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This course of Advanced Trigonometry encompasses all aspects traditionally included in studying of this subject :

Section 1. Introduction to Trigonometry

Section 2. Trigonometric Functions

Section 3. Trigonometric Identities

Section 4. Trigonometric Equations

Section 5. Trigonometric Inequalities

Section 6. Trigonometric Geometry

Section 7. Trigonometry and Complex Numbers

Section 8. Trigonometric Calculus

Each section contains certain number of video-recorded lessons, each with detail textbook-like description, which you can download and study off-line.

Most lessons are accompanied by multiple choice tests.

In addition, sections have exercises intended for you to apply your skills and knowledge to solve problems or prove theorems. In some cases these exercises contain answers for you to check your results. Important is to get these answers by yourself. Every such exercise has a subsequent review lesson, where these problems are analyzed in detail.

This introductory lecture is about the history of Trigonometry, its main subject - the angles, ways to measure them, initial definitions of simple operations with angles and basic functions of them.

In particular, the definition of trigonometric functions for angles in right triangles is presented.

Important exercises on basic concepts of trigonometry - values of trigonometric functions for main types of angles of right triangles and construction problems on using elementary trigonometric functions.

This video reviews all exercises offered for lesson Trigonometry 1.1. - Basic Concepts

This lecture introduces a concept of a unit circle on a coordinate plane.

Using this concept of a unit circle, we can define trigonometric functions SIN and COS for any angles, positive and negative, not limited to any range, measured in degrees or radians.

From functions SIN and COS we define all other trigonometric functions - TAN, COT, SEC and CSC.

We also demonstrate an equivalence of this definition of SIN and COS to a definition of these functions for angles in a right triangle as a ratio between its sides.

Illustrative exercises on Unit Circle and Full Definition of Trigonometric Functions

This video reviews all exercises offered for lesson Trigonometry 1.2. - Unit Circle

Discussed methodology of derivation of the values of all trigonometric functions for most frequently occurred angles in trigonometric problems. This methodology has a simple foundation of a concept of a unit circle, Phifagorean Theorem and simple geometric properties of triangles.

Values of trigonometric functions for angles outside the first quadrant.

Values of trigonometric functions for angles outside the first quadrant.

We present a list of simple trigonometric identities with detailed proof of each.

Based on a previous lecture, these exercises allow deeper understanding of relationship between trigonometric functions

Review of exercises with simple trigonometric identities

Connection between sides, angles and trigonometric functions of these angles in right triangles:

In patricular, the following equalities are presented:

Equalities for a cathetus:*a = c·sin α**a = c·cos β**a = b·tan α**a = b·cot β*

(with similar equalities for another cathetus)

Equalities for a hypotenuse:*c = a / sin α**c = a / cos β*

(with similar equalities for another cathetus)

Solved are simple problems related to usage of trigonometric functions to determine characteristics of right triangles.

Review of simple problems related to usage of trigonometric functions to determine characteristics of right triangles.

Major properties of function y=SIN(x) are presented and proven. Discussed a graphical representation of this function.

Major properties of function y=COS(x) are presented and proven. Discussed a graphical representation of this function.

Major properties of function y=TAN(x) are presented and proven. Discussed a graphical representation of this function.

Major properties of function y=COT(x) are presented and proven. Discussed a graphical representation of this function.

Major properties of function y=SEC(x) are presented and proven. Discussed a graphical representation of this function.

Major properties of function y=CSC(x) are presented and proven. Discussed a graphical representation of this function.

For each of six trigonometric functions (** SIN**,

*A. Is it odd, even or neither?*

*B. Is it periodical and, if yes, what is the period?*

*C. Graph it.*

*D. What kind of symmetry, if any, does its graph have?*

*E. Does it equal to zero at some points and, if it does, where?*

*F. Does it have asymptotes and, if does, where?*

*G. What are its maximum and minimums and at what values of an argument?*

Review exercises on properties of trigonometric functions

Exercises on graphic representation of trigonometric functions combined with linear graph transformation techniques

Review of exercises on graphical representation of trigonometric functions

In this lecture we introduce a general concept of an inverse functions, including the relationship between domain and range of an original function with domain and range of its inverse copunterpart. Also included the graphical representation af an inverse function.

Six inverse trigonometric functions (arcsin, arccos, arctan, arccot, arcsec, arccsc) are defined and analyzed, including their domains, ranges, properties and graphs.

