103 Trigonometry Problems

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5. Solutions to Advanced Problems 167 A1 B3 s3 t2 B2 t3 s2 A2 s1 E B1 Figure 5.10. t1 F A3 Note that segment EF, the projection of segment B 2 B 3 onto the line A 2 A 3 , has length a 1 − t 3 cos A 2 − s 2 cos A 3 , and so Because 0
168 103 Trigonometry Problems Applying the law of sines to triangle A 1 A 2 A 3 , the last inequality reduces to 3∑ sin A i (cos A i − 2 cos A i+1 A i+2 ) ≥ 0, i=1 which follows directly from the next Lemma. Lemma Proof: Let ABC be a triangle. Then the Cyclic Sum ∑ sin A(cos A − 2 cos B cos C) = 0. cyc By the double-angle formulas, it suffices to show that ∑ sin 2A = 2 ∑ sin A cos A = 4 ∑ sin A cos B cos C. cyc cyc cyc Applying the addition and subtraction formulas gives sin A cos B cos C + sin B cos C cos A = cos C(sin A cos B + sin B cos A) = cos C sin(A + B) = cos C sin C. Hence 4 ∑ sin A cos B cos C cyc = 2 ∑ cyc (sin A cos B cos C + sin B cos C cos A) = 2 ∑ cos C sin C = ∑ sin 2C, cyc cyc as desired. 42. Let ABC be a triangle. Let x,y, and z be real numbers, and let n be a positive integer. Prove the following four inequalities. (a) [D. Barrow] x 2 + y 2 + z 2 ≥ 2yzcos A + 2zx cos B + 2xy cos C. (b) [J. Wolstenholme] x 2 + y 2 + z 2 ≥ 2(−1) n+1 (yz cos nA + zx cos nB + xy cos nC).
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About the Authors Titu Andreescu re
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Titu Andreescu University of Wiscon
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vi Contents Vectors 41 The Dot Prod
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viii Preface Throughout MOSP, full
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Abbreviations and Notation Abbrevia
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103 Trigonometry Problems
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3 Advanced Problems 1. Two exercise
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3. Advanced Problems 75 9. Find the
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