,.)"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 Solving Trigonometric Equations Solving Trigonometric Equations By: OpenStaxCollege Egyptian pyramids standing near a modern city (credit: Oisin Mulvihill) Thales of Miletus (circa 625–547 BC) is known as the founder of geometry The legend is that he calculated the height of the Great Pyramid of Giza in Egypt using the theory of similar triangles, which he developed by measuring the shadow of his staff Based on proportions, this theory has applications in a number of areas, including fractal geometry, engineering, and architecture Often, the angle of elevation and the angle of depression are found using similar triangles In earlier sections of this chapter, we looked at trigonometric identities Identities are true for all values in the domain of the variable In this section, we begin our study of trigonometric equations to study real-world scenarios such as the finding the dimensions of the pyramids 1/29 Solving Trigonometric Equations Solving Linear Trigonometric Equations in Sine and Cosine Trigonometric equations are, as the name implies, equations that involve trigonometric functions Similar in many ways to solving polynomial equations or rational equations, only specific values of the variable will be solutions, if there are solutions at all Often we will solve a trigonometric equation over a specified interval However, just as often, we will be asked to find all possible solutions, and as trigonometric functions are periodic, solutions are repeated within each period In other words, trigonometric equations may have an infinite number of solutions Additionally, like rational equations, the domain of the function must be considered before we assume that any solution is valid The period of both the sine function and the cosine function is 2π In other words, every 2π units, the y-values repeat If we need to find all possible solutions, then we must add 2πk, where k is an integer, to the initial solution Recall the rule that gives the format for stating all possible solutions for a function where the period is 2π : sin θ = sin(θ ± 2kπ) There are similar rules for indicating all possible solutions for the other trigonometric functions Solving trigonometric equations requires the same techniques as solving algebraic equations We read the equation from left to right, horizontally, like a sentence We look for known patterns, factor, find common denominators, and substitute certain expressions with a variable to make solving a more straightforward process However, with trigonometric equations, we also have the advantage of using the identities we developed in the previous sections Solving a Linear Trigonometric Equation Involving the Cosine Function Find all possible exact solutions for the equation cos θ = From the unit circle, we know that cos θ = π 5π θ= , 3 These are the solutions in the interval [0, 2π] All possible solutions are given by θ= π 5π ± 2kπ and θ = ± 2kπ 3 where k is an integer 2/29 Solving Trigonometric Equations Solving a Linear Equation Involving the Sine Function Find all possible exact solutions for the equation sin t = Solving for all possible values of t means that solutions include angles beyond the period π 5π of 2π From the [link], we can see that the solutions are t = and t = But the problem is asking for all possible values that solve the equation Therefore, the answer is t= π 5π ± 2πk and t = ± 2πk 6 where k is an integer How To Given a trigonometric equation, solve using algebra Look for a pattern that suggests an algebraic property, such as the difference of squares or a factoring opportunity Substitute the trigonometric expression with a single variable, such as x or u Solve the equation the same way an algebraic equation would be solved Substitute the trigonometric expression back in for the variable in the resulting expressions Solve for the angle Solve the Trigonometric Equation in Linear Form Solve the equation exactly: cos θ − = − 5, ≤ θ < 2π Use algebraic techniques to solve the equation cos θ − = − cos θ = − cos θ = − θ=π Try It Solve exactly the following linear equation on the interval [0, 2π) : sin x + = x= 7π 11π 6, 3/29 Solving Trigonometric Equations Solving Equations Involving a Single Trigonometric Function When we are given equations that involve only one of the six trigonometric functions, their solutions involve using algebraic techniques and the unit circle (see [link]) We need to make several considerations when the equation involves trigonometric functions other than sine and cosine Problems involving the reciprocals of the primary trigonometric functions need to be viewed from an algebraic perspective In other words, we will write the reciprocal function, and solve for the angles using the function Also, an equation involving the tangent function is slightly different from one containing a sine or cosine function First, as we know, the period of tangent is π, not 2π Further, the domain of tangent is all real numbers with the exception of odd integer ...