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.

2 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. 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.

3 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.

4 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 .. 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.

5 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.

6 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.. 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.

7 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.

8 Examples of changes of coordinates: ..835 Equations of lines in the Parametric equations of lines.. 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.

9 107 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.

10 140 Example 2. 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.