Precalculus with Geometry and Trigonometry by Avinash Sathaye, Professor of Mathematics 1 Department of Mathematics, University of Kentucky Aryabhat¯ .a This book may be freely downloaded for personal use from the author’s web site www.msc.uky.edu/sohum/ma110/text/ma110 fa08.pdf Any commercial use must be preauthorized by the author. Send an email to [email protected] for inquiries. November 23, 2009 1Partially supported by NSF grant thru AMSP(Appalachian Math Science Partnership) ii Introduction. This book on Precalculus with Geometry and Trigonometry should be treated as simply an enhanced version of our book on College Algebra. Most of the topics that appear here have already been discussed in the Algebra book and often the text here is a verbatim copy of the text in the other book. We expect the student to already have a strong Algebraic background and thus the algebraic techniques presented here are more a refresher course than a first introduction. We also expect the student to be heading for higher level mathematics courses and try to supply the necessary connections and motivations for future use. Here is what is new in this book. In contrast with the Algebra book, we make a more extensive use of complex • numbers. We use Euler’s representation of complex numbers as well as the Argand diagrams extensively. Even though these are described and shown to be useful, we do not yet have tools to prove these techniques properly. They should be used as motivation and as an easy method to remember the trigonometric results. We have supplied a brief introduction to matrices and determinants. The idea • is to supply motivation for further study and a feeling for the Linear Algebra. In the appendix, we give a more formal introduction to the structure of real • numbers. While this is not necessary for calculations in this course, it is vital for understanding the finer concepts of Calculus which will be introduced in higher courses. We have also included an appendix discussing summation of series - both • finite and infinite, as well as a discussion of power series. While details of convergence are left out, this should generate familiarity with future techniques and a better feeling for the otherwise mysterious trigonometric and exponential functions. Contents 1 Review of Basic Tools. 1 1.1 Underlying field of numbers. ........................ 1 1.1.1 Working with Complex Numbers. 7 1.2 Indeterminates, variables, parameters . ..... 12 1.3 BasicsofPolynomials........................... 13 1.3.1 Rationalfunctions. 17 1.4 Workingwithpolynomials . 17 1.5 Examplesofpolynomialoperations. .. 21 Example 1.Polynomialoperations. 21 Example 2. Collecting coefficients. 22 Example 3. Usingalgebraforarithmetic. 22 Example 4.TheBinomialTheorem.. 23 Example 5. Substituting in a polynomial. 27 Example 6.Completingthesquare. 28 2 Solving linear equations. 31 2.1 Whatisasolution?............................ 32 2.2 One linear equation in one variable. .. 34 2.3 Several linear equations in one variable. ..... 35 2.4 Two or more equations in two variables. 36 2.5 Several equations in several variables. ..... 37 2.6 Solving linear equations efficiently. ... 38 Example 1. Manipulationofequations. 38 Example 2.Cramer’sRule........................ 39 Example 3. ExceptionstoCramer’sRule.. 41 Example 4. Cramer’s Rule with many variables. 42 3 The division algorithm and applications 45 3.1 Divisionalgorithminintegers. .. 45 Example 1. GCDcalculationinIntegers. 48 3.2 Aryabhat¯ .a algorithm: Efficient Euclidean algorithm. 49 Example 2. Kuttaka or Chinese Remainder Problem. 52 1 2 CONTENTS Example 3.MoreKuttakaproblems. 53 Example 4.Disguisedproblems. 54 3.3 Division algorithm in polynomials. .. 56 3.4 RepeatedDivision. ............................ 60 3.5 TheGCDandLCMoftwopolynomials. 62 Example 5. Aryabhat¯ .aalgorithmforpolynomials. 64 Example 6. Efficient division by a linear polynomial. 67 Example 7. Division by a quadratic polynomial. 69 4 Introduction to analytic geometry. 73 4.1 Coordinatesystems. ........................... 73 4.2 Geometry: Distanceformulas. 75 4.2.1 Connection with complex numbers. 77 4.3 Changeofcoordinatesonaline. 78 4.4 Changeofcoordinatesintheplane. 79 4.5 Generalchangeofcoordinates.. 81 4.5.1 DescriptionofIsometies. 81 4.5.2 Complex numbers and plane transformations. 82 4.5.3 Examples of complex transformations. 82 4.5.4 Examples of changes of coordinates: . 83 5 Equations of lines in the plane. 87 5.1 Parametricequationsoflines. 87 Examples. Parametricequationsoflines. 89 5.2 Meaning of the parameter t:....................... 91 Examples. Specialpointsonparametriclines.. 93 5.3 Comparison with the usual equation of a line. ... 95 5.4 Examples of equations of lines. 100 Example 1. Points equidistant from two given points. 