www.EngineeringBooksPDF.com www.EngineeringBooksPDF.com About the Authors Titu Andreescu received his BA, MS, and PhD from the West University of Timisoara, Romania The topic of his doctoral dissertation was "Research on Diophantine Analysis and Applications" Titu served as director of the MAA American Mathematics Competitions (1998-2003), coach of the USA International Mathematical Olympiad Team (IMO) for 10 years (19932002), director of the Mathematical Olympiad Summer Program (19952002) and leader of the USA IMO Team (1995-2002) In 2002 Titu was elected member of the IMO Advisory Board, the governing body of the international competition Titu received the Edyth May Sliffe Award for Distinguished High School Mathematics Teaching from the MAA in 1994 and a "Certificate of Appreciation" from the president of the MAA in 1995 for his outstanding service as coach of the Mathematical Olympiad Summer Program in preparing the US team for its perfect performance in Hong Kong at the 1994 International Mathematical Olympiad Zuming Feng graduated with a PhD from Johns Hopkins University with emphasis on Algebraic Number Theory and Elliptic Curves He teaches at Phillips Exeter Academy He also served as a coach of the USA IMO team (1997-2003), the deputy leader of the USA IMO Team (2000-2002), and an assistant director of the USA Mathematical Olympiad Summer Program (1999-2002) He is a member of the USA Mathematical Olympiad Committee since 1999, and has been the leader of the USA IMO team and the academic director of the USA Mathematical Olympiad Summer Program since 2003 He received the Edyth May Sliffe Award for Distinguished High School Mathematics Teaching from the MAA in 1996 and 2002 www.EngineeringBooksPDF.com Titu Andreescu Zuming Feng 103 Trigonometry Problems From the Training of the USA IMO Team Birkhäuser Boston • Basel • Berlin www.EngineeringBooksPDF.com Titu Andreescu University of Wisconsin Department of Mathematical and Computer Sciences Whitewater, WI 53190 U.S.A Zuming Feng Phillips Exeter Academy Department of Mathematics Exeter, NH 03833 U.S.A AMS Subject Classifications: Primary: 97U40, 00A05, 00A07, 51-XX; Secondary: 11L03, 26D05, 33B10, 42A05 Library of Congress Cataloging-in-Publication Data Andreescu, Titu, 1956103 trigonometry problems : from the training of the USA IMO team / Titu Andreescu, Zuming Feng p cm Includes bibliographical references ISBN 0-8176-4334-6 (acid-free paper) Trigonometry–Problems, exercises, etc I Title: One hundred and three trigonometry problems II Feng, Zuming III Title QA537.A63 2004 516.24–dc22 ISBN 0-8176-4334-6 2004045073 Printed on acid-free paper Birkhäuser ©2005 Birkhäuser Boston ® All rights reserved This work may not be translated or copied in whole or in part without the written permission of the publisher (Birkhäuser Boston, c/o Springer Science+Business Media Inc., Rights and Permissions, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden The use in this publication of trade names, trademarks, service marks and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights Printed in the United States of America SPIN 10982723 www.birkhauser.com www.EngineeringBooksPDF.com Contents Preface vii Acknowledgments ix Abbreviations and Notation xi Trigonometric Fundamentals Definitions of Trigonometric Functions in Terms of Right Triangles Think Within the Box You’ve Got the Right Angle Think Along the Unit Circle Graphs of Trigonometric Functions The Extended Law of Sines Area and Ptolemy’s Theorem Existence, Uniqueness, and Trigonometric Substitutions Ceva’s Theorem Think Outside the Box Menelaus’s Theorem The Law of Cosines Stewart’s Theorem Heron’s Formula and Brahmagupta’s Formula Brocard Points 1 10 14 18 19 23 28 33 33 34 35 37 39 www.EngineeringBooksPDF.com vi Contents Vectors The Dot Product and the Vector Form of the Law