Exercise on inverse trigonometric functions cover the following three issues for each such function:

1. What are the domain, co-domain (range) of a particular inverse trigonometric function?

2. Draw a graph of this function.

3. What are the values of this function for certain specific values of argument?

Review of exercises on properties of inverse trigonometric functions

Presented and rigorously proven are a few basic trigonometric identities. In particular, simple correspondence between sin(x) and cos(x), trivial for acute angles, is expanded to all angles.

Derived and proved for all types of angles a formula for cos(2x) in terms of cos(x) and/or sin(x)

In this lecture we present a proof of formula cos(x+y)=cos(x)cos(y)-sin(x)sin(y). First, geometrically, for acute angles, and then expanded to all angles in one particular case (other cases are analogous) using basic identities presented earlier in the course.

Also presented is a geometric proof for formula cos(x-y)=cos(x)cos(y)+sin(x)sin(y) for acute angles. Expansion to any angles is similar to above.

Formulas for sin, tan, cot of a sum of or a difference between angles are presented in this lecture.

Also, formulas for sin and cos of a tripple angle are addressed.

All formulas are rigorously proven.

Theese exercises are intended as an illustration of techniques used in trigonometry to simplify and transform trigonometric expressions and to assist in developing the logic behind these manipulations.

Review exercises that are intended as an illustration of techniques used in trigonometry to simplify and transform trigonometric expressions

Two conditional identities are presented that require some technical skills to be proven

Review of exercises with identities that are incrementally more involved, primarily on a technical side (need some accuracy in the proofs)

Basic trigonometric equations of type sin(x)=a, tan(x)=a etc. are presented with discussion on methodology of solving them. Simple examples are provided as an illustration.

Suggested methods to solve trigonometric equations for the following cases:

Linear transformation of an argument

Polynomial of a single trigonometric function

Function of a single trigonometric function

Equations with Mixed Trigonometric Functions

A relatively simple equation is analyzed from four different viewpoints with four solutions looking differently, by numerically identical

A few problems with trigonometric equations are offered, explained, analyzed and solved

**Problems with trigonometric equations are explained, analyzed and solved.**

A few more problems with trigonometric equations are presented, analyzed and solved.

**More problems with trigonometric equations are explained, analyzed and solved.**

Three examples of systems of two differential equations with two unknowns are presented to illustrate different methods of solving them.

**Problems with systems of trigonometric equations are explained, analyzed and solved.**

Concept of trigonometric inequality and examples of primitive inequalities

tan(x) ≤ a,

cot(x) ≤ a,

sin(x) ≤ a and

cos(x) ≤ a

Emphasis on graphical interpretation of inequalities.

Presented are the following two problems:

**Problem 1.1.**

Solve the following inequality

**sin(x)****·sin(2x) + cos(x)·cos(2x) > sin(2x) **

in the interval (**–*** π*,

**Problem 1.2.**

Solve the following inequality

**4****·****sin(x)****·sin(2x)·sin(3x) > sin(4x)**

(no restriction on argument * x*)

Presented, analyzed and solved are a few problems with trigonometric inequalities.

Three more problems on trigonometric inequalities

These analyzed and solved problem on trigonometric inequalities introduce a couple of new techniques to solve them.

**Law of Sines** is simple trigonometric identities that connect lengths of sides in a triangle with sines of opposite to these sides angles:

**a/sin(α) = b/sin(β) = c/sin(γ)**

**Law of Cosines** is an extension of Pythagorean Theorem to any triangle and connects the length of one side in a triangle with the lengths of two others and a cosine of an angle between them

**a²+b²−2·a·b·cos(γ) = c²**

Geometrical approach to prove important trigonometric inequalities:

* sin(x) < x < tan(x) *for

**cos(x) < sin(x)/x < 1 **

Four geometrical problems are presented with solutions based on trigonometry.

Four geometrical problems are presented with solutions based on trigonometry. Among them are these (and two others)

Let * H *be a point of intersection of altitudes in the triangle Δ

Express * s*(area of a triangle) in terms of

More problems that illustrate application of trigonometry to solving geometric problems

1. Prove that for any triangle the following equality is true.**a·b·cos²(γ/2) = p·(p−c)**

2. Prove that for any triangle the following equality is true.

**a·b·cos²(γ/2) = p·(p−c)**

3. Prove that the area of any triangle equals to

**s = ¼*** ·*[

4. Prove that for any triangle the following equality is true.

**8R³·h**_{a}**·h**_{b}**·h**_{c}**=a²·b²·c²**

Four more Geometry problems that can be solved using Trigonometry.

