## Pages

Showing posts with label trigonometric functions. Show all posts
Showing posts with label trigonometric functions. Show all posts

## 11.02.2019

### Borsch trigonometry. Addition.

I will not tell you recipes for cooking borsch, I will talk about mathematics. What is borsch? These are vegetables cooked in water according to a specific recipe. I will consider two source components (vegetable salad and water) and the final result is borsch. Geometrically, this can be thought of as a rectangle in which one side is a salad, the other side is water. The sum of these two sides is borsch. The diagonal and area of such a rectangle are mathematical concepts and are never used in recipes.

 Borsch trigonometry

How do salad and water turn into borsch? How does the sum turn into trigonometry? To understand this, linear angular functions will help us.

The linear angular functions

There are no linear angular functions in mathematics textbooks. But without them there can be no mathematics. Mathematical laws work whether we know them or not.

The linear angular functions are the laws of addition. This is how algebra turns into geometry, and geometry turns into trigonometry.

The main law of addition, which mathematicians do not like to talk about, requires that the summands have the same units of measurements. For salad, water and borsch it can be units of weight, volume, value or others.

The figure shows two levels of difference. The first level is differences in the field of numbers. This is what mathematicians do. The second level is the difference in the area of units of measurement (shown in square brackets). Physicists are doing this. We can understand the third level - the differences in the field of described objects. Different objects can describe the same number of identical units of measurements. How important this is, the trigonometry of borscht will show us. Here is how linear angular functions for borsch will look.

 The law of addition for borsch

Part of the water and part of the salad give one serving of borsch. Now we can see what will happen when the angle of linear angular functions changes.

 The angle is zero

The angle is zero. We have a salad but no water. You can’t cook borsch. The amount of borsch is also zero. This does not mean that zero borsch is equal to zero water. Zero borsch can be at zero salad (right angle).

 The angle is greater than zero, but less than the right angle

The angle is greater than zero, but less than forty-five degrees. We have a lot of salad, but not enough water. The result is a thick borsch.

The angle is forty-five degrees. We have an equal amount of water and lettuce. Perfect borsch (forgive me the cook, it's just math).

The angle is more than forty-five degrees, but less than ninety degrees. We have a lot of water and a little salad. Get liquid borsch.

 Right angle

Right angle. We have water. From the salad there were only memories. We cannot cook borscht. The amount of borscht is zero. In this case, drink water.

Something like this. If you're tired of borscht, check out a few other stories.

Percentage.

 Percentage

Cell division.

 Cell division

Two friends had their shares in the general business. After killing one of them, everything went to another.

The emergence of mathematics on our planet.

 The emergence of mathematics

All these stories in the language of mathematics are presented using linear angular functions. I will tell you separately about the real place of these functions in the structure of mathematics. In the meantime, we return to the trigonometry of borscht and consider the projections.

## 3.02.2017

### Inverse transformations

Last time we transformed the law of cosines and a Pythagorean theorem to the sum of a line segments. Now we will execute inverse transformations.

 Inverse transformations

In inverse transformations I made everything very simply. The minus sign from a formula disappeared. The problem is that we are not able to measure angles correctly. Than differ a angle 0 degrees from a angle of 180 degrees?

 Measurement of a angle
It is possible to assume that if a line segment one, then a angle is equal 0 degrees. If a line segments two, then a angle is equal 180 degrees.

The transformations executed by us show that the mathematics has no separate areas of mathematics: "arithmetic", "algebra", "geometry" or "trigonometry". The mathematics is a single whole.

The mathematics is DNA of the nature. Further we will continue to study a cosine gene in a cosine law.

If you liked the publication and you want to know more, help me with working on by other publications.

## 2.27.2017

### We use the law of cosines

I was always interested in a question: how the Pythagorean theorem turns into the sum of a line segments? What I speak about? Here you look.

 The Pythagorean theorem and the sum of a line segments
In geometry everything is very prime. The first time we draw a right triangle and we write down a Pythagorean theorem. The second time we draw two a line segments and we write down the sum of a line segments.

 Right triangle and two a line segments
How one formula turns into other formula? To see it, we use the law of cosines. We will draw the picture, we will write down conditions, we will execute transformations.

 Triangle and the law of cosines
 Right triangle and the Pythagorean theorem
 The law of cosines and sum of two a line segments
We use the law of cosines and turned the Pythagorean theorem into the sum of two a line segments. Further we will consider an inverse transformation.

If you liked the publication and you want to know more, help me with working on by other publications.

## 2.13.2017

### Transformations of trigonometric functions

 Transformations of trigonometric functions
This table shows how one trigonometric functions can be transformed to other trigonometric functions. Sin, cos, tan, cot, sec, csc - all these functions can be transformed.

