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The Problem Solvers cover material ranging from the elementary to the advanced and make excellent review books and textbook ...
The Problem Solvers cover material ranging from the elementary to the advanced and make excellent review books and textbook companions.
The Algebra & Trigonometry Problem Solver is the perfect resource for any class, any exam, and any problem!
WHAT THIS BOOK IS FOR
Students have generally found algebra and trigonometry difficult subjects to understand and learn. Despite the publication of hundreds of textbooks in this field, each one intended to provide an improvement over previous textbooks, students of algebra and trigonometry continue to remain perplexed as a result of numerous subject areas that must be remembered and correlated when solving problems. Various interpretations of algebra and trigonometry terms also contribute to the difficulties of mastering the subject.
In a study of algebra and trigonometry, REA found the following basic reasons underlying the inherent difficulties of both math subjects:
No systematic rules of analysis were ever developed to follow in a step-by-step manner to solve typically encountered problems. This results from numerous different conditions and principles involved in a problem that leads to many possible different solution methods. To prescribe a set of rules for each of the possible variations would involve an enormous number of additional steps, making this task more burdensome than solving the problem directly due to the expectation of much trial and error.
Current textbooks normally explain a given principle in a few pages written by a mathematics professional who has insight into the subject matter not shared by others. These explanations are often written in an abstract manner that causes confusion as to the principle's use and application. Explanations then are often not sufficiently detailed or extensive enough to make the reader aware of the wide range of applications and different aspects of the principlebeing studied. The numerous possible variations of principles and their applications are usually not discussed, and it is left to the reader to discover this while doing exercises. Accordingly, the average student is expected to rediscover that which has long been established and practiced, but not always published or adequately explained.
The examples typically following the explanation of a topic are too few in number and too simple to enable the student to obtain a thorough grasp of the involved principles. The explanations do not provide sufficient basis to solve problems that may be assigned for homework or given on examinations.
Poorly solved examples such as these can be presented in abbreviated form which leaves out much explanatory material between steps, and as a result requires the reader to figure out the missing information. This leaves the reader with an impression that the problems and even the subject are hard to learn - completely the opposite of what an example is supposed to do.
Poor examples are often worded in a confusing or obscure way. They might not state the nature of the problem or they present a solution, which appears to have no direct relation to the problem. These problems usually offer an overly general discussion - never revealing how or what is to be solved.
Many examples do not include accompanying diagrams or graphs, denying the reader the exposure necessary for drawing good diagrams and graphs. Such practice only strengthens understanding by simplifying and organizing algebra and trigonometry processes.
Students can learn the subject only by doing the exercises themselves and reviewing them in class, obtaining experience in applying the principles with their different ramifications.
In doing the exercises by themselves, students find that they are required to devote considerable more time to algebra and trigonometry than to other subjects, because they are uncertain with regard to the selection and application of the theorems and principles involved. It is also often necessary for students to discover those "tricks" not revealed in their texts (or review books) that make it possible to solve problems easily. Students must usually resort to methods of trial and error to discover these "tricks," therefore finding out that they may sometimes spend several hours to solve a single problem.
When reviewing the exercises in classrooms, instructors usually request students to take turns in writing solutions on the boards and explaining them to the class. Students often find it difficult to explain in a manner that holds the interest of the class, and enables the remaining students to follow the material written on the boards. The remaining students in the class are thus too occupied with copying the material off the boards to follow the professor's explanations.
This book is intended to aid students in algebra and trigonometry overcome the difficulties described by supplying detailed illustrations of the solution methods that are usually not apparent to students. Solution methods are illustrated by problems that have been selected from those most often assigned for class work and given on examinations. The problems are arranged in order of complexity to enable students to learn and understand a particular topic by reviewing the problems in sequence. The problems are illustrated with detailed, step-by-step explanations, to save the students large amounts of time that is often needed to fill in the gaps that are usually found between steps of illustrations in textbooks or review/outline books.