of Cosines The Cauchy–Schwarz Inequality Radians and an Important Limit Constructing Sinusoidal Curves with a Straightedge Three Dimensional Coordinate Systems Traveling on Earth Where Are You? De Moivre’s Formula 41 46 47 47 50 51 55 57 58 Introductory Problems 63 Advanced Problems 73 Solutions to Introductory Problems 83 Solutions to Advanced Problems 125 Glossary 199 Further Reading 211 www.EngineeringBooksPDF.com Preface This book contains 103 highly selected problems used in the training and testing of the U.S International Mathematical Olympiad (IMO) team It is not a collection of very difficult, impenetrable questions Instead, the book gradually builds students’ trigonometric skills and techniques The first chapter provides a comprehensive introduction to trigonometric functions, their relations and functional properties, and their applications in the Euclidean plane and solid geometry This chapter can serve as a textbook for a course in trigonometry This work aims to broaden students’ view of mathematics and better prepare them for possible participation in various mathematical competitions It provides in-depth enrichment in important areas of trigonometry by reorganizing and enhancing problem-solving tactics and strategies The book further stimulates interest for the future study of mathematics In the United States of America, the selection process leading to participation in the International Mathematical Olympiad (IMO) consists of a series of national contests called the American Mathematics Contest 10 (AMC 10), the American Mathematics Contest 12 (AMC 12), the American Invitational Mathematics Examination (AIME), and the United States of America Mathematical Olympiad (USAMO) Participation in the AIME and the USAMO is by invitation only, based on performance in the preceding exams of the sequence The Mathematical Olympiad Summer Program (MOSP) is a four-week intensive training program for approximately 50 very promising students who have risen to the top in the American Mathematics Competitions The six students representing the United States of America in the IMO are selected on the basis of their USAMO scores and further testing that takes place during MOSP www.EngineeringBooksPDF.com viii Preface Throughout MOSP, full days of classes and extensive problem sets give students thorough preparation in several important areas of mathematics These topics include combinatorial arguments and identities, generating functions, graph theory, recursive relations, sums and products, probability, number theory, polynomials, functional equations, complex numbers in geometry, algorithmic proofs, combinatorial and advanced geometry, functional equations, and classical inequalities Olympiad-style exams consist of several challenging essay problems Correct solutions often require deep analysis and careful argument Olympiad questions can seem impenetrable to the novice, yet most can be solved with elementary high school mathematics techniques, cleverly applied Here is some advice for students who attempt the problems that follow • Take your time! Very few contestants can solve all the given problems • Try to make connections between problems An important theme of this work is that all important techniques and ideas featured in the book appear more than once! • Olympiad problems don’t “crack” immediately Be patient Try different approaches Experiment with simple cases In some cases, working backwards from the desired result is helpful • Even if you can solve a problem, read the solutions They may contain some ideas that did not occur in your solutions, and they may discuss strategic and tactical approaches that can be used elsewhere The solutions are also models of elegant presentation that you should emulate, but they often obscure the tortuous process of investigation, false