Precalculus with Geometry and Trigonometry

1 Precalculus with Geometry andTrigonometryby Avinash Sathaye, Professor of Mathematics1 Department of Mathematics, University of Kentucky book may be freely downloaded for personal use from the author s web commercial use must be preauthorized by the an email to for 23, 20091 Partially supported by NSF grant thru AMSP(Appalachian Math Science Partnership) book onPrecalculus with Geometry and Trigonometryshould betreated as simply an enhanced version of our book onCollege Algebra. Most ofthe topics that appear here have already been discussed in the Algebra book andoften the text here is a verbatim copy of the text in the other expect the student to already have a strong Algebraic background and thus thealgebraic techniques presented here are more a refresher course than a firstintroduction. We also expect the student to be heading for higher level mathematicscourses and try to supply the necessary connections and motivations for future is what is new in this book.

2 In contrast with the Algebra book, we make a more extensive use of complexnumbers. We use Euler s representation of complex numbers as well as theArgand diagrams extensively. Even though these are described and shown tobe useful, we do not yet have tools to prove these techniques properly. Theyshould be used as motivation and as an easy method to rememberthetrigonometric results. We have supplied a brief introduction to matrices and determinants. The ideais 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 realnumbers. While this is not necessary for calculations in this course, it is vitalfor understanding the finer concepts of Calculus which will be introduced inhigher courses. We have also included an appendix discussing summation of series - bothfinite and infinite, as well as a discussion of power series. While details ofconvergence are left out, this should generate familiaritywith futuretechniques and a better feeling for the otherwise mysterious trigonometric andexponential Review of Basic field of.

3 Working with Complex Numbers.. Indeterminates, variables, parameters .. Basics of Polynomials .. Rational functions.. Working with polynomials .. Examples of polynomial operations.. 21 Example 1. Polynomial operations.. 21 Example 2. Collecting coefficients.. 22 Example 3. Using algebra for arithmetic.. 22 Example 4. The Binomial .. 23 Example 5. Substituting in a polynomial.. 27 Example 6. Completing the square.. 282 Solving linear What is a solution? .. One linear equation in one variable.. Several linear equations in one variable.. Two or more equations in two variables.. Several equations in several variables.. Solving linear equations efficiently.. 38 Example 1. Manipulation of equations.. 38 Example 2. Cramer s Rule.. 39 Example 3. Exceptions to Cramer s Rule.. 41 Example 4. Cramer s Rule with many variables.. 423 The division algorithm and Division algorithm in integers.

4 45 Example 1. GCD calculation in Integers.. algorithm: Efficient Euclidean algorithm.. 49 Example 2. Kuttaka or Chinese Remainder Problem.. 5212 CONTENTSE xample 3. More Kuttaka problems.. 53 Example 4. Disguised problems.. Division algorithm in polynomials.. Repeated Division.. The GCD and LCM of two polynomials.. 62 Example 5. algorithm for polynomials.. 64 Example 6. Efficient division by a linear polynomial.. 67 Example 7. Division by a quadratic polynomial.. 694 Introduction to analytic Coordinate systems.. Geometry : Distance formulas.. Connection with complex numbers.. Change of coordinates on a line.. Change of coordinates in the plane.. General change of coordinates.. Description of Isometies.. Complex numbers and plane transformations.. Examples of complex transformations.. Examples of changes of coordinates: ..835 Equations of lines in the Parametric equations of lines.

5 87 Examples. Parametric equations of lines.. Meaning of the parametert: .. 91 Examples. Special points on parametric lines.. Comparison with the usual equation of a line.. Examples of equations of lines.. 100 Example 1. Points equidistant from two given points.. 102 Example 2. Right angle triangles.. 1036 Special study of Linear and Quadratic Linear Polynomials.. Factored Quadratic Polynomial.. 107 Interval on real line.. The General Quadratic Polynomial.. Examples of Quadratic polynomials.. 1137 Plane algebraic curves .. What is a function? .. Modeling a function.. Inverse Functions.. 124 CONTENTS38 The Circle Basics.. Parametric form of a circle.. Application to Pythagorean Triples.. 133 Pythagorean of.. Examples of equations of a circle.. 137 Example 1. Intersection of two circles.. 138 Example 1a. Complex intersection of two circles.. 140 Example 2.

