trigonometric functions – problem solving approach (hàm số lượng giác tiếp cận theo hướng giải quyết vấn đề) bởi a. panchishkin và e. shavgulidze

,.)"Ji . ',""" ., " TRIGONOMETRIC FUNCTIONS / l II s, A. A. I1aHlJRmKlIH, E. T. llIaBryJIR)J,3e TPl1rOHOMETPI1QECRI1E <DYHRUl1lJ B 3A,D;AQAX ~I3AaTeJlhCTHO «Hayrca: MOCRBa ~ I 1 , 1 I -, I A. Panchishkin E.Shavgulidze ' TRIGONOMETRIC FUNCTIONS (Problem-50lving Approach) Mir Publishers Moscow' Translated from Russian by Leonid Levant First published 1988 Revised from the 1986 Russian edition Ha anaAuiic1:0M nsune Printed in the Union of Soviet Socialist Republics I; t /1 I r I ISBN 5-03-000222-7 © 1I3p;aTeJIhcTBo «HayKa., I'aaanaa penasnaa .pHaHKO-MaTeMaTHQeCKOii nareparypu, 1986 © English translation, Mir Publishers, 1988 FroD1 the J\uthors By tradition, trigonometry is an important component of mathematics courses at high school, and trigonometry questions are always set at oral and written examina- tions to those entering universities, engineering colleges, and teacher-training institutes. The aim of this study aid is to help the student to mas- ter the basic techniques of solving difficult problems in trigonometry using appropriate definitions and theorems from the school course of mathematics. To present the material in a smooth way, we have enriched the text with some theoretical material from the textbook Algebra and Fundamentals of Analysis edited by Academician A. N. Kolmogorov and an experimental textbook of the same title by Professors N.Ya. Vilenkin, A.G. Mordko- vich, and V.K. Smyshlyaev, focussing our attention on the application of theory to solution of problems. That is why our book contains many worked competition problems and also some problems to be solved independ- ently (they are given at the end of each chapter, the answers being at the end of the book). Some of the general material is taken from Elementary Mathematics by Professors G.V. Dorofeev, M.K. Potapov, and N.Kh. Rozov (Mir Publishers, Moscow, 1982), which is one of the best study aids on mathematics for pre- college students. We should like to note here that geometrical problems which can be solved trigonometrically and problems involving integrals with trigonometric functions are not considered. At present, there are several problem hooks on mathe- matics (trigonometry included) for those preparing to pass their entrance examinations (for instance, Problems 6 From the Authors at Entrance Examinations in Mathematics by Yu.V. Nes- terenko, S.N. Olekhnik, and M.K. Potapov (Moscow, Nauka, 1983); A Collection of Competition Problems in Mathematics with Hints and Solutions edited by A.I. Pri- Iepko (Moscow, Nauka, 1986); A Collection of Problems in Mathematics for Pre-college Students edited by A. I. Pri- lepko (Moscow, Vysshaya Shkola, 1983); A Collection of Competition Problems in Mathematics for Those Entering Engineering Institutes edited by M.1. Skanavi (Moscow, Vysshaya Shkola, 1980). Some problems have been bor- rowed from these for our study aid and we are grateful to their authors for the permission to use them. The beginning of a solution to a worked example is marked by the symbol and its end by the symbol ~. The symbol ~ indicates the end of the proof of a state- ment. Our book is intended for high-school and pre-college students. We also hope that it will be helpful for the school children studying at the "smaller" mechanico- mathematical faculty of Moscow State University. From the Authors Contents 5 Chapter 1. Definitions and Basic Properties of Trigono- metric Functions 9 1.1. Radian Measure of an Arc. Trigonometric Circle 9 1.2. Definitions of the Basic Trigonometric Func- tions 18 1.3. Basic Properties of Trigonometric Functions 23 1.4. Solving the Simplest Trigonometric Equations. Inverse Trigonometric Functions 31 Problems 36 Chapter 2. Identical Transformations of Trigonometric Expressions 41 2.1. Addition Formulas 41 2.2. Trigonometric Identities for Double, Triple, and Half Arguments 55 2.3. Solution of Problems Involving Trigonometric Transformations 63 Problems 77 Chapter 3. Trigonometric Equations and Systems of Equations 80 3.1. General 80 3.2. Principal Methods of Solving Trigonometric