starts, inspiration, and attention to detail that led to them When you read the solutions, try to reconstruct the thinking that went into them Ask yourself, “What were the key ideas? How can I apply these ideas further?” • Go back to the original problem later, and see whether you can solve it in a different way Many of the problems have multiple solutions, but not all are outlined here • Meaningful problem-solving takes practice Don’t get discouraged if you have trouble at first For additional practice, use the books on the reading list www.EngineeringBooksPDF.com Acknowledgments Thanks to Dorin Andrica and Avanti Athreya, who helped proofread the original manuscript Dorin provided acute mathematical ideas that improved the flavor of this book, while Avanti made important contributions to the final structure of the book Thanks to David Kramer, who copyedited the second draft He made a number of corrections and improvements Thanks to Po-Ling Loh, Yingyu Gao, and Kenne Hon, who helped proofread the later versions of the manuscript Many of the ideas of the first chapter are inspired by the Math and Math teaching materials from the Phillips Exeter Academy We give our deepest appreciation to the authors of the materials, especially to Richard Parris and Szczesny “Jerzy” Kaminski Many problems are either inspired by or adapted from mathematical contests in different countries and from the following journals: ã High-School Mathematics, China ã Revista Matematica Timiásoara, Romania We did our best to cite all the original sources of the problems in the solution section We express our deepest appreciation to the original proposers of the problems www.EngineeringBooksPDF.com 200 103 Trigonometry Problems Binomial Coefficient n n! = , k k!(n − k)! the coefficient of x k in the expansion of (x + 1)n Cauchy–Schwarz Inequality For any real numbers a1 , a2 , , an , and b1 , b2 , , bn (a12 + a22 + · · · + an2 )(b12 + b22 + · · · + bn2 ) ≥ (a1 b1 + a2 b2 + · · · + an bn )2 , with equality if and only if and bi are proportional, i = 1, 2, , n Ceva’s Theorem and Its Trigonometric Form Let AD, BE, CF be three cevians of triangle ABC The following are equivalent: (i) AD, BE, CF are concurrent; (ii) |AF | |BD| |CE| · · = 1; |F B| |DC| |EA| (iii) sin ABE sin BCF sin CAD · · = sin EBC sin F CA sin DAB Cevian A cevian of a triangle is any segment joining a vertex to a point on the opposite side Chebyshev’s Inequality Let x1 , x2 , , xn and y1 , y2 , , yn be two sequences of real numbers such that x1 ≤ x2 ≤ · · · ≤ xn and y1 ≤ y2 ≤ · · · ≤ yn Then (x1 + x2 + · · · + xn )(y1 + y2 + · · · + yn ) ≤ x1 y1 + x2 y2 + · · · + xn yn n Let x1 , x2 , , xn and y1 , y2 , , yn be two sequences of real numbers such that x1 ≥ x2 ≥ · · · ≥ xn and y1 ≥ y2 ≥ · · · ≥ yn Then (x1 + x2 + · · · + xn )(y1 + y2 + · · · + yn ) ≥ x1 y1 + x2 y2 + · · · + xn yn n www.EngineeringBooksPDF.com Glossary 201 Chebyshev Polynomials Let {Tn (x)}∞ n=0 be the sequence of polynomials such that T0 (x) = 1, T1 (x) = x, and Ti+1 = 2xTi (x) − Ti−1 (x) for all positive integers i The polynomial Tn (x) is called the nth Chebyshev polynomial Circumcenter The center of the circumscribed circle or sphere Circumcircle A circumscribed circle Convexity A function f (x) is concave up (down) on [a, b] ⊆ R if f (x) lies under (over) the line connecting (a1 , f (a1 )) and (b1 , f (b1 )) for all a ≤ a1 < x < b1 ≤ b Concave up and down functions are also called convex and concave, respectively If f is concave up on an interval [a, b] and λ1 , λ2 , , λn are nonnegative numbers with sum equal to 1, then λ1 f (x1 ) + λ2 f (x2 ) + · · · + λn f (xn ) ≥ f (λ1 x1 + λ2 x2 + · · · + λn xn ) for any x1 , x2 , , xn in the interval [a, b] If the function is concave down, the inequality is reversed This is Jensen’s inequality Cyclic Sum Let n be a positive integer Given a