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For Madge Goldman who has done so much to bring so many people together I.M. Gelfand Mark Saul Trigonometry Birkhauser Boston • Basel • Berlin I.M. Gelfand Mark Saul Department of Mathematics Bronxville Schools Rutgers University Bronxville, NY 10708 Piscataway, NJ 08854-8019 U.S.A. U.S.A. Library of Congress Cataloging-in-Publication Data Gel'fand, I.M. (Izrail' Moiseevich) Trigonometry I I.M. Gel'fand, Mark Saul. p. em. ISBN 0-8176-3914-4 (alk. paper) I. Trigonometry, Plane. I. Saul, Mark E. II. Title. QA533.G45 1999 516.24'2-dc21 99-32245 CIP Printed on acid-free paper © 2001 I.M. Gelfand Birkhiiuser All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Birkhauser Boston, c/o Springer-Verlag New York, Inc., 175 Fifth Avenue, New York, NY 10010, 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 of general descriptive names, trade names, trademarks, etc., in this publication, even if the former are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone. ISBN 0-8176-3914-4 SPIN 10536370 ISBN 3-7643-3914-4 Typeset in J.!\TEX by Martin Stock, Cambridge, MA Printed and bound by Hamilton Printing, Rensselaer, NY Printed in the United States of America 9 8 7 6 5 4 3 2 1 Contents Preface ix 0 Trigonometry 1 1 What is new about trigonometry? . 1 2 Right triangles . 6 3 The Pythagorean theorem . 7 4 Our best friends (among right triangles) 9 5 Our next best friends (among right triangles) . 10 6 Some standard notation . 12 1 Trigonometric Ratios in a Triangle 21 1 Definition of sin a . 21 2 Find the hidden sine . 24 3 The cosine ratio . 26 4 A relation between the sine and the cosine 28 5 A bit of notation . 28 6 Another relation between the sine and the cosine 29 7 Our next best friends (and the sine ratio) .. 30 8 What is the value of sin 90°? .......... 32 9 An exploration: How large can the sum be? .. 33 10 More exploration: How large can the product be? 34 11 More names for ratios . 35 2 Relations among Trigonometric Ratios 41 1 The sine and its relatives 41 2 Algebra or geometry? . 44 3 A remark about names 45 4 An identity crisis? .... 46 5 Identities with secant and cosecant 48 6 A lemma ..... 50 7 Some inequalities . 51 vi Contents 8 Calculators and tables . 52 9 Getting the degree measure of an angle from its sine . 53 10 Solving right triangles . 55 11 Shadows .............. 56 12 Another approach to the sine ratio 57 3 Relationships in a Triangle 67 1 Geometry of the triangle •••• 0 0 0 • 0 0 67 2 The congruence theorems and trigonometry 68 3 Sines and altitudes 69 4 Obtuse triangles . 70 5 The Law of Sines 71 6 The circumradius .,. 74 7 Area of a triangle 75 8 Two remarks 78 9 Law of cosines 78 4 Angles and Rotations 91 1 Measuring rotations . 0 •••••••• ' 91 2 Rotation and angles . 0 • 0 •••• 0 • 93 3 Trigonometric functions for all angles 94 4 Calculations with angles of rotations 98 5 Odd and even functions 0 •• 0 •• 0 101 5 Radian Measure 103 1 Radian measure for angles and rotations 103 2 Radian measure and distance . 109 3 Interlude: How to explain radian measure to your brother and sister . 113 4 Radian measure and calculators . 114 5 An important graph 115 6 Two small miracles 117 6 The Addition Formulas 123 1 More identities .. 123 2 The addition formulas . 125 3 Proofs of the addition formulas 126 4 A first beautiful proof . 127 5 A second beautiful proof .... ·130 Contents vii 7 Trigonometric Identities 139 1 Extending the identities . 139 2 The Principle of Analytic Continuation: Higher mathematics to the rescue . 139 3 Back to our identities . 141 4 A formula for tan (a+ {3) . 143 5 Double the angle . 145 6 Triple the angle . 148 7 Derivation of the formulas for sin a /2 and cos a /2 . 149 8 Another formula for tan aj2 151 9 Products to sums 152 10 Sums to products . 154 8 Graphs of Trigonometric Functions 173 1 Graphing the basic sine curve . 173 2 The period of the function y = sin x 175 3 Periods of other sinusoidal curves 176 4 The amplitude of a sinusoidal curve 178 5 Shifting the sine . 179 6 Shifting and stretching . 182 7 Some special shifts: Half-periods . 183 8 Graphing the tangent and cotangent functions 187 9 An important question about sums of sinusoidal functions . 188 10 Linear combinations of sines and cosines . 189 11 Linear combinations of sinusoidal curves with the same frequency . 192 12 Linear combinations of functions with different frequencies 194 13 Finding the period of a sum of sinusoidal curves with different periods . 