## 2.10.2017

### Trigonometrical circle of a tangent

Last time I drew for you a unit circle of cotangent. Now you can look at a trigonometrical circle of tangent.

 Trigonometrical circle of a tangent
There is no sense here. There are angles and values of tangents. If you want to understand sense of a tangent and cotangent, look here.

## 2.08.2017

### Trigonometric table

 Trigonometric table are crazy
This trigonometrical table is an example of mathematical marasmus. I am touched by exact values of a sine and cosine. No comments.

If mathematicians do not know what is trigonometric functions, let read here. If mathematicians are not able to divide into zero, let study. I like the idea of this table. I do not like its contents. I corrected this trigonometrical table.

 Trigonometric table

The most popular angles are highlighted with blue color. 0, 30, 45, 60, 90 degrees most often occur in textbooks.Common fractions will be useful to pupils to fight against teachers. Decimal fractions will be useful physics and to engineers to calculations.

In this trigonometrical table there are no cotangents (cot, cotan, cotg, ctg, ctn). Anything terrible. There are useful formulas which will help you.

 Useful formulas

Sin 0, 15, 22.5, 30, 45, 60, 67.5, 75, 90, 120, 135, 150, 180 degrees in this table.
Cos 0, Pi/12, pi/8, pi/6, pi/4, pi/3, 3/8 pi, 5/12 pi, pi/2, 2/3 pi, 3/4 pi, 5/6 pi, 1 pi radians.
Tan pi/2 radians or 90 degrees it makes sense.

### Cotangent 210 degrees

Today we will consider a cotangent of 210 degrees. We will draw a unit disk and we will write values of cotangents. Here what turned out.
 The unit circle and cotangent
Value of a cotangent of 210 degrees same, as at a cotangent of 30 degrees. We take mathematics in hand and we check. The cotangent is the cosine divided into a sine. Values of a sine and cosine of 30 and 210 degrees can be taken here.

 Cotangent 210 degrees
We already know how to divide fraction into fraction. Formulas of transformation of angles of trigonometric functions.

 Formulas of transformation of angles
These formulas will be very useful to you.

## 2.07.2017

### Signs of trigonometric functions

All trigonometric functions can have different signs. Signs of trigonometric functions depend on the coordinate system. In mathematician the Cartesian coordinate system is accepted. Four quadrants of a Cartesian coordinate system define signs of trigonometric functions.

 Signs of trigonometric functions
You do not speak to mathematicians, but remember. Signs are not property of trigonometric functions. Without coordinate system signs will not be. If to measure corners on another, signs will be others. In other coordinate system signs can be others.Signs of trigonometric functions are a property of the chosen coordinate system. Functions are considered in mathematician only in the Cartesian coordinate system.

## 1.30.2017

### Decomposition on items

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

Lesson 16

Decomposition on items

If only the result of addition is known and items are unknown, then the sum can be spread out to items by means of the linear angular functions.

 Decomposition of the sum on items

Transformation of a square of quantity to the work of two sums (see an example above), it is possible to execute with application of decomposition of result of multiplication to multipliers and items.

 Transformation of a square to the work of the sums

Similitudes can be useful when studying various natural phenomena to their best comprehension. Let's review an example of reproduction of living beings.

Asexual reproduction of live organisms can be described by means of decomposition of the sum on items. As a result of division of an organism A two self-contained organisms B and C.

 Asexual reproduction

Decomposition on items with corners about 45 degrees, is characteristic of unicells. For metaphytes the range of an angle of decomposition varies in wider limits (a vegetative reproduction, budding, fragmentation). The unit of measure at decomposition can be considered a physical body of an organism.

The beginning of life (zero) of similar organisms can be considered the moment of division of a parental organism. The termination of life (unit) can be considered characteristic division or death.

Sexual reproduction is described by means of multiplication. The moment of emergence of sexual reproduction can be described by means of the linear angular functions. At simultaneous reproduction of organisms A and B there was the common stream C which had signs of two parents.

 Emergence of sexual reproduction

What have to be the angles of decomposition for emergence of the common offspring? The most probable candidates in "invented" sexual reproduction it is a larger cage and a virus. The virus breeds in a cage. Along with cell division there was a division of a virus. The result was a new body. Or two body - male M and female W.

 Male and female
Molecule DNA which is available both for a cage, and for a virus could act as a basis for addition (unit of measure).

It is only one of a set of options of possible succession of events. From the moment of emergence of life on Earth until emergence of sexual reproduction in the Nature was enough time for the most different experiments.

Closing part

Further studying of properties of units of measure will help to understand better and more precisely to describe by mathematical methods various phenomena in the world around.

The separate ideas published in this work will be considered in more detail in the subsequent publications.