function f of n variables, define the cyclic sum of variables (x1 , x2 , , xn ) as f (x1 , x2 , , xn ) = f (x1 , x2 , , xn ) + f (x2 , x3 , , xn , x1 ) cyc + · · · + f (xn , x1 , x2 , , xn−1 ) De Moivre’s Formula For any angle α and for any integer n, (cos α + i sin α)n = cos nα + i sin nα www.EngineeringBooksPDF.com 202 103 Trigonometry Problems From this formula, we can easily derive the expansion formulas of sin nα and cos nα in terms of sin α and cos α Euler’s Formula (in Plane Geometry) Let O and I be the circumcenter and incenter, respectively, of a triangle with circumradius R and inradius r Then |OI |2 = R − 2rR Excircles, or Escribed Circles Given a triangle ABC, there are four circles tangent to the lines AB, BC, CA One is the inscribed circle, which lies in the interior of the triangle One lies on the opposite side of line BC from A, and is called the excircle (escribed circle) opposite A, and similarly for the other two sides The excenter opposite A is the center of the excircle opposite A; it lies on the internal angle bisector of A and the external angle bisectors of B and C Excenters See Excircles Extended Law of Sines In a triangle ABC with circumradius equal to R, |BC| |CA| |AB| = = = 2R sin A sin B sin C Gauss’s Lemma Let p(x) = an x n + an−1 x n−1 + · · · + aa x + a0 be a polynomial with integer coefficients All the rational roots (if there are any) of p(x) can be written in the reduced form m n , where m and n are divisors of a0 and an , respectively Gergonne Point If the incircle of triangle ABC touches sides AB, BC, and CA at F, D, and E, then lines AD, BE, and CF are concurrent, and the point of concurrency is called the Gergonne point of the triangle www.EngineeringBooksPDF.com Glossary 203 Heron’s Formula The area of a triangle ABC with sides a, b, c is equal to [ABC] = s(s − a)(s − b)(s − c), where s = (a + b + c)/2 is the semiperimeter of the triangle Homothety A homothety (central similarity) is a transformation that fixes one point O (its center) and maps each point P to a point P for which O, P , P are collinear and the ratio |OP | : |OP | = k is constant (k can be either positive or negative); k is called the magnitude of the homothety Homothetic Triangles Two triangles ABC and DEF are homothetic if they have parallel sides Suppose that AB DE, BC EF , and CA F D Then lines AD, BE, and CF concur at a point X, as given by a special case of Desargues’s theorem Furthermore, some homothety centered at X maps triangle ABC onto triangle DEF Incenter The center of an inscribed circle Incircle An inscribed circle Jensen’s Inequality See Convexity Kite A quadrilateral with its sides forming two pairs of congruent adjacent sides A kite is symmetric with one of its diagonals (If it is symmetric with both diagonals, it becomes a rhombus.) The two diagonals of a kite are perpendicular to each other For example, if ABCD is a quadrilateral with |AB| = |AD| and |CB| = |CD|, then ABCD is a kite, and it is symmetric with respect to the diagonal AC www.EngineeringBooksPDF.com 204 103 Trigonometry Problems Lagrange’s Interpolation Formula Let x0 , x1 , , xn be distinct real numbers, and let y0 , y1 , , yn be arbitrary real numbers Then there exists a unique polynomial P (x) of degree at most n such that P (xi ) = yi , i = 0, 1, , n This polynomial is given by n P (x) = i=0 yi (x − x0 ) · · · (x − xi−1 )(x − xi+1 ) · · · (x − xn ) (xi − x0 ) · · · (xi − xi−1 )(xi − xi+1 ) · · · (xi − xn ) Law of Cosines In a triangle ABC, |CA|2 = |AB|2 + |BC|2 − 2|AB| · |BC| cos ABC, and analogous equations hold for |AB|2 and |BC|2 Median formula This is also called the length of the median formula Let AM be a median in triangle ABC Then 2|AB|2 + 2|AC|2 − |BC|2 |AM|2 = Minimal Polynomial We call a polynomial p(x) with integer coefficients irreducible if p(x) cannot be written as a product