[...]... Determine the inverse tangent of a number 2 Determine the inverse sine and cosine of a number using the graphing calculator 3 Identify the domain and range of the inverse sine, cosine, and tangent functions Activity 6.4 Objective: Project Activity 6.5 Objectives: Solving a Murder 609 1 Determine the measure of all sides and angles of a right triangle How Stable Is That Tower? 614 1 Solve problems using... the final selling price 4 a The input is any number and the output is the square of the number b The square of a number is the input and the output is the number 5 a In the following table, elevation is the input and amount of snowfall is the output ELEVATION (IN FEET) SNOWFALL (IN INCHES) 2000 4 3000 6 4000 9 5000 12 Activity 1.1 Parking Problems b In the preceding table, snowfall is the input and. .. work, both in and out of class It is the belief of the authors that students learn mathematics best when they are actively involved in solving problems that are meaningful to them The text is primarily a collection of situations drawn from real life Each situation leads to one or more problems By answering a series of questions and solving each part of the problem, you will be using and learning one or... Getting Started with the TI-83/TI-84 Plus Family of Calculators A-35 Selected Answers A-51 Glossary A-79 Index I-1 Preface Our Vision Mathematics in Action: Algebraic, Graphical, and Trigonometric Problem Solving, Fourth Edition, is intended to help college mathematics students gain mathematical literacy in the real world and simultaneously help them build a solid foundation for future study in mathematics. .. classroom learning with real-world applications • Prepare effectively for further college work in mathematics and related disciplines • Learn to work in groups as well as independently • Increase knowledge of mathematics through explorations with appropriate technology • Develop a positive attitude about learning and using mathematics • Build techniques of reasoning for effective problem solving • Learn... using the Mathematics in Action approach for the first time • Skills worksheets for topics with which students typically have difficulty • Sample chapter tests and final exams for individual and group assessment • Sample journal topics for each chapter • Learning in groups with questions and answers for instructors using collaborative learning for the first time • Incorporating technology, including... of similar right triangles using proportions 3 Determine the sine, cosine, and tangent of an angle using a right triangle 4 Determine the sine, Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2009, Article ID 393245, 9 pages doi:10.1155/2009/393245 Review Article T-Stability Approach to Variational Iteration Method for Solving Integral Equations R. Saadati, 1 S. M. Vaezpour, 1 and B. E. Rhoades 2 1 Department of Mathematics and Computer Science, Amirkabir University of Technology, 424 Hafez Avenue, Tehran 15914, Iran 2 Department of Mathematics, Indiana University, Bloomington, IN 47405-7106, USA Correspondence should be addressed to B. E. Rhoades, rhoades@indiana.edu Received 16 February 2009; Accepted 26 August 2009 Recommended by Nan-jing Huang We consider T-stability definition according to Y. Qing and B. E. Rhoades 2008 and we show that the variational iteration method for solving integral equations is T-stable. Finally, we present some text examples to illustrate our result. Copyright q 2009 R. Saadati et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction and Preliminaries Let X, · be a Banach space and T a self-map of X.Letx n1 fT, x n be some iteration procedure. Suppose that FT, the fixed point set of T, is nonempty and that x n converges to apointq ∈ FT.Let{y n }⊆X and define n y n1 − fT, y n . If lim n 0 implies that lim y n q, then the iteration procedure x n1 fT, x n is said to be T-stable. Without loss of generality, we may assume that {y n } is bounded, for if {y n } is not bounded, then it cannot possibly converge. If these conditions hold for x n1 Tx n , that is, Picard’s iteration, then we will say that