Gratitude

I express sincere gratitude to the parents and the daughter Inna for financial support of my working on with mathematics.

Subject of occupations:
TRIGONOMETRIC FUNCTIONS IN A RECTANGLE
Subject of the previous lesson
The linear angular functions

Lesson 15

As a result of addition of two different quantities the third quantity turns out. At addition of change occurs in a number domain, the area of units of measure does not change. Addition is possible only for parallel quantities with identical units of measure. Addition reflects the quantitative changes of quantities.

5а+3а=(5+3)а=8а

For realization of addition of two different quantities with units of measure in different scales (the corner of scale of units of measure is not equal to zero), it is necessary to change the scale of units of measure so that the scale corner between them equaled to zero. At the same time does not matter, the first item, the second or both changes at once.

It is impossible to put two identical numbers with different units of measure as the result does not make sense.

5а+5b=5(a+b)

Transformation of result of addition of pieces to the parties of a rectangle looks so.

Items can be presented as the party of a rectangle, then a half of perimeter of a rectangle is result of addition. For any sum it is possible to define the linear angular functions if items are known.

At the following lesson we will consider
Decomposition on items

## 9.03.2016

### The linear angular functions

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

Lesson 14

The linear angular functions

If to consider terminating trigonometric functions as coordinates of points of a unit circle in a Cartesian coordinate system, then the linear angular functions are coordinates of points of the chord connecting circle cross points to a coordinate. The sum of coordinates of any point of this chord is always equal to unit.

 The linear angular functions

In mathematician of a concept, similar to the linear angular functions, are used since ancient times - it is division whole on a part. In the modern world an analog are percent.

The explanation for readers of this website. For myself I called the linear angular functions "linos" and "loses". How I thought up these names? Took designation of a sine and cosine. Visually they quite well differ. In each designation I replaced the first letter with the Latin letter "l" from the word "line". It turned out quite nicely. But to solve to you. Whether these functions in mathematician will get accustomed and as they will be called - time will show. I just offer one more mathematical tool for the description of reality.

At the following lesson we will consider

### Division

Subject of occupations:
TRIGONOMETRIC FUNCTIONS IN A RECTANGLE
Subject of the previous lesson
Decomposition on factors

Lesson 13

Division

Contrary to the standard opinion, division is not mathematical operation. This solution of a standard task of finding of one of factors if other factor and result of multiplication is known. In ancient Babylon the fraction was considered as result of multiplication of number to inverse other number. Even in the modern mathematics there is no division of one fraction into other fraction, this operation is replaced with multiplication of a dividend to fraction, inverse to a divider.

Division can be considered as a projection of result of multiplication along one of factors. For example, length is a projection of the area along width, width is a projection of the area along length.

The most interesting in this plan is the speed which is measured by the lengthiest, divided into time. If to assume that length is result of multiplication of two perpendicular directions of time, then speed is a projection of length (the area of time) along one of the directions of time. For a comprehension of the nature and an essence of light velocity, this approach can be the very useful.

At the following lesson we will consider
The linear angular functions

### Decomposition on factors

Subject of occupations:
TRIGONOMETRIC FUNCTIONS IN A RECTANGLE
Subject of the previous lesson
Zero and infinity

Lesson 12

Decomposition on factors

Mathematical operation, opposite on sense to multiplication, decomposition on factors is. It is carried out with application of the infinite trigonometric functions.

 Decomposition on factors
The most prime example of decomposition on factors at an angle in 45 degrees is root squaring. As both factors in this case are identical, as result of decomposition it is accepted to write down only one of factors.

Decomposition on factors can be applied when only the result of multiplication is known and any of factors is not known. Units of measure as a result decomposition on factors should be selected intuitively so that as a result of their multiplication the tentative unit of measure turned out. The quantity of spacelike dimensions in units of measure of factors at decomposition can be a miscellaneous. For example, three-dimensional volume can be spread out to one-dimensional factors by means of two operations of decomposition, one of options looks so:

 Decomposition of volume

In this example corners α and β are not bound among themselves. If to display volume in a cube (a=b=c), then α≈35° is a angle between the diagonal of a cube and diagonal of the basis, β=45 ° is a angle between the diagonal of the basis and its party.

At the following lesson we will consider
Division

## 8.19.2016

### Zero and infinity

Subject of occupations:
TRIGONOMETRIC FUNCTIONS IN A RECTANGLE
Subject of the previous lesson
Distinctions between multiplication and addition

Lesson 11

Zero and infinity

If the angle is equal to zero or 90°, then the two dimensional rectangle disappears and there is a one-dimensional piece. From here the sense of infinity follows: as if we did not change the party of a rectangle, it will never turn into a piece. Unit divided into zero is not equal to infinity. Infinitesimal size is not equal to unit divided into infinity.