Equations 87 3.3. Solving Trigonometric Equations and Systems of Eqnations in Several Unknowns 101 Problems 109 Chapter 4. Investigating Trigonometric Functions 11:1 4.1. Graphs of Basic Trigonometric Functions 11:1 4.2. Computing Limits 126 8 Contents 4.3. Investigating Trigonometric Functions with the Aid of a Derivative 132 Problems 146 Chapter 5. Trigonometric Inequalities 149 5.1. Proving Inequalities Involving Trigonometric Functions 149 5.2. Solving Trigonometric Inequalities 156 Problems 162 Answers 163 Chapter 1 Definitions and Basic Properties of Trigonometric Functions 1.1. Radian Measure of an Arc. Trigonometric Circle 1. The first thing the student should have in mind when studying trigonometric functions consists in that the arguments of these functions are real numbers. The pre- college student is sometimes afraid of expressions such as sin 1, cos 15 (but not sin 1°, cos 15°), cos (sin 1), and 110 cannot answer simple questions whose answer becomes obvious if the sense of these expressions is understood. 'When teaching a school course of geometry, trigonomet- ric functions are first introduced as functions of an angle (even only of an acute angle). In the subsequent study, the notion of trigonometric function is generalized when functions of an arc are considered. Here the study is not confined to the arcs enclosed within the limits of one complete revolution, that is, from 0° to 360°; the student is encountered with arcs whose measure is expressed by any number of degrees, both positive and negative. The next essential step consists in that the degree (or sexage- simal) measure is converted to a more natural radian measure. Indeed, the division of a complete revolution into 360 parts (degrees) is done by tradition (the division into other number of parts, say into 100 parts, is also used). Radian measure of angles is based on measuring the length of arcs of a circle. Here, the unit of measure- ment is one radian which is defined as a central angle subtended in a circle by an arc whose length is equal to the radius of the circle. Thus, the radian measure of an angle is the ratio of the arc it subtends to the radius of the circle in which it is the central angle; also called circular measure. Since the circumference of a circle of a unit radius is equal to 2n, the length of the arc of 360° is equal to 2n radians. Consequently, to 180° there corre- spend n radians. To change from degrees to radians and [...]... t) and cos (sin t) are decreasing functions on the interval [0, n/2] ~ 4 Relation Between Trigonometric Functions of One and the Same Argument If for a fixed value of the argument the value of a trigonometric function is known, then, under certain conditions, we can find the values of other trigonometric functions Here, the most important relationship is the principal trigonometric identity (see Sec... equal to 2 ~ 2-01644 18 1 Properties of Trigonometric Functions 1.2 Definitions of the Basic Trigonometric Functions 1 The Sine and Cosine Defined Here, recall that in school textbooks the sine and cosine of a real number t E H is defined with the aid of a trigonometric mapping P: H.~ S Definition Let the mapping P associate a number t E R with the poi II t PI on th c trigonometric circle Then the ordinate... 592653589793238462643) To solve similar problems, it is sufficient to use far less accurate approximations, but they should be written in 16 1 Properties of Trigonometric Functions the form of strict inequalities of type 3.1 . present the material in a smooth way, we have enriched the text with some theoretical material from the textbook Algebra and Fundamentals of Analysis edited by Academician A. N. Kolmogorov and an experimental textbook of the same title by. 1. Properties of Trigonometric Functions vice versa, it suffices to remember that the relation be- tween the degree and radian measures of an arc is of proportional nature. Example 1.1.1. How many degrees are contained in the arc. degrees, both positive and negative. The next essential step consists in that the degree (or sexage- simal) measure is converted to a more natural radian measure. Indeed, the division of a complete revolution into 360 parts (degrees)