of two polynomials with integer coefficients neither of which is a constant Suppose that the number α is a root of a polynomial q(x) with integer coefficients Among all polynomials with integer coefficients with leading coefficient (i.e., monic polynomials with integer coefficients) that have α as a root, there is one of smallest degree This polynomial is the minimal polynomial of α Let p(x) denote this polynomial Then p(x) is irreducible, and for any other polynomial q(x) with integer coefficients such that q(α) = 0, the polynomial p(x) divides q(x); that is, q(x) = p(x)h(x) for some polynomial h(x) with integer coefficients Orthocenter of a Triangle The point of intersection of the altitudes www.EngineeringBooksPDF.com Glossary 205 Periodic Function A function f (x) is periodic with period T > if T is the smallest positive real number for which f (x + T ) = f (x) for all x Pigeonhole Principle If n objects are distributed among k < n boxes, some box contains at least two objects Power Mean Inequality Let a1 , a2 , , an be any positive numbers for which a1 + a2 + · · · + an = For positive numbers x1 , x2 , , xn we define M−∞ = min{x1 , x2 , , xk }, M∞ = max{x1 , x2 , , xk }, M0 = x1a1 x2a2 · · · xnan , Mt = a1 x1t + a2 x2t + · · · + ak xkt 1/t , where t is a nonzero real number Then M−∞ ≤ Ms ≤ Mt ≤ M∞ for s ≤ t Rearrangement Inequality Let a1 ≤ a2 ≤ · · · ≤ an ; b1 ≤ b2 ≤ · · · ≤ bn be real numbers, and let c1 , c2 , , cn be any permutations of b1 ≤ b2 ≤ · · · ≤ bn Then a1 bn + a2 bn−1 + · · · + an b1 ≤ a1 c1 + a2 c2 + · · · + an cn ≤ a1 b1 + a2 b2 + · · · + an bn , with equality if and only if a1 = a2 = · · · = an or b1 = b2 = · · · = bn Root Mean Square–Arithmetic Mean Inequality For positive numbers x1 , x2 , , xn , x12 + x22 + · · · + xk2 x1 + x2 + · · · + xk ≥ n n www.EngineeringBooksPDF.com 206 103 Trigonometry Problems The inequality is a special case of the power mean inequality Schur’s Inequality Let x, y, z be nonnegative real numbers Then for any r > 0, x r (x − y)(x − z) + y r (y − z)(y − x) + zr (z − x)(z − y) ≥ Equality holds if and only if x = y = z or if two of x, y, z are equal and the third is equal to The proof of the inequality is rather simple Because the inequality is symmetric in the three variables, we may assume without loss of generality that x ≥ y ≥ z Then the given inequality may be rewritten as (x − y) x r (x − z) − y r (y − z) + zr (x − z)(y − z) ≥ 0, and every term on the left-hand side is clearly nonnegative The first term is positive if x > y, so equality requires x = y, as well as zr (x − z)(y − z) = 0, which gives either x = y = z or z = Sector The region enclosed by a circle and two radii of the circle Stewart’s Theorem In a triangle ABC with cevian AD, write a = |BC|, b = |CA|, c = |AB|, m = |BD|, n = |DC|, and d = |AD| Then d a + man = c2 n + b2 m This formula can be used to express the lengths of the altitudes and angle bisectors of a triangle in terms of its side lengths Trigonometric Identities sin2 a + cos2 a = 1, + cot2 a = csc2 a, tan2 x + = sec2 x www.EngineeringBooksPDF.com Glossary 207 Addition and Subtraction Formulas: sin(a ± b) = sin a cos b ± cos a sin b, cos(a ± b) = cos a cos b ∓ sin a sin b, tan a ± tan b tan(a ± b) = , ∓ tan a tan b cot a cot b ∓ cot(a ± b) = cot a ± cot b Double-Angle Formulas: sin 2a = sin a cos a = tan a , + tan2 a cos 2a = cos2 a − = − sin2 a = − tan2 a , + tan2 a tan a ; − tan2 a cot2 a − cot 2a = cot a tan 2a = Triple-Angle Formulas: sin 3a = sin a − sin3 a, cos 3a = cos3 a − cos a, tan 3a = tan a − tan3 a − tan2 a Half-Angle Formulas: a 2 a cos a tan a cot sin2 − cos a , + cos a = , − cos a sin a = = , sin a + cos a + cos a sin a = = sin a − cos a = www.EngineeringBooksPDF.com 208 103 Trigonometry Problems Sum-to-Product Formulas: a+b a−b cos , 2 a+b a−b cos a + cos b = cos cos , 2 sin(a + b) tan