788 CHAPTER 8 Applications of Trigonometry As an example of this property, it would be impossible to construct a triangle with sides of lengths 3, 4, and 10. See Figure 12. a = 3 c = 10 b = 4 No triangle is formed. Figure 12 y A c 908 (0, 0) B D x a b C (a, 0) (c cos B, c sin B) x y Figure 13 Derivation of the Law of Cosines To derive the law of cosines, let ABC be any oblique triangle. Choose a coordinate system so that vertex B is at the origin and side BC is along the positive x-axis. See Figure 13. Let 1x, y2 be the coordinates of vertex A of the triangle. Then the following hold true for angle B, whether obtuse or acute. sin B = y c and cos B = x c Definition of sine and cosine y = c sin B and x = c cos B Here x is negative when B is obtuse. Thus, the coordinates of point A become 1c cos B, c sin B2. Point C in Figure 13 has coordinates 1a, 02, AC has length b, and point A has coordinates 1c cos B, c sin B2. We can use the distance formula to write an equation. d = 21x2 - x122 + 1y 2 - y122 Distance formula b = 21c cos B - a22 + 1c sin B - 022 Substitute. b2 = 1c cos B - a22 + 1c sin B22 Square each side. b2 = 1c2 cos2 B - 2ac cos B + a22 + c2 sin2 B b2 = a2 + c21cos2 B + sin2 B2 - 2ac cos B Properties of real numbers b2 = a2 + c2112 - 2ac cos B Fundamental identity b2 = a2 + c2 - 2ac cos B Law of cosines This result is one of three possible forms of the law of cosines. In our work, we could just as easily have placed vertex A or C at the origin. This would have given the same result, but with the variables rearranged. Multiply; 1x - y22 = x2 - 2xy + y2 Law of Cosines In any triangle ABC, with sides a, b, and c, the following hold true. a2 =b2 +c2 −2bc cos A b2 =a2 +c2 −2ac cos B c2 =a2 +b2 −2ab cos C That is, according to the law of cosines, the square of a side of a triangle is equal to the sum of the squares of the other two sides, minus twice the product of those two sides and the cosine of the angle included between them.
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