Algebra
This is about the fundamental ideas of Algebra, understanding why and how Algebra works.
The focus is on general methods for solving algebraic equations allows one to know how to solve any problem.
Algebraic operations are used modify the form of the equation by making the same changes to both sides of =. The equality is not upset if the same changes are made to both sides of =.
A polynomial in one variable x is defined and its essential properties are discussed.
The Remainder Theorem is explained.
Methods finding factors of polynomials and Newton’s method for finding zeros of polynomial (factors) are presented.
Cramer’s Rule is the straightforward way to find solutions by determinants of algebraic equations. How to find solutions of linear equations by addition, subtraction or substitution is explained. The formula solving quadratic equations is derived.
The Binomial Theorem shows how to expand (a+x)n when a and n are any numbers, positive, negative, integral or fractional. When a=1 expansions of (1+x)n provide useful approximations.
The Exponential bx and Logarithmic Functions logb x were created to solve problems not solvable by known functions.
Many problems are simplified when a rational function, the ratio of two polynomials, is decomposed into a sum of partial fractions with denominators of lower degree.
The mathematical induction method of proof by induction has many uses such as proving theorems.
Arithmetic and geometric progressions are discussed.
Matrix algebra allows one to write and process equations efficiently.
The text In this modest mathematics text, we have tried hard to write in plain English. We do not use the phrase this is obvious for a good reason. Nothing is obvious to a person learning any subject.
Most Algebra books are forests full of trees that make it very difficult for the reader to know what is important in a forest. This text contains very few trees.
Trigonometry
Trigonometric functions are periodic. This is why they are immensely important.
Originally, the primary object of the mathematics of plane trigonometry was a study of the relationships between sides and angles of a triangle, which evolved into definitions of trigonometric functions and a method of solving plane triangles. Today solving plane triangles only play a minor role in modern engineering and scientific activities.
A plane triangle has three sides and three angles in a plane. If the values of any three of these six parts is a given, then solving the triangle means finding the values of the other three parts. In order to solve triangles the functions of angular magnitude, sine and cosine, were introduced. Later Plane Trigonometry was extended to include investigation of the properties of functions in addition to the sine and cosine, such as tangent, cotangent, secant, cosecant, and their inverses.
Angles are defined, and the degree and radian units are explained.
Pythagoras’ Theorem is explained and the Pythagorean Identities are derived.
Solving Right Triangles is a straightforward process – 4 cases.
Triangle Laws allow for straightforward solutions of oblique triangles.
In order to work with the trigonometric functions of any angle the trigonometric sine and cosine functions are defined by using a circle of radius 1.
The Inverse Trigonometric Functions are defined. The angle of a trigonometric function is the inverse of that function.
Equations for circular functions of two or more angles are derived.
The derivatives of the sine and cosine are derived.
The integrals of the sine and cosine are derived.
The cosh and sinh hyperbolic functions are derived.