196 14 A discovery of Monsieur Fourier . 197 9 Inverse Functions and Trigonometric Equations 207 1 Functions and Inverse Functions 207 2 Arcsin: The inverse function to sin 209 3 Graphing inverse functions . 214 4 Trigonometric equations . 217 5 A more general trigonometric equation . 220 6 More complicated trigonometric equations . 222 Preface In a .sense, trigonometry sits at the center of high school mathematics. It originates in the study of geometry when we investigate the ratios of sides in similar right triangles, or when we look at the relationship between a chord of a circle and its arc. It leads to a much deeper study of periodic functions, and of the so-called transcendental functions, which cannot be described using finite algebraic processes. It also has many applications to physics, astronomy, and other branches of science. It is a very old subject. Many of the geometric results that we now state in trigonometric terms were given a purely geometric exposition by Euclid. Ptolemy, an early astronomer, began to go beyond Euclid, using th~ geometry of the time to construct what we now call tables of values of trigonometric functions. Trigonometry is an important introduction to calculus, where one stud­ ies what mathematicians call analytic properties of functions. One of the goals of this book is to prepare you for a course in calculus by directing your attention away from particular values of a function to a study of the function as an object in itself. This way of thinking is useful not just in calculus, but in many mathematical situations. So trigonometry is a part of pre-calculus, and is related to other pre-calculus topics, such as exponential and logarithmic functions, and complex numbers. The interaction of these topics with trigonometry opens a whole new landscape of mathematical results. But each of these results is also important in its own right, without being "pre-" anything. We have tried to explain the beautiful results of trigonometry as simply and systematically as possible. In many cases we have found that simple problems have connections with profound and advanced ideas. Sometimes we have indicated these connections. In other cases we have left them for you to discover as you learn more about mathematics. X Preface About the exercises: We have tried to include a few problems of each "routine" type. If you need to work more such problems, they are easy to find. Most of our problems, however, are more challenging, or exhibit a new aspect of the technique or object under discussion. We have tried to make each exercise tell a little story about the mathematics, and have the stories build to a deep understanding. We will be happy if you enjoy this book and learn something from it. We enjoyed writing it, and learned a lot too. Acknowledgments The authors would like to thank Martin Stock, who took a very ragged manuscript and turned it into the book you are now holding. We also thank Santiago Samanca and the late Anneli Lax for their reading of the manu­ script and correction of several bad blunders. We thank Ann Kostant for her encouragement, support, and gift for organization. We thank the students of Bronxville High School for their valuable classroom feedback. Finally, 'We thank Richard Askey for his multiple readings of the manuscripts, for cor­ recting some embarrassing gaffes, and for making important suggestions that contributed significantly to the content of the book. Israel M. Gelfand Mark Saul March 20, 2001 Chapter 0 Trigonometry f. Gr. r p[y wvo- v triangle + - {tHp[a measurement. - Oxford English Dictionary In this chapter we will look at some results in geometry that set the stage for a study of trigonometry. 1 What is new about trigonometry? Two of the most basic figures studied in geometry are the triangle and the circle. Tngonometry will tell us more than we learned in geometry about each of these figures. For example, in geometry we learn that if we know the lengths of the three sides of a triangle, then the measures of its angles are completely determined1 (and, in fact, almost everything else about the triangle is de­ termined). But, except for a few very special triangles, geometry does not tell us how to compute the measures of the angles, given the measures of the sides. Example 1 The measures of the sides of a triangle are 6, 6, and 6 cen­ timeters.
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