## 8.11.2016

### Multiplication

Subject of occupations:
TRIGONOMETRIC FUNCTIONS IN A RECTANGLE
Subject of the previous lesson
Variable units of measure

Lesson 8

Multiplication

As a result of multiplication of two different quantities the third quantity turns out. At multiplication of change occurs both in a number domain, and in the field of units of measure. Multiplication is possible only for perpendicular quantities. Multiplication reflects high-quality changes of quantities.

Multiplication of the parties of a rectangle allows to receive the area.

 Multiplication of the parties of a rectangle

For transition from the area of a rectangle to the simple area it is necessary to divide the square into itself. In this case two options of algebraic transformations are possible. It is necessary to highlight that the recommendation of mathematicians about reduction of identical quantities in numerator and a denominator of fraction is consciously controlled for identification of quantity of the received result. The first option of transformations looks as follows.

 First option of the unit area

The tangent and cotangent are connected by the inverse proportion. The parties of a rectangle can change somehow, from infinitely big quantity to infinitesimal. If change of the parties is carried out with keeping of the inverse proportion, then the area of a rectangle will remain invariable.

The second option of transformations is presented in an algebraic form and in physical units of measure.

 Second option of the unit area

This transformation shows that as a result of multiplication of two different units the third unit turns out. Multiplication is an interaction of two perpendicular units of measure as a result of which the new unit of measure turns out. As any quantity is result of multiplication of number and a unit of measure, under figure "one" can the number, a unit of measure or result of their interaction is meant.

Perpendicular units of measure are not connected among themselves in any way and can have the arbitrariest scale. The scale of units of measure of factors scales work units of measure, but does not influence a process essence – multiplication leads to high-quality change of initial units of measure.

The area is a result of interaction of two perpendicular measurements of length. If to multiplication inches by meters or millimeters on meters, then the area will be expressed in inches on meter or millimeters on meter. Traditionally, the area can be expressed in identical units of measure of length and width. In practice frames with different scales on a vertical and a horizontal are often used.

At determination of the area of a square the size of the party can be built in the second degree. But it does not mean at all that the party of a square is multiplied by itself. The area can still be determined only by multiplication of length to width, just at a square they have identical numerical values. We never multiply rectangle length by length or width on width because the result of such actions does not make sense. The involution is a multiplication of the different perpendicular quantities having identical numerical values and units of measure.

Algebraic transformations of multiplication of two to a square can be written down so:

 Algebraic transformations

If to consider that quantity a at a quadrating is multiplied by itself, then the inverse transformation of a square in work of two sums will be impossible. As it is possible to execute a similitude, is shown in the partition Decomposition of sume below.

Diagonal of a rectangle and angle between the diagonal and the party are padding characteristics of the interacting quantities.

At the following lesson we will consider
Examples of multiplication

## 8.08.2016

### Variable units of measure

Subject of occupations:
TRIGONOMETRIC FUNCTIONS IN A RECTANGLE
Subject of the previous lesson
Quantity

Lesson 7

Variable units of measure

Example of variable units of measure are exchange rates of the different countries. The main world currencies are in continuous fluctuation. They are connected by laws of the inverse symmetry – rise in price of one currency means reduction in cost of other currency.

One more example. Why people never celebrate such significant day how "The Middle of Life"? Because nobody knows the precise duration of characteristic life. After death such important parameter as "life expectancy", ceases to change. Similar indeterminacy is characteristic of all living beings.

If to take duration of existence of an alive organism for unit, then the description in variable units of measure will be the most precise mathematical description of any life. As the beginning of this unit of measure it is necessary to consider the conception moment (at sexual manifolding), the end is the death. It is necessary to notice that increase in life expectancy of concrete organisms is not a priority in natural selection.

Quantity, the back symmetric life expectancy, the age is. The age allows to express the variable characterizing life in constant units of measure of time.

From the point of view of mathematics, emergence of life can be considered as transition from constant units of measure (inanimate nature) to variables (wildlife).

