2 + ( c + x ) 2 . (2.15) Thus, substituting the expression for h 2 in equation (2.12) into equation (2.15) gives b 2 = a 2 x 2 + ( c + x ) 2 = a 2 x 2 + c 2 + 2 cx + x 2 = a 2 + c 2 + 2 cx . But we see from Figure 2.2.1(a) that x = a cos (180 B ), and we know from Section 1.5 that cos (180 B ) =− cos B . Thus, x =− a cos B and so b 2 = a 2 + c 2 2 ca cos B . (2.16) So for both acute and obtuse triangles we have proved formula (2.10) in the Law of Cosines. Notice that the proof was for
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Unformatted text preview: B acute and obtuse. By similar arguments for A and C we get the other two formulas. QED Note that we did not prove the Law of Cosines for right triangles, since it turns out (see Exercise 15) that all three formulas reduce to the Pythagorean Theorem for that case. The Law of Cosines can be viewed as a generalization of the Pythagorean Theorem. Also, notice that it sufces to remember just one of the three formulas (2.9)-(2.11), since the other two can be obtained by cycling through the letters a , b , and c . That is, replace a by b , replace b by c , and replace c by a (likewise for the capital letters). One cycle will give you the second formula, and another cycle will give you the third....
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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.

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