6 Line joining through the intersection of two circles.. 141 Example 3. Circle through three given points.. 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.. 1479 Trigonometric parameterization of a circle.. 149 Definition. Trigonometric Functions.. Basic Formulas for the Trigonometric Functions.. Connection with the usual Trigonometric Functions.. Important formulas.. Using Trigonometry .. Proofs I.. Proofs II.. 18110 Looking closely at a Introductory examples.. near its points.. near its points.. Analyzing a general curvey=f(x) near a point (a, f(a)).. The slope of the tangent, calculation of the derivative.. Derivatives of more complicated functions.

7 General power and chain .. Using the derivatives for approximation.. 202 Linear .. 20311 Root Newton s Method .. Limitations of the Newton s Method .. 2114 CONTENTS12 An analysis of 2 as a real number.. Idea of a Real Number.. Summation of series.. A basic formula.. Using the basic formula.. On the exponential and logarithmic functions.. Infinite series and their use.. Inverse functions by series .. Decimal expansion of a Rational Number.. Matrices and determinants: a quick introduction.. 241 Chapter 1 Review of Basic field of may be described as the science of process of using mathematics to analyze the physical universe often consists ofrepresenting events by a set of numbers, converting the lawsof physical change intomathematical functions and equations and predicting or verifying the physicalevents by evaluating the functions or solving the begin by describing various types of numbers in use.

8 During thousands of years,mathematics has developed many systems of numbers. Even when some of theseappear to be counterintuitive or artificial, they have proved to be increasingly usefulin developing advanced solution be useful, our numbers must have a few fundamental should be able to performthe four basic operations of algebra: addition subtraction multiplication and division (except by 0)and produce well defined numbers as set of numbers having all these properties is said to bea fieldof numbers (orconstants). Depending on our intended use, we work with different fields of is a description of fields of numbers that we typically most natural idea of numbers comes from simplecounting 1,2,3, and these form the set of natural numbers often denotedby 1. REVIEW OF BASIC are not yet good enough to make a field since subtractionlike 2 5 isundefined. To fix the subtraction property, we can add in the zero 0 andnegative produces the set of integers,ZZ ={0, 1, 2, }.

9 These are still deficient because the division does not cannot divide1 by 2 and get an integer natural next step is to introduce the fractionsmnwherem, nare integersandn6= 0. You probably remember the explanation in terms of picking upparts; thus38-th of a pizza is three of the eight slices of one now have a natural field at hand, the so-called field of rational numbersQ={mn|m, n ZZ, n6= 0} numbers .One familiar way of thinking of numbers is as decimalnumbers, say something like which is nothing but a rational number234567100,000whose denominator is a power of simple sounding idea took several hundred years to develop and be accepted, because theidea of a negative count is hard to imagine. If we think of a number representing money owned,then a negative number can easily be thought of as money owed!The concept of negative numbersand zero was developed and expanded in India where a negativenumber is called which alsomeans debt!

10 The word used for a positive number, is similarlydhana which means point is that even though the idea of certain numbers sounds unrealistic or impossible, oneshould keep an open mind and accept and use them as needed. They can be useful and somebodymay find a good interpretation for them some , the number 10 can and is often replaced by other convenient numbers. The computerscientists often prefer 2 in place of 10, leading to the binary numbers, or they also use 8 or 16 inother contexts, leading to octal or hexadecimal Number Theory, it is customary to replace 10 by some primepand study the is also possible and sometimes convenient to choose a product of suitable numbers, rather thanpower of a single is interesting to consider numbers of the forma1+a22!+a33!+ wherea1is an arbitrary integer, while 0 a2<2, 0 a3<3 and so on. Thus, you can verify that:179= 1 +12!+23!+14!+15!+46! FIELD OF , rational numbers whose denominators are powers of 10 are calleddecimal calculator, especially a primitive one, deals exclusively with such quickly realizes that even something as simple as13runs into problems ifwe insist on using only decimal numbers.