Picard’s iteration is T-stable. Theorem 1.1 see 1. Let X, · be a Banach s pace and T a self-map of X satisfying Tx − Ty ≤ L x − Tx α x − y 1.1 for all x,y ∈ X,whereL ≥ 0, 0 ≤ α<1. Suppose that T has a fixed point p. Then, T is Picard T-stable. Various kinds of analytical methods and numerical methods 2–10 were used to solve integral equations. To illustrate the basic idea of the method, we consider the general 2 Fixed Point Theory and Applications nonlinear system: L u t N u t g t , 1.2 where L is a linear operator, N is a nonlinear operator, and gt is a given continuous function. The basic character of the method is to construct a functional for the system, which reads u n1 x u n x t 0 λ s Lu n s N u n s − g s ds, 1.3 where λ is a Lagrange multiplier which can be identified optimally via variational theory, u n is the nth approximate solution, and u n denotes a restricted variation; that is, δu n 0. Now, we consider the Fredholm integral equation of second kind in the general case, which reads u x f x λ b a K x, t u t dt, 1.4 where Kx, t is the kernel of the integral equation. There is a simple iteration formula for 1.4 in the form u n1 x f x λ b a K x, t u n t dt. 1.5 Now, we show that the nonlinear mapping T, defined by u n1 x T u n x f x λ b a K x, t u n t dt, 1.6 is T-stable in L 2 a, b. First, we show that the nonlinear mapping T has a fixed point. For m, n ∈ N we have T u m x − T u n x u m1 x − u n1 x λ b a K x, t u m t − u n t dt ≤ | λ | b a K 2 x, tdxdt 1/2 u m x − u n x . 1.7 Fixed Point Theory and Nguyen Huu Thong et al Tạp chí KHOA HỌC ĐHSP TPHCM A NEW PROBABILISTIC ALGORITHM FOR SOLVING NONLINEAR EQUATIONS SYSTEMS NGUYEN HUU THONG*, TRAN VAN HAO** ABSTRACT In this paper, we consider a class of optimization problems having the following characteristics: there exists a fixed number k (1≤k[...]... 6 Grosan C., Abraham A (2008), A New Approach for Solving Nonlinear Equations Systems , IEEE Transactions on Systems, Man, and Cybernetics, Part A 38(3), 698-714 7 Trần Văn Hao and Nguyễn Hữu Thông (2007), “Search via Probability Algorithm for Engineering Optimization Problems”, In Proceedings of XIIth International Conference on Applied Stochastic Models and Data Analysis (ASMDA2007), Chania, Crete,... is very efficient for solving nonlinear equations systems PDS algorithm has the abilities to overcome local optimal solutions and to obtain global optimal solutions Many optimization problems have very narrow feasible domains that require the algorithm having an ability to search values of two or more consecutive digits simultaneously to find a feasible solution We study this case and the results will... Trust-Region Methods Philadelphia, PA: SIAM 3 Denis J E and Wolkowicz H (1993), “Least change secant methods, sizing, and shifting”, SIAM J Numer Anal., vol 30, pp 1291–1314 4 Denis J E (1967), “On Newton’s method and nonlinear simultaneous replacements”, SIAM J Numer Anal., vol 4, pp 103–108 5 Effati S and Nazemi A R (2005), A new method for solving a system of the nonlinear equations , Appl Math Comput., vol... Remarks: For each problem, solutions that are found by PDS algorithm dominate solutions of [6] That means, solutions of [6] are dominated and NOT Pareto optimal solutions! PDS algorithm is very efficient for solving equations systems The algorithm has the abilities to overcome local optimal solutions and to obtain global optimal solutions 7 Conclusions We consider a class of optimization problems having... [7], but the probabilities of [7] are only relevant to the problems having no many local optimums In this paper we build new probabilities to control changes of values of the solution and design the PDS algorithm for solving single-objective optimization problems For application of PDS algorithm we transform the nonlinear equations system into a single-objective optimization problem PDS algorithm is very... book: Recent Advances in Stochastic Modeling and Data Analysis, editor: Christos H Skiadas, publisher: World Scientific Publishing Co Pte Ltd, 454 – 463 8 Hentenryck V., McAllester D and Kapur D (1997), Solving polynomial systems using a branch and prune approach”, SIAM J Numer Anal., vol 34, no 2, pp 797–827 9 Morgan A P (1987), Solving Polynomial Systems Using Continuation for Scientific and Engineering... reported in the next paper We also compare Search via Probability algorithm of papers [7] with PDS algorithm of this paper for solving engineering optimization problems REFERENCES 1 16 Broyden C G (1965), A class of methods for solving nonlinear simultaneous equations , Math Comput., vol 