102 Example 2.Rightangletriangles. 103 6 Special study of Linear and Quadratic Polynomials. 105 6.1 LinearPolynomials.. .. .. 105 6.2 FactoredQuadraticPolynomial. 107 Interval notation. Intervalsonrealline. 107 6.3 TheGeneralQuadraticPolynomial. 110 6.4 Examples of Quadratic polynomials. 113 7 Functions 115 7.1 Planealgebraiccurves . .. .. 115 7.2 Whatisafunction?............................ 117 7.3 Modelingafunction. ........................... 120 7.3.1 InverseFunctions.. 124 CONTENTS 3 8 The Circle 129 8.1 CircleBasics. ............................... 129 8.2 Parametricformofacircle. 131 8.3 Application to Pythagorean Triples. ... 133 Pythagorean Triples. Generationof. 133 8.4 Examples of equations of a circle. 137 Example 1. Intersectionoftwocircles.. 138 Example 1a. Complex intersection of two circles. 140 Example 2. Line joining through the intersection of two circles. 141 Example 3. Circlethroughthreegivenpoints. 142 Example 4. Exceptions to a circle through three points. 142 Example 5. Smallest circle with a given center meeting a given line. 143 Example 6. Circle with a given center and tangent to a given line. 144 Example 7. The distance between a point and a line. 145 Example 8. Half plane defined by a line. 147 9 Trigonometry 149 9.1 Trigonometric parameterization of a circle. ...... 149 Definition. TrigonometricFunctions.. 153 9.2 Basic Formulas for the Trigonometric Functions. ...... 157 9.3 Connection with the usual Trigonometric Functions. ....... 161 9.4 Importantformulas.. .. .. 162 9.5 Usingtrigonometry.. .. .. 174 9.6 ProofsI................................... 180 9.7 ProofsII. ................................. 181 10 Looking closely at a function 187 10.1 Introductoryexamples. 187 Parabola. Analysisnearitspoints.. 187 Circle. Analysisnearitspoints.. 191 10.2 Analyzing a general curve y = f(x) near a point (a, f(a)). ...... 192 10.3 The slope of the tangent, calculation of the derivative. ........ 194 10.4 Derivatives of more complicated functions. ...... 198 10.5 Generalpowerandchainrules.. 199 10.6 Using the derivatives for approximation. ..... 202 Linear Approximation. Examples. .................. 203 11 Root finding 209 11.1 Newton’sMethod ............................. 209 11.2 LimitationsoftheNewton’sMethod . 211 4 CONTENTS 12 Appendix 213 12.1 An analysis of √2asarealnumber.. 213 12.2 IdeaofaRealNumber.. 215 12.3 Summationofseries. .. .. 217 12.3.1 Abasicformula. .. .. 218 12.3.2 Usingthebasicformula. 218 12.4 On the exponential and logarithmic functions. ...... 222 12.5 Infinite series and their use. 225 12.6 Inversefunctionsbyseries . 233 12.7 Decimal expansion of a Rational Number. 238 12.8 Matrices and determinants: a quick introduction. ....... 241 Chapter 1 Review of Basic Tools. 1.1 Underlying field of numbers. Mathematics may be described as the science of manipulating numbers. The process of using mathematics to analyze the physical universe often consists of representing events by a set of numbers, converting the laws of physical change into mathematical functions and equations and predicting or verifying the physical events by evaluating the functions or solving the equations. We begin by describing various types of numbers in use. During thousands of years, mathematics has developed many systems of numbers. Even when some of these appear to be counterintuitive or artificial, they have proved to be increasingly useful in developing advanced solution techniques. To be useful, our numbers must have a few fundamental properties. We should be able to perform the four basic operations of algebra: addition • subtraction • multiplication and • division (except by 0) • and produce well defined numbers as answers. Any set of numbers having all these properties is said to be a field of numbers (or constants). Depending on our intended use, we work with different fields of numbers. Here is a description of fields of numbers that we typically use. 1. Rational numbers Q . The most natural idea of numbers comes from simple counting 1, 2, 3, and these form the set of natural numbers often denoted ··· by IN. 1 2 CHAPTER 1. REVIEW OF BASIC TOOLS. These are not yet good enough to make a field since subtraction like 2 5 is − undefined. To fix the subtraction property, we can add in the zero 0 and negative numbers. 1 This produces the set of integers, ZZ= 0, 1, 2, . { ± ± ···} These are still deficient because the division does not work. You cannot divide 1 by 2 and get an integer back. m The natural next step is to introduce the fractions n where m, n are integers and n = 0. You probably remember the explanation in terms of picking up 6 3 parts; thus 8 -th of a pizza is three of the eight slices of one pizza. We now have a natural field at hand, the so-called field of rational numbers m Q = m, n ZZ, n =0 { n | ∈ 6 } 2.