Trigonometry. Gelfand I.M., Lvovsky S.M., Toom A.L.

M.: 2003. - 200 p.

Approved by the Ministry of Education of the Russian Federation as a teaching aid in trigonometry for students in grades 10 of general education institutions.

This book, written by a group of authors under the leadership of one of the greatest mathematicians of the 20th century, Academician I.M. Gelfand, is designed to refute the popular opinion about trigonometry as a boring and incomprehensible section of the school mathematics course. The reader is invited to look at a familiar subject in a new way. The presentation, accompanied by a large number of tasks, starts “from scratch” and reaches material that goes quite far beyond the scope of the school curriculum; trigonometric formulas are illustrated with examples from physics and geometry.

A separate chapter is devoted to typical methods for solving trigonometric problems offered in entrance exams to higher educational institutions.

The book will be an indispensable assistant for high school students, teachers, parents and anyone interested in mathematics.

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Table of contents
1. First introduction to trigonometry 7
§ 1. How to measure steepness 7
1.1. Sine 7
1.2. Measuring angles 9
§ 2. Tangent 11
§ 3. Cosine 13
§ 4. Small angles 16
2. Initial properties of trigonometric functions 21
§ 5. Clocks, or a modern view of trigonometry. 21
5.1. Clocks and processes 21
5.2. Speed ​​24
§ 6. Definition of trigonometric functions 26
6.1. Tangent axis 31
6.2. Signs of trigonometric functions 32
§ 7. The simplest formulas 34
§ 8. Periods of trigonometric functions 36
§ 9. Reduction formulas 40
§ 10. The simplest trigonometric equations.... 45
§ 11. Graphs of sine and cosine 55
§ 12. Graphs of tangent and cotangent 62
§ 13. What is sin x + cos x 65?
3. Solving triangles 67
§ 14. Cosine theorem 67
§ 15. Around the area of ​​a triangle 71
§ 16. Theorem of sines 76
4. Addition formulas and their consequences 81
§ 17. Vectors 81
17.1. Directed segments and vectors 81
17.2. Vector addition 87
17.3. Subtraction and multiplication by the number 90
17.4. About vectors in physics 94
§ 18. Dot product 95
§ 19. Trigonometric addition formulas 99
§ 20. Auxiliary angle formula, or addition of oscillations of equal frequency 105
§ 21. Double, triple and half angles 111
§ 22. Converting a product into a sum and a sum into a product 118
§ 23. Derivatives of trigonometric functions.... 126
5. Trigonometry for applicants 137
§ 24. How to solve trigonometric equations.... 137
§ 25. Selection of numbers on the trigonometric circle 151
§ 26. How to solve trigonometric inequalities. . . 159
§ 27. Repetition problems 165
6. Complex numbers 168
§ 28. What are complex numbers 168
§ 29. Modulus and argument of the complex number 173
§ 30. Exponential function and Euler's formula 182
Answers and directions to some problems 189
Subject index 196

This book, written by a group of authors under the leadership of Academician I. M. Gelfand, one of the greatest mathematicians of the 20th century, is designed to refute the popular opinion about trigonometry as a boring and incomprehensible section of the school mathematics course. The reader is invited to look at a familiar subject in a new way. The presentation, accompanied by a large number of tasks, starts “from scratch” and reaches material that goes quite far beyond the scope of the school curriculum; trigonometric formulas are illustrated with examples from physics and geometry. A separate chapter is devoted to typical methods for solving trigonometric problems offered in entrance exams to higher educational institutions. The book will be an indispensable assistant for high school students, teachers, parents and anyone interested in mathematics.

On our website you can download the book "Trigonometry" Israel Gelfand, Sergey Lvovsky, Andrey Toom for free and without registration in fb2, rtf, epub, pdf, txt format, read the book online or buy the book in the online store.

Name: Trigonometry. 2003.

This book, written by a group of authors under the leadership of one of the greatest mathematicians of the 20th century, Academician I.M. Gelfand, is designed to refute the popular opinion about trigonometry as a boring and incomprehensible section of the school mathematics course. The reader is invited to look at a familiar subject in a new way. The presentation, accompanied by a large number of tasks, starts “from scratch” and reaches material that goes quite far beyond the scope of the school curriculum; trigonometric formulas are illustrated with examples from physics and geometry.
A separate chapter is devoted to typical methods for solving trigonometric problems offered in entrance exams to higher educational institutions.
The book will be an indispensable assistant for high school students, teachers, parents and anyone interested in mathematics.

What is trigonometry? Almost all high school students will say boring and useless formulas. However, we want to dissuade you from this.
To take a fresh look at trigonometry, we talk about it from scratch. Therefore, it is better to read the manual from the very beginning and consecutively, although you will need some information. Most likely, you already know.
Our definitions are equivalent to those from school textbooks, but do not always coincide verbatim with them.
There is no need to strive to solve all the problems from the book (we deliberately placed them with a reserve), but it is worth solving a number of problems after each paragraph. If the tasks for a paragraph don’t work out at all, it means you haven’t learned something, and it makes sense to re-read this paragraph.
More difficult problems are marked with an asterisk, more difficult text is printed in small font. You can skip all this on your first reading.

Table of contents
1. First introduction to trigonometry 7
§ 1. How to measure steepness 7
1.1. Sine 7
1.2. Measuring angles 9
§ 2. Tangent 11
§ 3. Cosine 13
§ 4. Small angles 16
2. Initial properties of trigonometric functions 21
§ 5. Clocks, or a modern view of trigonometry 21
5.1. Clocks and processes 21
5.2. Speed ​​24
§ 6. Definition of trigonometric functions 26
6.1. Tangent axis 31
6.2. Signs of trigonometric functions 32
§ 7. The simplest formulas 34
§ 8. Periods of trigonometric functions 36
§ 9. Reduction formulas 40
§ 10. The simplest trigonometric equations 45
§ 11. Graphs of sine and cosine 55
§ 12. Graphs of tangent and cotangent 62
§ 13. What is sin x + cos x 65?
3. Solving triangles 67
§ 14. Cosine theorem 67
§ 15. Around the area of ​​a triangle 71
§ 16. Theorem of sines 76
4. Addition formulas and their consequences 81
§ 17. Vectors 81
17.1. Directed segments and vectors 81
17.2. Vector addition 87
17.3. Subtraction and multiplication by the number 90
17.4. About vectors in physics 94
§ 18. Dot product 95
§ 19. Trigonometric addition formulas 99
§ 20. Auxiliary angle formula, or addition of oscillations of equal frequency 105
§ 21. Double, triple and half angles 111
§ 22. Converting a product into a sum and a sum into a product 118
§ 23. Derivatives of trigonometric functions 126
5. Trigonometry for applicants 137
§ 24. How to solve trigonometric equations 137
§ 25. Selection of numbers on the trigonometric circle 151
§ 26. How to solve trigonometric inequalities 159
§ 27. Repetition problems 165
6. Complex numbers 168
§ 28. What are complex numbers 168
§ 29. Modulus and argument of the complex number 173
§ 30. Exponential function and Euler's formula 182
Answers and directions to some problems 189
Subject index 196

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