At the following lesson we will consider
Multiplication

## 8.05.2016

### Quantity

Subject of occupations:
TRIGONOMETRIC FUNCTIONS IN A RECTANGLE
Subject of the previous lesson
The infinite trigonometric functions

Lesson 6

Quantity

 Tangent and cotangent

Mathematical interpretation of the drawing given above will look as follows.

 Quantity

Both options of representation of quantity are absolutely equal and yield identical result. On a numerical axis the parity of numbers and units of measure looks so.

 Numbers on a straight line

In this case zero and unit divided into zero act as the horizons which cannot be reached by means of numbers. For further transformation of quantity to a look habitual to us, to the abstract mathematical concepts it is necessary to enter homocentrism elements.

If we consider that across units of measure settle down, and numbers are perpendicular them, then quantity will have two options of representation.

 Two options of quantity

The numerical axis takes the following form.

 Numerical axis with units of measure

Introduction of the following element of a homocentrism allows to pass to habitual representation of quantity. If we consider that the unit of measure always remains to a constant, then for the adequate description of quantity it is necessary to enter inverse numbers. The inverse symmetry of numbers is result of transition from variable units of measure to constants.

 Quantity in constant units of measure

The numerical axis will be transformed as follows.

 Numbers and inverse numbers

In this case the numerical axis is represented without imposing of a reflecting symmetry. Unit acts as a point of the inverse symmetry. By the inverse symmetry numbers and units of measure in any quantity are connected. At invariable quantity, decrease of number leads to increase in a unit of measure, increase in number – to decrease of a unit of measure. The algebra of similitudes looks so.

 Decrease of number Increase in number

Here elements, the falling into number domains, are represented in round brackets. Elements, the falling into areas of units of measure, are represented in square brackets.

At the following lesson we will consider
Variable units of measure

## 8.03.2016

### The infinite trigonometric functions

Subject of occupations:
TRIGONOMETRIC FUNCTIONS IN A RECTANGLE
Subject of the previous lesson
Terminating trigonometric functions

Lesson 5

THE INFINITE TRIGONOMETRIC FUNCTIONS

If to divide rectangle elements with simple diagonal into terminating trigonometric functions, the infinite trigonometric functions will turn out.

 Transition from terminating trigonometric functions to the infinite

The tangent and cotangent can be presented as the party of a rectangle when as a unit of measure of length length of the perpendicular party is accepted. This geometrical representation of a tangent and cotangent can be considered as visual display of interaction between numbers (the infinite trigonometric functions) and units of measure (unit) as a result of which size appears. Other option of geometrical representation of size is shown in earlier published work.

 The tangent and cotangent

Similar representation of numbers and units of measure possesses property of a rectangular symmetry. Value of a angle the alpha in this case characterizes a dependence angle between number and a unit of measure.

At the following lesson we will consider
Quantity

### Terminating trigonometric functions

Subject of occupations:
TRIGONOMETRIC FUNCTIONS IN A RECTANGLE
Subject of the previous lesson
Three main types of trigonometric functions

Lesson 4

TERMINATING TRIGONOMETRIC FUNCTIONS

The sine and cosine are rather well studied, their values cannot be more unit. If to divide elements of a rectangle into length of diagonal, lengths of the parties will accept values of a sine and a cosine. Also the parties of a rectangle are diagonal projections in the perpendicular directions.

 Sine and cosine

Names of all trigonometric functions depend on the line of the beginning of measurement of a angle. The same line defines the direction of projection. Dependence between legs and a hypotenuse in a rectangular triangle, known as "Pythagorean theorem" (for length units of measure, not the bound to a hypotenuse) or "Pythagorean trigonometric identity" (when hypotenuse length is accepted to a unit of measure of length), is an integral part of properties of a rectangle.

If we project simple diagonal on the parties of a rectangle, then we will receive two projections of diagonal expressed through different angles. If we project the same parties on diagonal, then we will receive the diagonal length as the sum of two projections of the parties.

 Pythagorean trigonometric identity

The Pythagorean theorem is a dependence between diagonals and the parties of a rectangle.

