## 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

## 7.25.2016

### Trigonometric functions in a rectangle

Published on 7 July, 2016
"The Papers of Independent Authors"
Volume 36 of p. 46-69

Annotation

Representation of trigonometric functions in a rectangle allows to unite algebra, geometry and physics in a single whole.

Short analysis of trigonometric functions

Usually trigonometric functions of plane angle are defined in a rectangular triangle as ratios of the parties of this triangle [a reference to the source in the printing edition].

 Trigonometric functions in a triangle

If to accept the definitions of trigonometric functions entered by mathematicians in a rectangular triangle, then values of these functions depend only on a ratio of the sizes of the parties of a triangle. The size of angles in a rectangular triangle is in the range of trigonometrical angles. Trigonometric functions have no signs and do not possess periodic. These properties are homocentrism elements.

The homocentrism mathematics is a mathematics in which the result depends on the option of the relative mathematics accepted by us or on our opinion. Striking examples of a homocentrism in mathematician are: division of numbers on positive and the negative, decimal numeration, numbers and inverse numbers, Cartesian coordinate system, etc.

In Cartesian coordinate system trigonometric functions are defined as coordinates of points of a unit circle [a reference to the source in the printing edition].

 Trigonometric functions in Cartesian coordinate system

This definition is possible only because for any point of a circle of coordinate are used as padding elements for creation of a rectangular triangle. For cross points of a circle and coordinate creation of a triangle is impossible.

For any point of the plane in Cartesian coordinate system trigonometric functions can be defined as the relation of coordinates of this point or the relation of coordinates of a point to distance from a point to the center of a frame. An exception is the cross point of coordinate (the center of a frame) for which trigonometric functions cannot be defined. This fact is congenital defect of Cartesian coordinate system. If it is necessary to define trigonometric functions for the point coinciding with the center of a frame, then the frame needs to be displaced aside.

In Cartesian coordinate system trigonometric functions are the relative and depend on the relative positioning of the plane on which the considered points, and frames are located. Periodicity of trigonometric functions is result of rotation of a piece around the center of a frame. Signs of trigonometric functions depend on the positive direction of coordinate accepted by us. All this result homocentrism of views of trigonometric functions.

Signs of trigonometric functions "+" and "-" serve for orientation in space of Cartesian coordinate system. In mathematical formulas with use of trigonometric functions, the sign "minus" at function automatically changes addition for a subtraction or a subtraction on addition. In formulas it is possible to do without the negative values of trigonometric functions. It will allow to reduce twice amount of values of trigonometric functions, but will increase quantity of formulas.

At the following lesson we will consider
Transformations of a triangle to a rectangle

## 7.12.2016

### Measurement of plane angles

Published on 7 July, 2016
"The Papers of Independent Authors"
Volume 36 of p. 43-45

Annotation

Plane angles need to be divided into trigonometrical angles and rotation angles.

Measurement of angles in a solid geometry

In a solid geometry «angle between a straight line and the plane is called angle between this line and its projection (orthogonal) to the plane. If the straight line to be a perpendicular plane, then angle between it and the plane is considered (my italic type – Nikolaj Khizhnjak) equal 90°, and between a parallel straight line and the plane – equal 0°". [1, p. 308].

As appears from definition, the angle between a straight line and the plane cannot be more than 90°. Angles more than 90° fall into to angles of rotation [2, p. 81]. In a solid geometry it is not accepted to describe by means of angles rotation of a straight line around a point of its crossing with the plane.

If the straight line is parallel to the plane (or belongs to the plane), then it is considered, that the size of the angle has zero value. Actually there is no vertex of angle – a cross point of a straight line and the plane. If the perpendicular plane straight line, then a trace turns into a point which coincides with a cross point of a straight line and the plane. Actually there is no second ray forming a angle. In this case the angle is considered right angle.

Measurement of angles in a planimetrics

On analogies to a solid geometry, a plane angle between two straight lines (line segments) it has to be measured as a angle between a point on one direct and its projection on other straight line with vertex of angle in a cross point of these straight lines.

This rule allows to divide plane angle into trigonometrical angles (from 0° to 90°) and rotation angles. The fundamental difference between these angles consists in amount of the objects on the two dimensional plane forming a angle. The trigonometrical angle is measured between two different objects, the rotation angle is measured between two provisions (initial and terminating) one object. For trigonometrical angles the time factor is absent, for rotation angles the time factor is obligatory: tentative situation, rotation time, terminating situation.

At measurement of angles between two parallel straight lines the angle cannot be defined because there is no cross point of straight lines. At measurement of a corner between two perpendicular straight lines the angle cannot also be defined because the projection of a point coincides with a cross point of these straight lines.

Conclusion: zero and right angles cannot have just the same mathematical properties as other angles. In more detail these distinctions will be described by consideration of trigonometric functions.

Angles and distances

The trigonometrical angle on the properties is similar to distance between two points, the rotation angle is similar to the passable way between the same points. If driving is carried out in a straight line, then the passable path is equal to distance between initial and terminating points of a way. If rotation is carried out on a angle to 90°, then the angle of rotation is equal to a trigonometrical angle. The rotation angle cannot be less trigonometrical angle, the passable path cannot be less than a distance between points of the beginning and end of this way. The distance between two points can be in the numerical range from zero ad infinitum. Compliance between angles from 0° to 90° and numbers from zero ad infinitum is established by a trigonometric function of a tangent of these angles.

The passable path is also expressed in the numerical range from zero ad infinitum even if the distance between an initial and finishing point of the passable way is equal to zero. In the same way the rotation angle can reach infinitely great values, at the same time the trigonometrical angle between the beginning and the end of rotation can be equal to zero. In this case it is possible to establish direct analogy between number and size of a angle as it is accepted in a calculus.

Various mathematical tools, for example, a Pythagorean theorem are used to calculation of distance between initial and terminating points of the passable path. Reduction formulas of angles are applied to definition of a angle between the beginning and the end of rotation.

Conclusion: plane angle on the properties are similar to distances on the plane.

Literature

1. Zabelyshynskaia M.Y."Mathematician. Educational and practical reference book" Kharkiv, Ranock, 2010.
2. Khizhnjak Nikolaj «Bases of mathematics», The Papers of Independent Authors, vol. 19, 2011.