Math lessons

CONTENT

• ANALYTICAL GEOMETRY IN THE PLANE
• TRIGONOMETRIC FUNCTIONS
• COMBINATORICS
• STATISTICS
• ELEMENTS OF PROBABILITY THEORY

I. ANALYTICAL GEOMETRY IN THE PLANE.

§ 1. Pythagoras' Theorem

1. Formula for distance in a coordinate system

2. Coordinates and length of a vector in a coordinate system

3. Perpendicularity of vectors

4. Orientation of a plane

§ 2. Circle

1. Equation of a circle

2. Equation x² + y² - 2ax - 2by + c = 0

3. Circle determined by three points

4. Inequality x² + y² - 2ax - 2by + c > 0

§ 3. Line

1. Graph of a function

2. Graph of a polynomial f(x) = ax + b

3. Two basic problems about a line

4. Implicit equation of a line

§ 4. Some problems about a line and a first-degree polynomial of two variables.

1. Minimum and maximum of the polynomial Ax + By + C on a convex polygon

2. Inequality Ax + By + C > 0

3. Parametric equations of a line.

4. Conditions for perpendicularity and parallelism of lines

5. Distance from a point to a line

6. Intersection of a line and a circle. Tangent to a circle.

7. Bisectors of pairs of lines.

§ 5. Conic Sections

1. Ellipse

2. Hyperbola

3. Parabola

4. Set of the zeros of a second-degree polynomial of two variables

§ 6. Translation of the coordinate system with applications to conic sections

1. Translation of the coordinate system in the plane

2. Equation Ax² + Cy² + Dx + Ey + F = 0

3. Vertex equation of a conic section

4. Equilateral hyperbola.

Review of some facts and formulas from analytical geometry in the plane.

II. TRIGONOMETRIC FUNCTIONS

§ 1. Trigonometric Functions

1. Exponential mapping of a line to a circle. Numerical circle.
2. Definition of trigonometric functions.
3. Evenness of cosine, oddness of sine, and periodicity of cosine and sine.
4. Addition formulas for sine and cosine functions.
5. Calculation and tables of trigonometric functions.

§ 2. Trigonometry

1. Angle.
2. Definition of angle's trigonometric functions. Measurement of angles.
3. Right triangle.
4. Oblique triangle.
5. Some applications of trigonometry.

§ 3. Scalar Product of Vectors

1. Scalar product of vectors.
2. Orthogonal projection of a vector onto a line and a plane.
3. Rotation of the coordinate system in the plane.
4. Coordinate system transformations in the plane.
5. Elements of analytical geometry in space.

§ 4. Graphic Representation of Trigonometric Functions

1. Sinusoid.
2. Cosine wave.
3. Graph of the function A sin(at + p).
4. Tangent and cotangent functions.
5. Converting the product of trigonometric functions into a sum. Trigonometric polynomials.

§ 5. Some Applications of Trigonometric Functions

1. Polar coordinates.
2. Equation of conic sections in polar coordinates.
3. Harmonic motion.
4. Euler's formula.

Overview of basic formulas for trigonometric functions.

III. COMBINATORICS. STATISTICS. ELEMENTS OF PROBABILITY THEORY

§ 1. Combinatorics

1. Introduction
2. Theorem of consecutive counting
3. Permutations and variations
4. Combinations
5. Binomial theorem

§ 2. Statistics

1. Introduction
2. Frequency distribution
3. Mean (Arithmetic mean)
4. Standard deviation
5. Additivity of relative frequency

§ 3. Elementary Probability Theory

1. Introduction
2. Probability as relative frequency
3. Probability on a finite set
4. Cartesian product of the probability space

Overview of some formulas from combinatorics, statistics, and probability

Solutions to the exercises

Appendix: Reference tables (logarithms, 10% function, trigonometric functions, conversion of degrees to radians)