a + tan b = cos a cos b sin a + sin b = sin Difference-to-Product Formulas: a−b a+b cos , 2 a−b a+b cos a − cos b = −2 sin sin , 2 sin(a − b) tan a − tan b = cos a cos b sin a − sin b = sin Product-to-Sum Formulas: sin a cos b = sin(a + b) + sin(a − b), cos a cos b = cos(a + b) + cos(a − b), sin a sin b = − cos(a + b) + cos(a − b) Expansion Formulas n n cosn−1 α sin α − cosn−3 α sin3 α n + cosn−5 α sin5 α − · · · , n n cos nα = cosn α − cosn−2 α sin2 α n + cosn−4 α sin4 α − · · · sin nα = Viète’s Theorem Let x1 , x2 , , xn be the roots of polynomial P (x) = an x n + an−1 x n−1 + · · · + a1 x + a0 , where an = and a0 , a1 , , an ∈ C Let sk be the sum of the products of the xi taken k at a time Then an−k sk = (−1)k ; an www.EngineeringBooksPDF.com Glossary 209 that is, x1 + x2 + · · · + xn = − x1 x2 + · · · + xi xj + xn−1 xn = an−1 ; an an−2 ; an x1 x2 · · · xn = (−1)n www.EngineeringBooksPDF.com a0 an www.EngineeringBooksPDF.com Further Reading Andreescu, T.; Feng, Z., 101 Problems in Algebra from the Training of the USA IMO Team, Australian Mathematics Trust, 2001 Andreescu, T.; Feng, Z., 102 Combinatorial Problems from the Training of the USA IMO Team, Birkhäuser, 2002 Andreescu, T.; Feng, Z., USA and International Mathematical Olympiads 2003, Mathematical Association of America, 2004 Andreescu, T.; Feng, Z., USA and International Mathematical Olympiads 2002, Mathematical Association of America, 2003 Andreescu, T.; Feng, Z., USA and International Mathematical Olympiads 2001, Mathematical Association of America, 2002 Andreescu, T.; Feng, Z., USA and International Mathematical Olympiads 2000, Mathematical Association of America, 2001 Andreescu, T.; Feng, Z.; Lee, G.; Loh, P., Mathematical Olympiads: Problems and Solutions from around the World, 2001–2002, Mathematical Association of America, 2004 Andreescu, T.; Feng, Z.; Lee, G., Mathematical Olympiads: Problems and Solutions from around the World, 2000–2001, Mathematical Association of America, 2003 www.EngineeringBooksPDF.com 212 103 Trigonometry Problems Andreescu, T.; Feng, Z., Mathematical Olympiads: Problems and Solutions from around the World, 1999–2000, Mathematical Association of America, 2002 10 Andreescu, T.; Feng, Z., Mathematical Olympiads: Problems and Solutions from around the World, 1998–1999, Mathematical Association of America, 2000 11 Andreescu, T.; Kedlaya, K., Mathematical Contests 1997–1998: Olympiad Problems from around the World, with Solutions, American Mathematics Competitions, 1999 12 Andreescu, T.; Kedlaya, K., Mathematical Contests 1996–1997: Olympiad Problems from around the World, with Solutions, American Mathematics Competitions, 1998 13 Andreescu, T.; Kedlaya, K.; Zeitz, P., Mathematical Contests 1995–1996: Olympiad Problems from around the World, with Solutions, American Mathematics Competitions, 1997 14 Andreescu, T.; Enescu, B., Mathematical Olympiad Treasures, Birkhäuser, 2003 15 Andreescu, T.; Gelca, R., Mathematical Olympiad Challenges, Birkhäuser, 2000 16 Andreescu, T.; Andrica, D., 360 Problems for Mathematical Contests, GIL Publishing House, 2003 17 Andreescu, T.; Andrica, D., Complex Numbers from A to Z, Birkhäuser, 2004 18 Beckenbach, E F.; Bellman, R., An Introduction to Inequalities, New Mathematical Library, Vol 3, Mathematical Association of America, 1961 19 Coxeter, H S M.; Greitzer, S L., Geometry Revisited, New Mathematical Library, Vol 19, Mathematical Association of America, 1967 20 Coxeter, H S M., Non-Euclidean Geometry, The Mathematical Association of America, 1998 21 Doob, M., The Canadian Mathematical Olympiad 1969–1993, University of Toronto Press, 1993 22 Engel,A., Problem-Solving Strategies, Problem Books in Mathematics, Springer, 1998 www.EngineeringBooksPDF.com Further Reading 213 23 Fomin, D.; Kirichenko, A., Leningrad Mathematical Olympiads 1987–1991, MathPro