103 Trigonometry Problems

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3 Advanced Problems 1. Two exercises on sin k ◦ sin(k + 1) ◦ : (a) Find the smallest positive integer n such that 1 sin 45 ◦ sin 46 ◦ + 1 sin 47 ◦ sin 48 ◦ +···+ 1 sin 133 ◦ sin 134 ◦ = 1 sin n ◦ . (b) Prove that 1 sin 1 ◦ sin 2 ◦ + 1 sin 2 ◦ sin 3 ◦ +···+ 1 cos 1◦ sin 89 ◦ = sin 90◦ sin 2 1 ◦ . 2. Let ABC be a triangle, and let x be a nonnegative real number. Prove that a x cos A + b x cos B + c x cos C ≤ 1 2 (ax + b x + c x ). 3. Let x,y,z be positive real numbers.
74 103 Trigonometry Problems (a) Prove that if x + y + z = xyz; x √ 1 + x 2 + y √ 1 + y 2 + z √ ≤ 3√ 3 1 + z 2 2 (b) Prove that x 1 − x 2 + y 1 − y 2 + z 1 − z 2 ≥ 3√ 3 2 if 0
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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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2 103 Trigonometry Problems First w
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4 103 Trigonometry Problems triangl
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6 103 Trigonometry Problems By sett
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8 103 Trigonometry Problems Q A B P
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10 103 Trigonometry Problems for co
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12 103 Trigonometry Problems Furthe
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14 103 Trigonometry Problems (b) By
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16 103 Trigonometry Problems 15 10
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18 103 Trigonometry Problems such a
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20 103 Trigonometry Problems A A B
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5 Solutions to Advanced Problems 1.
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5. Solutions to Advanced Problems 1
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or 5. Solutions to Advanced Problem
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It follows that ∑ 4 sin 3 A cos(B
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200 103 Trigonometry Problems Binom
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202 103 Trigonometry Problems From
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204 103 Trigonometry Problems Lagra
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206 103 Trigonometry Problems The i
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208 103 Trigonometry Problems Sum-t
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Further Reading 1. Andreescu, T.; F
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Further Reading 213 23. Fomin, D.;
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