 Zero and infinity

Difference between elements in these inequalities same as difference between the point lying on a straight line and the point which is not lying on a straight line.

Multiplication to zero and division by zero do not fall into to mathematical operations with numbers, they are carried out in the field of units of measure. It is possible to call these values of trigonometric functions non-numerical.

In addition to the materials about multiplication and division by zero explained earlier it is necessary to add the following. In positional notation zero designates lack of number of the particular category. Lack of number number cannot be. Here zero is similar to punctuation marks in writing which have the graphic form, but are not said when reading.

Generally zero should be understood as lack of the considered unit of measure. For example, zero value of a angle means that the angle is absent. Division by zero should be considered as need of introduction of a unit of measure, perpendicular to already existing, for the further solution of a task. Division by zero does not mean automatic transition to multiplication. For example, it is impossible to describe turn of a piece in one-dimensional space, for this purpose it is necessary to enter padding measurement and to consider a task in two dimensional space.

At the following lesson we will consider
Decomposition on factors

## 8.14.2016

### Distinctions between multiplication and addition

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

Lesson 10

Distinctions between multiplication and addition

Multiplication is a change of quality, that is change of units of measure. Contrary to the standard opinion, multiplication cannot be presented in the form of addition. When multiplication is substituted for addition, mathematical properties of multiplication are used. With units of measure substitution looks so:

 Multiplication substitution by addition

Algebraic expressions with use of letters show distinction between addition and multiplication:

 Multiplication in algebra

If at addition items have identical numerical values, then it is possible to apply mathematical model of multiplication for addition. At the same time we assume that units of measure at multiplication do not change:

 Addition replacement with multiplication

In mathematician possess separate properties of units of measure:

- number systems of numbers – it is impossible to addition the numbers presented in different number systems, a question of a possibility of multiplication of numbers in different number systems the author it was not studied;

- denominators of common fractions – it is impossible to addition fractions with different denominators, at multiplication of fractions denominators are multiplied;

- alphabetic references in algebraic expressions – it is impossible to addition numbers with different designations, at multiplication new designation of result turns out;

- legends of functions (for example, trigonometrical).

At the following lesson we will consider
Zero and infinity

### Examples of multiplication

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

Lesson 9

Examples of multiplication

 Multiplication in space

At interaction of two one-dimensional spaces (in figure D is quantity of spacelike dimensions) the two dimensional space turns out. At addition to interaction of perpendicular measurement, the quantity of spacelike dimensions of result increases. It is necessary to understand any quantities participating in multiplication as spacelike dimensions. Multiplication of two one-dimensional lengths gives the two dimensional area, multiplication of the two dimensional area at one-dimensional length gives three-dimensional volume and so on. The possibility of receiving four-dimensional space as a result of multiplication of two two dimensional spaces is theoretically possible, but demands specification from the point of view of physical reality.

 Examples of multiplication

Voltage is result of interaction of current intensity and resistance. If to consider current intensity not as one-dimensional quantity and as two dimensional (the area of current), then as a result of interaction with a resistance power will turn out. Units of measure of the specified quantities (Volts, Ampere, Ohms, Watts) are household (convenient in daily use), but demand mathematical representation for comprehension of an essence of the physical processes expressed by these units of measure. The system of physical quantities offered R.O. di Bartini can be considered as attempt of transition from household units of measure to mathematical. Interaction between quantities is carried out due to the processes studied in physics.

The arrangement of threads in the perpendicular direction allows to receive cloth. Interaction between threads is carried out at the expense of frictional force. If in the perpendicular directions to arrange reinforcing rod stock, then it is possible to receive a two dimensional reinforcing grid or a three-dimensional reinforcing framework. Interaction of separate rod stock in clusters is provided with electric welding or a knitting wire. Plywood is the fibers of a wood neveer located perpendicularly. Interaction is provided by pasting of separate layers of wood neveer.

At multiplication the quantity of goods on unit price, as a result turns out the cost of a consignment of goods (sum). It is an example of application of mathematical model of multiplication to the virtual units of measure. In this case physical interaction between goods and the price is absent.

Sexual reproduction it is possible to present in multiplication form. As a result of interaction of a male and a female there is a posterity which has genetic signs of two parents. Partial transfer of genetic signs from each of parents can be described by means of the linear angular functions.

For more precise reflection of reality in mathematician it is necessary to consider three stages of multiplication: beginning of multiplication, interaction process, end of multiplication. In electric circuits the switch is the device which operates interaction. Creation of cloth, plywood, a reinforcing grid is the beginning of interaction. Physical destruction of these objects is the end of interaction.

At the following lesson we will consider
Distinctions between multiplication and addition

## 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