Press, 1994 24 Fomin, D.; Genkin, S.; Itenberg, I., Mathematical Circles, American Mathematical Society, 1996 25 Graham, R L.; Knuth, D E.; Patashnik, O., Concrete Mathematics, AddisonWesley, 1989 26 Gillman, R., A Friendly Mathematics Competition, The Mathematical Association of America, 2003 27 Greitzer, S L., International Mathematical Olympiads, 1959–1977, New Mathematical Library, Vol 27, Mathematical Association of America, 1978 28 Holton, D., Let’s Solve Some Math Problems, A Canadian Mathematics Competition Publication, 1993 29 Kazarinoff, N D., Geometric Inequalities, New Mathematical Library, Vol 4, Random House, 1961 30 Kedlaya, K; Poonen, B.; Vakil, R., The William Lowell Putnam Mathematical Competition 1985–2000, The Mathematical Association of America, 2002 31 Klamkin, M., International Mathematical Olympiads, 1978–1985, New Mathematical Library, Vol 31, Mathematical Association of America, 1986 32 Klamkin, M., USA Mathematical Olympiads, 1972–1986, New Mathematical Library, Vol 33, Mathematical Association of America, 1988 33 Kürschák, J., Hungarian Problem Book, volumes I & II, New Mathematical Library, Vols 11 & 12, Mathematical Association of America, 1967 34 Kuczma, M., 144 Problems of the Austrian–Polish Mathematics Competition 1978–1993, The Academic Distribution Center, 1994 35 Kuczma, M., International Mathematical Olympiads 1986–1999, Mathematical Association of America, 2003 36 Larson, L C., Problem-Solving Through Problems, Springer-Verlag, 1983 37 Lausch, H The Asian Pacific Mathematics Olympiad 1989–1993, Australian Mathematics Trust, 1994 38 Liu, A., Chinese Mathematics Competitions and Olympiads 1981–1993, Australian Mathematics Trust, 1998 www.EngineeringBooksPDF.com 214 103 Trigonometry Problems 39 Liu, A., Hungarian Problem Book III, New Mathematical Library, Vol 42, Mathematical Association of America, 2001 40 Lozansky, E.; Rousseau, C Winning Solutions, Springer, 1996 41 Mitrinovic, D S.; Pecaric, J E.; Volonec, V Recent Advances in Geometric Inequalities, Kluwer Academic Publisher, 1989 42 Savchev, S.; Andreescu, T Mathematical Miniatures, Anneli Lax New Mathematical Library, Vol 43, Mathematical Association of America, 2002 43 Sharygin, I F., Problems in Plane Geometry, Mir, Moscow, 1988 44 Sharygin, I F., Problems in Solid Geometry, Mir, Moscow, 1986 45 Shklarsky, D O; Chentzov, N N;Yaglom, I M., The USSR Olympiad Problem Book, Freeman, 1962 46 Slinko, A., USSR Mathematical Olympiads 1989–1992, Australian Mathematics Trust, 1997 47 Szekely, G J., Contests in Higher Mathematics, Springer-Verlag, 1996 48 Taylor, P J., Tournament of Towns 1980–1984, Australian Mathematics Trust, 1993 49 Taylor, P J., Tournament of Towns 1984–1989, Australian Mathematics Trust, 1992 50 Taylor, P J., Tournament of Towns 1989–1993, Australian Mathematics Trust, 1994 51 Taylor, P J.; Storozhev, A., Tournament of Towns 1993–1997, Australian Mathematics Trust, 1998 52 Yaglom, I M., Geometric Transformations, New Mathematical Library, Vol 8, Random House, 1962 53 Yaglom, I M., Geometric Transformations II, New Mathematical Library, Vol 21, Random House, 1968 54 Yaglom, I M., Geometric Transformations III, New Mathematical Library, Vol 24, Random House, 1973 www.EngineeringBooksPDF.com ... of a circle between points A and B www.EngineeringBooksPDF.com 103 Trigonometry Problems www.EngineeringBooksPDF.com www.EngineeringBooksPDF.com Trigonometric Fundamentals Definitions of Trigonometric... This technique will certainly be helpful in solving problems www.EngineeringBooksPDF.com 18 103 Trigonometry Problems such as Introductory Problems 19(b), 20(b), 23(a) and (d), 27(b), and 28(b)... 58 Introductory Problems 63 Advanced Problems 73 Solutions to Introductory Problems 83 Solutions to Advanced Problems 125 Glossary 199 Further Reading 211 www.EngineeringBooksPDF.com Preface