Four more Geometry problems that can be solved using Trigonometry.

Among others, the following problem is offered.

Given a parallelogram * ABCD *with an acute angle ∠

The purpose of this lesson is not only to describe the expansion of a set of real numbers towards complex ones, but to justify the necessity of this and to introduce the complex numbers in a more rigorous fashion than traditionally presented to many students.

We will follow the evolution of a concept of *number* from *natural* to *integer*, to *rational*, to *real* and explain the necessity of this evolution in a quest for symmetry and harmony of our theory. That same quest will lead us to introduction of *complex* numbers.

Basic definitions that are used to create a new set of numbers are presented in details, as well as main operations with complex numbers and laws that direct these operations.

This lesson is a more formal, abstract, more rigorous definition of *complex* numbers and operations with them.

The need for more rigorous definition of *complex* numbers lies in a simple fact that, defining any operation on two objects, we have to properly define a **set** these objects and the result of operation on them belong to.

If* r *is a

How to perform

What

These and other questions can be answered by rigorously constructing a new **set** of *complex* numbers that properly represents all *real* numbers, all *imaginary* numbers and the results of operations on them. This is a subject of this lesson.

*Vector* representation of *complex* numbers in Cartesian and polar coordinates.*Modulus* and *phase* *angle*.

DeMoivre's formula.

An important trigonometric problem that can be solved using the complex numbers and their representation in trigonometric form in polar coordinates

In this lecture we will present an example of solving trigonometric problems using the presentation of complex numbers in trigonometric form.

Calculate the following sums using** complex**

**B**** = sin(x) + sin(2x) + sin(3x) + sin(4x) +...+ sin(Nx)**

Defining the operation of raising of a *real* number into the *imaginary* power (**Euler's formula**) and proving the validity of this definition.

A few problems related to applications of Euler's formula

Analysis and solution of the few problems related to applications of Euler's formula

Brief recap of main concepts of Calculus - *Limits*, *Derivatives*, *Integrals* - for the purpose of analyzing the behavior of trigonometric functions in subsequent lessons.

Definitions and main characteristics of these concepts are duscussed in a very short format.

Three functions, ** sin(x)**,

**Welcome to Advanced Trigonometry!**

Learning is not just accumulation of information. Educational process, especially for such a subject as Mathematics, is an active two-way street. The teacher provides you with information and expects something in return – your full attention, participation, and effort, without which anything you get will soon be forgotten without any traces.

In return for your efforts you will be rewarded with Knowledge and develop your abilities to approach many new problems in a creative, analytical way, which leads to success in any field you will be working.

** **

**WHAT'S IN THE COURSE**

This course of Advanced Trigonometry contains:

8 major sections dedicated to broad spectrum of topics of Trigonometry

30 video-recorded theoretical lessons distributed among these sections (up to 30 min each)

28 multiple choice students' tests with detail explanation of correct answers available as parents' resources

19 video-recorded practical exercise lessons that offer problems and their review (up to 30 min each)

Overall, more than 20 hours of video-recorded lessons!

**IMPORTANT NOTES**

Resources, available for supervising parents, contain answers to all multiple choice tests with detail reasoning why these answers are correct. These tests are very important and we encourage all parents to make sure that the answers provided by students are not only correct, but have logical reasoning behind them similar to considerations offered in answers provided in the resources.

Practical exercises are presented as two separate items each - a description and an explanation and analysis in the following lesson. They are intended for students to apply their skills and knowledge to solve problems or prove theorems by themselves and then check it against the explanation provided in the course. In some cases these exercises contain answers in the description itself to check the results. It is important to go through these answers prior to watching the subsequent lesson with full analysis and explanation.

**COURSE GOAL**

The primary goal of this course is to present the students with rigorous definitions, properties, relationships and dependencies among different objects studied in Trigonometry.

Perhaps the most interesting part of this course is the kind of problems offered as tests or exercises. They require not only factual knowledge of the information provided in theoretical lectures, but also certain degree of creativity, ingenuity and analytical thinking and, as such, are intended to develop these skills, which will be useful in any future profession. Development of these qualities in students is an extremely important goal of this course.