At the following lesson we will consider
The infinite trigonometric functions

### Three main types of trigonometric functions

Subject of occupations:
TRIGONOMETRIC FUNCTIONS IN A RECTANGLE
Subject of the previous lesson
Angular symmetry in a rectangle

Lesson 3

Three main types of trigonometric functions

Trigonometric functions are the dependences between angles and numbers in a rectangle expressed in own units of measure. Own unit of measure is one of characteristics of object, accepted as a unit of measure.

In a rectangle it is possible to allocate three main types of trigonometric functions:

- the infinite trigonometric functions – a tangent and a cotangent, geometrically it:

a) the size of one of the parties of a rectangle at simple value of length of other party, is reflected by communication of numbers and units of measure;

b) the sizes of the parties of a rectangle at simple area, reflect laws for multiplication;

- terminating trigonometric functions – the sine and a cosine, geometrically are the sizes of the parties of a rectangle at simple diagonal, reflect projective properties of space;

- the linear angular functions – the linear sine (lin) and the linear cosine (los), geometrically is the sizes of the parties of a rectangle at a simple semiperimeter, reflect laws of addition.

Other types of trigonometric functions in this work are not considered as they did not draw attention of the author. All trigonometric functions establish particular connection between angles and numbers. It is quite possible that the mathematics of some extraterrestrial civilizations can be constructed on angles in the same way as at us it is constructed on numbers.

Expression of the trial trigonometric functions through the parties of a rectangle.

 Expression of the trial trigonometric functions through the parties of a rectangle

The main ratios on types of trigonometric functions.

 The main ratios on types of trigonometric functions

Values of trigonometric functions of some angles.

 Values of trigonometric functions of some angles

Expression of one trigonometric functions through others.

 Expression of one trigonometric functions through others

The explanation for visitors of this website. If to look at formulas of a tangent and a cotangent (the last picture), then can seem that between a sine with a cosine and linos with loses there is no difference. But you do not hurry with conclusions. If two fractions are equal, it does not mean at all that their numerators and denominators are equal. Let's review an example. Two thirds are equal to four sixth. But two it is not equal to four, three it is not equal to six.

In more detail the trial trigonometric functions are considered below.

At the following lesson we will consider
Terminating trigonometric functions

## 8.01.2016

### Triangle and rectangle

Subject of occupations:
TRIGONOMETRIC FUNCTIONS IN A RECTANGLE
At the first lesson we saw off
Short analysis of trigonometric functions

Lesson 2

Transformations of a triangle to a rectangle

If to combine two rectangular triangles on diagonal so that the rectangle turned out, then the hypotenuse of a triangle will turn into rectangle diagonal, and the parties of a triangle will turn into the parties of a rectangle. The trigonometrical relations of a triangle turn into the trigonometrical relations of a rectangle.

 Transformations of a triangle to a rectangle

Angular symmetry in a rectangle

Diagonal of a rectangle divides a right angle into two trigonometrical angles. The name of a trigonometric function depends on what angle we will take for definition of its numerical value. At the same time the numerical result does not depend on our choice.

 Angular symmetry in a rectangle

Names of trigonometric functions and the name of angles possess properties of an angular symmetry. At the same time a symmetry of AND functions of angles are inseparably linked among themselves. If we take the symmetric function and the symmetric angle, then the result will remain invariable. We apply an angular symmetry twice. The similar situation turns out in a plane geometry. If twice to apply a reflecting symmetry, then nothing will change. In algebra analog of an angular and reflecting symmetry is multiplication to minus unit. If an algebraic expression to increase unit by minus twice, the algebraic expression will remain the same.

Reflecting symmetry, the inverse symmetry, angular symmetry, multiplication to minus unit are manifestations of the same law of a symmetry under different conditions.

Considering a symmetry of trigonometric functions, they can be united pairwise in separate groups, each of which expresses particular type of dependence between angles and numbers.

At the following lesson we will consider
Three main types of trigonometric functions