- ANALYTICAL GEOMETRY IN THE PLANE
- TRIGONOMETRIC FUNCTIONS
- ELEMENTS OF PROBABILITY THEORY
I. ANALYTICAL GEOMETRY IN THE PLANE.¶
§ 1. Pythagoras' Theorem¶
Formula for distance in a coordinate system
Coordinates and length of a vector in a coordinate system
Perpendicularity of vectors
Orientation of a plane
§ 2. Circle¶
Equation of a circle
Equation x² + y² - 2ax - 2by + c = 0
Circle determined by three points
Inequality x² + y² - 2ax - 2by + c > 0
§ 3. Line¶
Graph of a function
Graph of a polynomial f(x) = ax + b
Two basic problems about a line
Implicit equation of a line
§ 4. Some problems about a line and a first-degree polynomial of two variables.¶
Minimum and maximum of the polynomial Ax + By + C on a convex polygon
Inequality Ax + By + C > 0
Parametric equations of a line.
Conditions for perpendicularity and parallelism of lines
Distance from a point to a line
Intersection of a line and a circle. Tangent to a circle.
Bisectors of pairs of lines.
§ 5. Conic Sections¶
Set of the zeros of a second-degree polynomial of two variables
§ 6. Translation of the coordinate system with applications to conic sections¶
Translation of the coordinate system in the plane
Equation Ax² + Cy² + Dx + Ey + F = 0
Vertex equation of a conic section
Review of some facts and formulas from analytical geometry in the plane.
II. TRIGONOMETRIC FUNCTIONS¶
§ 1. Trigonometric Functions¶
- Exponential mapping of a line to a circle. Numerical circle.
- Definition of trigonometric functions.
- Evenness of cosine, oddness of sine, and periodicity of cosine and sine.
- Addition formulas for sine and cosine functions.
- Calculation and tables of trigonometric functions.
§ 2. Trigonometry¶
- Definition of angle's trigonometric functions. Measurement of angles.
- Right triangle.
- Oblique triangle.
- Some applications of trigonometry.
§ 3. Scalar Product of Vectors¶
- Scalar product of vectors.
- Orthogonal projection of a vector onto a line and a plane.
- Rotation of the coordinate system in the plane.
- Coordinate system transformations in the plane.
- Elements of analytical geometry in space.
§ 4. Graphic Representation of Trigonometric Functions¶
- Cosine wave.
- Graph of the function A sin(at + p).
- Tangent and cotangent functions.
- Converting the product of trigonometric functions into a sum. Trigonometric polynomials.
§ 5. Some Applications of Trigonometric Functions¶
- Polar coordinates.
- Equation of conic sections in polar coordinates.
- Harmonic motion.
- Euler's formula.
Overview of basic formulas for trigonometric functions.
III. COMBINATORICS. STATISTICS. ELEMENTS OF PROBABILITY THEORY¶
§ 1. Combinatorics¶
- Theorem of consecutive counting
- Permutations and variations
- Binomial theorem
§ 2. Statistics¶
- Frequency distribution
- Mean (Arithmetic mean)
- Standard deviation
- Additivity of relative frequency
§ 3. Elementary Probability Theory¶
- Probability as relative frequency
- Probability on a finite set
- 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)