570 CHAPTER 8 Applications of Trigonometric Functions OBJECTIVES 1 Solve SAS Triangles (p. 571) 2 Solve SSS Triangles (p. 571) 3 Solve Applied Problems (p. 572) 8.3 The Law of Cosines Now Work the ‘Are You Prepared?’ problems on page 573. • Trigonometric Equations (Section 7.3, pp. 493–498) • Distance Formula (Section 1.2, pp. 13–15) PREPARING FOR THIS SECTION Before getting started, review the following: Case 3 Two sides and the included angle are known (SAS). Case 4 Three sides are known (SSS). Figure 35 C a c b (b, 0) (a cos C, a sin C) O (a) Angle C is acute y x y x C B A a c O (a cos C, a sin C) b (b, 0) (b) Angle C is obtuse B A THEOREM Law of Cosines The square of one side of a triangle equals the sum of the squares of the other two sides, minus twice their product times the cosine of their included angle. In the previous section, the Law of Sines was used to solve Case 1 (SAA or ASA) and Case 2 (SSA) of an oblique triangle. In this section, the Law of Cosines is derived and used to solve Cases 3 and 4. Observe that if the triangle is a right triangle (so that, say, C 90 = ° ), formula (1) becomes the familiar Pythagorean Theorem: c a b . 2 2 2 = + That is, the Pythagorean Theorem is a special case of the Law of Cosines. THEOREM Law of Cosines For a triangle with sides a b c , , and opposite angles A B C , , , respectively, c a b ab C 2 cos 2 2 2 = + − (1) b a c ac B 2 cos 2 2 2 = + − (2) a b c bc A 2 cos 2 2 2 = + − (3) Proof Only formula (1) is proved here. Formulas (2) and (3) can be proved using the same argument. Begin by strategically placing a triangle on a rectangular coordinate system so that the vertex of angle C is at the origin and side b lies along the positive x -axis. Regardless of whether C is acute, as in Figure 35(a), or obtuse, as in Figure 35(b), the vertex of angle B has coordinates a C a C cos, sin . ( ) The vertex of angle A has coordinates b, 0 . ( ) Use the distance formula to compute c .2 c b a C a C b ab C a C a C b ab C a C C a b ab C cos 0 sin 2 cos cos sin 2 cos cos sin 2 cos 2 2 2 2 2 2 2 2 2 2 2 2 2 2 ( ) ( ) ( ) = − + − = − + + = − + + = + − ■ Each of formulas (1), (2), and (3) may be stated in words as follows:
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