**TARGET AUDIENCE**

The course is intended for high school level students who are either involved in home schooling or are just interested in deepening their mathematical knowledge as a tool to develop their creativity, logic, analytical thinking and, arguably, general intelligence.

Those students who successfully complete this course will excel not only in Trigonometry, but in other mathematical topics and all aspects of their lives that involve deep understanding, thinking and decision making.

**COURSE REQUIREMENTS AND PREREQUISITES**

Strong algebraical and geometrical skills are assumed for students studying Trigonometry. In addition, knowledge of complex numbers, limits and derivatives would be useful. For those not familiar with the concepts of complex numbers, limits and derivatives the course contains introductory lectures, which introduces these concepts to a level sufficient to understand their application to Trigonometry.

**COURSE TOPICS**

This course of Advanced Trigonometry encompasses all aspects traditionally included in studying of this subject :

Section 1. Introduction to Trigonometry

(5 theoretical video lessons with detail notes and tests, 4 exercises with full review)

Basic Concepts

Unit Circle

Basic Angles

Simple Identities

Right Triangle

Section 2. Trigonometric Functions

(8 theoretical video lessons with detail notes and tests, 3 exercises with full review)

Properties of function y=SIN(x)

Properties of function y=COS(x)

Properties of function y=TAN(x)

Properties of function y=COT(x)

Properties of function 2.5 y=SEC(x)

Properties of function y=CSC(x)

Review of Properties of Trigonometric Functions

Inverse Trigonometric Functions

Section 3. Trigonometric Identities

(4 theoretical video lessons with detail notes and tests, 2 exercises with full review)

Basic Identities

Formula for cos(2x)

Formula for cos(x+y)

Other formulas for (x+y)

Section 4. Trigonometric Equations

(3 theoretical video lessons with detail notes and tests, 3 exercises with full review)

Primitive Equations

Methodology

Four Solutions to One Problem

Section 5. Trigonometric Inequalities

(1 theoretical video lesson with detail notes and tests, 2 exercises with full review)

Overview of Trigonometric Inequalities

Section 6. Trigonometric Geometry

(3 theoretical video lessons with detail notes and tests, 3 exercises with full review)

Law of Sines

Law of Cosines

Geometrical Proofs of Trigonometric Identities

Section 7. Trigonometry and Complex Numbers

(4 theoretical video lessons with detail notes and tests, 2 exercises with full review)

Introduction to Complex Numbers

Advanced Complex Numbers

Complex Trigonometry

Euler's Formula

Section 8. Trigonometric Calculus

(2 theoretical video lessons with detail notes and tests)

Limit, Derivative, Integral

Calculus of Trigonometric Functions

- Teacher: Zinovy
- Areas of expertise: Math, Computers, Finance
- Education: Moscow Lomonosov University, School of Mathematics
- Interests: Teaching, Politics
- Skills: Specialized in Theory of Probabilities and Mathematical Statistics
- Associations:
- Issues I care about: Education, Political Science, Technology, Health

My name is Zinovy (Zor). I am a mathematician by education, a computer programmer and a financial specialist by profession and a teacher by inspiration. I was born and educated in the Soviet Union, graduated as a mathematician from a prestigious Moscow University at the time when its math school was one of the best in the world. Though, I consider my high school years to be at least as important because of a brilliant math teacher who established a solid foundation to my future math study. After graduation from university I was working in the computer industry developing software for statistical applications. At age of 30 I with my family immigrated to the United States and continued my career in New York as a computer specialist primarily in the financial industry, working for the best in the world financial information provider and then for the largest asset management company in the world. I think, many of our high schools do not provide an adequate level of knowledge in such an important subject as Mathematics and Physics. I am sure, many young students, especially those studying these subjects at home, would welcome an opportunity to deepen their knowledge, while teachers might be benefited from a course concentrated on rigorous presentation of ideas, logic and methods. After retirement I have decided to dedicate my efforts to education to transfer my knowledge of Mathematics and system of study to those young students who appreciate rigorousness, logic and creativity and cannot get it in schools for any reason. When approached by Lernsys, I realized the potential of cooperation between this company, serving as a conduit of knowledge, and a provider of quality education to homeschoolers. I hope that the high quality of educational material I can provide, combined with publishing capabilities of Lernsys, allowing it to distribute these materials to a broad audience of students, will open a new page in the system of education.

Each document in this resource contains answers to corresponding test questions.