103 Trigonometry Problems

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1. Trigonometric Fundamentals 17 10 8 6 4 2 -400 -300 -200 -100 100 200 300 -2 -4 -6 -8 -10 Figure 1.17. A function f(x)is concave up (down) on an interval [a,b] if the graph of f(x) lies under (over) the line connecting (a 1 ,f(a 1 )) and (b 1 ,f(b 1 )) for all a ≤ a 1
18 103 Trigonometry Problems such as Introductory Problems 19(b), 20(b), 23(a) and (d), 27(b), and 28(b) and (c). Because the main goal of this book is to introduce techniques in trigonometric computation rather than in functional analysis, we will present solutions without using Jensen’s inequality. On the other hand, we certainly do not want the reader to miss this important method. In the second solution of Introductory Problem 51 and the solution of Advanced Problem 39, we illustrate this technique. The Extended Law of Sines Let ABC be a triangle. It is not difficult to show that [ABC] = see the next section.) By symmetry, we have ab sin C 2 . (For a proof, [ABC] = ab sin C 2 = bc sin A 2 = ca sin B . 2 Dividing all sides of the last equation by abc 2 gives the law of sines: sin A a = sin B b = sin C c or a sin A = b sin B = c sin C . A D O B M C a Figure 1.18. The common ratio sin A has a significant geometric meaning. Let ω be the circumcircle of triangle ABC, and let O and R be the center and radius of ω, respectively. (See Figure 1.18.) Then ̸ BOC = 2̸ CAB. Let M be the midpoint of segment BC. Because triangle OBC is isosceles with |OB| =|OC| =R, it follows that OM ⊥ BC and ̸ BOM = ̸ COM = ̸ CAB. In the right triangle BMO, a |BM|=|OB| sin A; that is, sin A = 2|BM| sin A = 2|OB|=2R. Hence, we obtain the extended law of sines: In a triangle ABC with circumradius equal to R, a sin A = b sin B = c sin A = 2R.
Page 3 and 4: About the Authors Titu Andreescu re
Page 5 and 6: Titu Andreescu University of Wiscon
Page 7 and 8: vi Contents Vectors 41 The Dot Prod
Page 9 and 10: viii Preface Throughout MOSP, full
Page 12 and 13: Abbreviations and Notation Abbrevia
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3 Advanced Problems 1. Two exercise
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5 Solutions to Advanced Problems 1.
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Further Reading 1. Andreescu, T.; F
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