The staff of REA considers algebra and trigonometry subjects that are best learned by allowing students to view the methods of analysis and solution techniques. This learning approach is similar to that practiced in various scientific laboratories, particularly in the medical fields.
In using this book, students may review and study the illustrated problems at their own pace; students are not limited to the time such problems receive in the classroom.
When students want to look up a particular type of problem and solution, they can readily locate it in the book by referring to the index that has been extensively prepared. It is also possible to locate a particular type of problem by glancing at just the material within the boxed portions. Each problem is numbered and surrounded by a heavy black border for speedy identification.
Chapter 1: Fundamental Algebraic Laws and Operations
Chapter 2: Least Common Multiple / Greatest Common Divisor
Chapter 3: Sets and Subsets
Chapter 4: Absolute Values
Chapter 5: Operations with Fractions
Chapter 6: Base, Exponent, Power
Chapter 7: Roots and Radicals
Simplification and Evaluation of Roots
Rationalizing the Denominator
Operations with Radicals
Chapter 8: Algebraic Addition, Subtraction, Multiplication, Division
Chapter 9: Functions and Relations
Chapter 10: Solving Linear Equations
Unknown in Numerator
Unknown in Numerator and/or Denominator
Unknown Under Radical Sign
Chapter 11: Properties of Straight Lines
Slopes, Intercepts, and Points of Given Lines
Finding Equations of Lines
Chapter 12: Linear Inequalities
Solving Inequalities and Graphing
Inequalities with Two Variables
Inequalities Combined with Absolute Values
Chapter 13: Systems of Linear Equations and Inequalities
Solving Equations in Two Variables and Graphing
Solving Equations in Three Variables
Solving Systems of Inequalities and Graphing
Chapter 14: Determinants and Matrices
Determinants of the Second Order
Determinants and Matrices of Third and Higher Order
Chapter 15: Factoring Expressions and Functions
Chapter 16: Solving Quadratic Equations by Factoring
Equations with Radicals
Solving by Completing theSquare
Chapter 17: Solutions by Quadratic Formula
Coefficients with Integers, Fractions, Radicals, and Variables
Interrelationships of Roots: Sums; Products
Determining the Character of Roots
Chapter 18: Solving Quadratic Inequalities
Chapter 19: Graphing Quadratic Equations / Conics and Inequalities
Circles, Ellipses, and Hyberbolas
Chapter 20: Systems of Quadratic Equations
Quadratic/Quadratic (Conic) Combinations
Chapter 21: Equations and Inequalities of Degree Greater than Two
Chapter 22: Progressions and Sequences
Chapter 23: Mathematical Induction
Chapter 24: Factorial Notation
Chapter 25: Binomial Theorem / Expansion
Chapter 26: Logarithms and Exponentials
Functions and Equations
Chapter 27: Trigonometry
Angles and Trigonometric Functions
Chapter 28: Inverse Trigonometric Functions
Chapter 29: Trigonometric Equations
Finding Solutions to Equations
Proving Trigonometric Identities
Chapter 30: Polar Coordinates
Chapter 31: Vectors and Complex Numbers
Rectangular and Polar/Trigonometric Forms of Complex Numbers
Operations with Complex Numbers
Chapter 32: Analytic Geometry
Points of Line Segments
Distances Between Points and in Geometrical Configurations
Circles, Arcs, and Sectors
Chapter 33: Permutations
Chapter 34: Combinations
Chapter 35: Probability
Chapter 36: Series
Chapter 37: Decimal / Factional Conversions / Scientific Notation
Chapter 38: Areas and Perimeters
Chapter 39: Angles of Elevation, Depression and Azimuth
Chapter 40: Motion
Chapter 41: Mixtures / Fluid Flow
Chapter 42: Numbers, Digits, Coins, and Consecutive Integers
Chapter 43: Age and Work
Chapter 44: Ratio, Proportions, and Variations
Ratios and Proportions
Joint and Combined Direct-Inverse Variation
Chapter 45: Costs
Chapter 46: Interest and Investments
Chapter 47: Problems in Space