19, no 92, pp Using the Explorer IDE After a plot has been selected, you can change its properties via the Plot Properties dialog accessible from the right-mouse button within the plot window. The Plot Prop- erties dialog allows you to set generic, axis, and signal attributes. Generic plot attributes include the plot type as well as display of titles and/or subti- tles. Axis properties of the plot allow you to set the axis styles for both the X- and Y-axis, including labels and tic-mark styles. Verilog-A Explorer IDE 197 Verilog-A Explorer IDE Signal properties allow you to edit the description of the signals displayed in the leg- end box, as well as the data format and drawing attributes. D.3.2 Creating a New Designs Starting a new design follows essentially the same procedure as previously outlined, but with the addition to creating a new circuit and/or Verilog-A file(s). From the main Explorer menu, select File->New, which raises the following dialog box: If you select a circuit file, the workspace will be cleared of any open files. If you select a Verilog-A file, it is assumed that it is associated with any existing circuit design open within the workspace. In both cases, a new file is created and initialized with a template file of the appropriate type. If you prefer your own template files, change the path of the template via the respective Editor Properties dialog accessible via the right mouse button. 198 Verilog-A HDL Appendix E Spice Quick Reference E.1 Introduction Spice is a general-purpose circuit simulation program for nonlinear DC, nonlinear transient, and linear AC analysis. Originating from the University of California at Berkeley, is by far the best known and most widely used circuit simulator. It is availa- ble in for a wide variety of computer platforms, in both commercial and proprietary derivatives of the original version. Newer versions of Spice offer many extensions, but the input format for circuit descriptions reflect the original batch-oriented program architecture. This appendix overviews the Spice input format, or netlist files including the fundamental types and analyses supported. Omitted for brevity are details regarding semiconductor device models and the various Spice options. Spice Quick Reference 199 Spice Quick Reference E.2 Circuit Netlist Description The netlist (also referred to as the input deck) consists of element lines which describes both the circuit topology and element values and control lines which describe analyses to be performed for Spice. The first card in the input deck must be a title card, and the last card must be the .END control line. The order of the remaining element and control lines is arbitrary. The input format is free format. Fields on an element or control line are separated by one or more blanks, commas, equal (=) sign, or a left or right parenthesis. A element or control line may be continued by placing a (+) in column 1 on the following line. Spice will continue reading beginning with column 2. Name fields must begin with a letter [a–z] and cannot contain any delimeters. Names within Spice netlists are considered case-insensitive 1 . An integer or a floating point number can be followed by one of the following scale factors: G = 1.0e9 MEG = 1.0e6 K = 1.0e3 MIL = 25.4e-4 M = 1.0e-3 U = 1.0e-6 N = l.0e-9 P = 1.0e-12 1. Names in Verilog are case-sensitive requiring a certain level of awareness for modelers in developing Verilog-A models that are case-independent for use within Spice netlists. 200 Verilog-A HDL Spice Quick Reference 201 E.3 Components Letters immediately following a Modeling with Trigonometric Equations Modeling with Trigonometric Equations By: OpenStaxCollege The hands on a clock are periodic: they repeat positions every twelve hours (credit: “zoutedrop”/Flickr) Suppose we charted the average daily temperatures in New York City over the course of one year We would expect to find the lowest temperatures in January and February and highest in July and ... with solving trigonometric equations • • • • • • Solving Trigonometric Equations I Solving Trigonometric Equations II Solving Trigonometric Equations III Solving Trigonometric Equations IV Solving. . .Solving Trigonometric Equations Solving Linear Trigonometric Equations in Sine and Cosine Trigonometric equations are, as the name implies, equations that involve trigonometric. .. Solving Trigonometric Equations V Solving Trigonometric Equations VI Key Concepts • When solving linear trigonometric equations, we can use algebraic techniques just as we solving algebraic equations