Е.С.Кочетков, Алгебра және элементар функциялар 9-сынып, 1-бөлім, 1974

Chapter I. Linear equations and inequalities

§ 1. Identities

§ 2. Equations

§ 3. Linear functions and their graphs

§ 4. Linear equations

§ 5. Solving the equation mx = n graphically

§ 6. Equations reducible to linear equations

§ 7. Equations with the unknown under the absolute value sign

§ 8. Interval method

§ 9. Inequalities

§ 10. Main properties of numerical inequalities

§ 11. Termwise addition and subtraction of inequalities

§ 12. Termwise multiplication of inequalities

§ 13. Compound inequalities

§ 14. True and false inequalities

§ 15. Some methods for proving inequalities

§ 16. The theorem on arithmetic and geometric means

§ 17. Theorems on constant sum and constant product

§ 18. Approximate values of a number. Property of the absolute value of a sum

§ 19. Addition and subtraction of approximate numbers

§ 20. Equivalent inequalities and their properties

§ 21. Linear inequalities

§ 22. Solving the inequality mx > n graphically

§ 23. Systems of linear inequalities

§ 24. Fractional-linear functions

§ 25. Inequalities with the unknown under the absolute value sign

§ 26. System of two linear equations with two unknowns. Consistent and inconsistent systems

§ 27. Determinant of the second order

§ 28. Condition for a second-order determinant to be zero

§ 29. Main and auxiliary determinants of a system of two linear equations with two unknowns

§ 30. Cramer's rule

§ 31. Case when the main determinant of a system is zero and at least one auxiliary determinant is nonzero

§ 32. Case when the main determinant and both auxiliary determinants of a system are zero

§ 33. Solving a system of two linear equations with two unknowns using basic properties of equations

§ 34. Solving systems of linear equations graphically

Review problems

Chapter II. Real numbers

§ 35. Rational numbers

§ 36. Operations with rational numbers

§ 37. Geometric representation of rational numbers

§ 38. Writing rational numbers as decimal fractions

§ 39. Finding square roots of rational numbers

§ 40. Commensurable and incommensurable segments

§ 41. Length of a segment incommensurable with the unit length

§ 42. Real numbers

§ 43. Comparing real numbers

§ 44. Geometric representation of real numbers

§ 45. Decimal approximations of real numbers

§ 46. Addition of real numbers

§ 47. Multiplication of real numbers

§ 48. Subtraction and division of real numbers

Review problems

Chapter III. Quadratic trinomial

§ 49. Completing the square in a quadratic trinomial

§ 50. Quadratic equations

§ 51. Special forms of quadratic equations

§ 52. Vieta's theorem

§ 53. Studying the signs of roots of a quadratic equation by its coefficients

§ 54. Factoring a quadratic trinomial into linear factors

§ 55. Constructing a quadratic equation from given roots

§ 56. Biquadratic equations

§ 57. Graph of a quadratic function

§ 58. Examples of constructing graphs of quadratic functions

§ 59. Characteristic points of a parabola

§ 60. Extremum value of the function y = ax² + bx + c

§ 61. Quadratic inequalities

§ 62. Examples of solving quadratic inequalities

§ 63. Solving some systems of equations

§ 64. Solving some systems of equations graphically

§ 65. Irrational equations

§ 66. Examples of solving irrational equations

§ 67. From the history of algebra development

Review problems

Chapter IV. Power with rational exponent. Power function

§ 68. Power with natural exponent. Raising a product and quotient to a power

§ 69. Multiplying and dividing powers with the same base

§ 70. Comparing powers

§ 71. Powers with zero and negative exponents

§ 72. Properties of powers with integer exponents

§ 73. n = 1, 2, 3 болғандағы y = xⁿ функциясы

§ 74. The function y = xⁿ for n = −1 and n = −2

§ 76. nth root of a positive number a

§ 77. Arithmetic value of a root

§ 78. nth root of a negative number −a

§ 79. Extracting roots from a product and quotient

§ 80. Extracting a root from a power. Raising a root to a power. Extracting a root from a root

§ 81. Taking a factor out of a radical and bringing a factor under a radical

§ 82. Multiplying and dividing roots

§ 83. Rationalizing the denominator of a fraction

§ 84. Power of a positive number with a positive fractional exponent

§ 85. Main properties of powers of a positive number with positive fractional exponent

§ 86. Power of a positive number with a negative fractional exponent

§ 87. The function y = xʳ for r = 1/2 and r = −1/3

§ 88. General properties of power functions

Review problems

Chapter V. Trigonometric functions

§ 89. Concept of a vector and an axis

§ 90. Projection of a vector on an axis

§ 91. Free and bound vectors

§ 92. Coordinates of a vector on the plane

§ 93. Generalization of the concepts of angle and arc

§ 94. Theorem on the ratio of a vector's coordinates to its length

§ 95. Definition of trigonometric functions of an angle

§ 96. Trigonometric circle. Tangent and cotangent axes

§ 97. Constructing an angle from given trigonometric function values

§ 98. Values of trigonometric functions of some angles

§ 99. Evenness of trigonometric functions

§ 100. Periodicity of sin φ and cos φ

§ 101. Periodicity of tg φ and ctg φ

§ 102. On periodic functions

§ 103. Variation of sin φ and cos φ

§ 104. Variation of tg φ and ctg φ

§ 105. Tables of trigonometric function values

§ 106. Using trigonometric tables to find an angle from trigonometric function values

§ 107. Radian measure of angles and arcs

§ 108. Trigonometric functions of a numerical argument

§ 109. Algebraic relations between trigonometric functions of the same argument

§ 110. Finding other trigonometric function values of an angle from one value

§ 111. Reduction formulas

§ 112. Determining trigonometric functions of any angle using tables

§ 113. Graph of y = sin x

§ 114. Graph of y = cos x

§ 115. Graphs of y = tg x and y = ctg x

§ 116. Proving trigonometric identities

§ 117. Arcsine of a number a

§ 118. Equation sin x = a

§ 119. Arccosine of a number a

§ 120. Equation cos x = a

§ 121. Arctangent and arccotangent of a number a

§ 122. Equations tg x = a and ctg x = a

§ 123. More complex trigonometric equations

§ 124. Homogeneous equations

§ 125. Solving trigonometric equations graphically

§ 126. Trigonometric inequalities

Review problems

Chapter VI. Number sequences

§ 127. Number sequences and ways of defining them. Finite and infinite sequences

§ 128. Monotone sequences

§ 129. Bounded and unbounded number sequences

§ 130. Limit of an infinite number sequence

§ 131. Examples

§ 132. Convergent and divergent number sequences

§ 133. Monotone and bounded sequences

§ 134. The number e

§ 135. Variable quantities and their limits

§ 136. Main theorems on limits

§ 137. q модулі 1-ден кіші болғандағы lim q^n

§ 138. What is circumference?

§ 139. Formula for finding circumference

§ 140. Finding approximate values of π

§ 141. Area of a disk

§ 142. Arithmetic progression

§ 143. Property of an arithmetic progression

§ 144. Sum of terms of an arithmetic progression

§ 145. Geometric progression. Formula for the general term of a geometric progression

§ 146. Property of a geometric progression with positive terms

§ 147. Sum of terms of a geometric progression

§ 148. Sum of an infinite decreasing geometric progression

Review problems

Answers to exercises