Pre-Calc Exam Notes 48

Pre-Calc Exam Notes 48 - in the following example Example...

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48 Chapter 2 General Triangles §2.2 Example 2.8 Case 2: Two sides and one opposite angle. Solve the triangle ABC given a = 18, A = 25 , and b = 30. Solution: In Example 2.2 from Section 2.1 we used the Law of Sines to show that there are two sets of solutions for this triangle: B = 44.8 , C = 110.2 , c = 40 and B = 135.2 , C = 19.8 , c = 14.4. To solve this using the Law of Cosines, Frst Fnd c by using the formula for a 2 : a 2 = b 2 + c 2 2 bc cos A 18 2 = 30 2 + c 2 2(30) c cos 25 c 2 54.38 c + 576 = 0 , which is a quadratic equation in c , so we know that it can have either zero, one, or two real roots (corresponding to the number of solutions in Case 2). By the quadratic formula, we have c = 54.38 ± r (54.38) 2 4(1)(576) 2(1) = 40 or 14.4 . Note that these are the same values for c that we found before. ±or c = 40 we get cos B = c 2 + a 2 b 2 2 ca = 40 2 + 18 2 30 2 2(40)(18) = 0.7111 B = 44.7 C = 110.3 , which is close to what we found before (the small difference being due to different rounding). The other solution set can be obtained similarly. Like the Law of Sines, the Law of Cosines can be used to prove some geometric facts, as
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Unformatted text preview: in the following example. Example 2.9 c d a b a b C D Figure 2.2.2 Use the Law of Cosines to prove that the sum of the squares of the diagonals of any parallelogram equals the sum of the squares of the sides. Solution: Let a and b be the lengths of the sides, and let the diago-nals opposite the angles C and D have lengths c and d , respectively, as in igure 2.2.2. Then we need to show that c 2 + d 2 = a 2 + b 2 + a 2 + b 2 = 2( a 2 + b 2 ) . By the Law of Cosines, we know that c 2 = a 2 + b 2 2 ab cos C , and d 2 = a 2 + b 2 2 ab cos D . By properties of parallelograms, we know that D = 180 C , so d 2 = a 2 + b 2 2 ab cos (180 C ) = a 2 + b 2 + 2 ab cos C , since cos (180 C ) = cos C . Thus, c 2 + d 2 = a 2 + b 2 2 ab cos C + a 2 + b 2 + 2 ab cos C = 2( a 2 + b 2 ) . QED...
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