Speech therapy portal / Parents / Circle cosine. The main values \u200b\u200bof trigonometric functions. Trigonometric circle. Single circle. Numerical circle. What it is

Circle cosine. The main values \u200b\u200bof trigonometric functions. Trigonometric circle. Single circle. Numerical circle. What it is

13.08.2020

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If we say simply, these are vegetables cooked in water by a special recipe. I will consider two source components (vegetable salad and water) and the finished result - borsch. Geometrically, this can be represented as a rectangle in which one side denotes a salad, the second side denotes water. The sum of these two sides will denote borsch. The diagonal and the area of \u200b\u200bsuch a "burst" rectangle are purely mathematical concepts and are never used in the recipes of boating borsch.

How are the salad and water turn into borsch in terms of mathematics? How can the sum of two segments be transformed into trigonometry? To understand this, we need linear angular functions.

In mathematics textbooks, you will not find anything about linear angular functions. But without them there can be no mathematicians. Laws of mathematics, as well as the laws of nature, work independently of whether we know about their existence or not.

Linear angular functions are the laws of addition. See how the algebra turns into geometry, and the geometry turns into trigonometry.

Is it possible to do without linear angular functions? It is possible, because mathematics still do without them. The trick of mathematicians is that they always tell us only about those challenges that they themselves can decide, and never tell about those tasks that they do not know how to decide. See. If we know the result of the addition and one term, to search for another complimentary, we use subtraction. Everything. We do not know other tasks and do not know how to solve. What to do in the event that only we are known for the result of addition and are not known both terms? In this case, the result of addition must be decomposed into two terms with linear angular functions. Then we already choose, how can one term may be, and linear angular functions show what the second term should be, so that the result of the addition was exactly what we need. Such pairs of terms can be an infinite set. IN everyday life We are perfectly accustomed without decomposition of the amount, we have enough subtraction. But in the scientific research of the laws of nature, the decomposition of the amount on the components can be very useful.

Another law of addition, about which mathematics do not like to speak (another of their trick), requires that the components had the same units of measurement. For lettuce, water and borschor, it may be a unit of measurement, volume, cost or unit of measurement.

The figure shows two levels of differences for mathematical. The first level is the differences in the field of numbers that are indicated a., b., c.. This is what mathematics are engaged. The second level is the differences in the field of units of measurement, which are shown in square brackets and indicated by the letter U.. Physics are engaged in this. We can understand the third level - differences in the field of described objects. Different objects may have the same number of identical units of measurement. As far as it is important, we can see the example of trigonometry of borscht. If we add lower indexes to the same designation of units of measurement of different objects, we can accurately say which mathematical value describes a specific object and how it changes over time or in connection with our actions. Letter W. I will refer water, letter S. Let the Salad and Letter B. - Borsch. This is how linear angular functions for borscht look like.

If we take some part of the water and some part of the salad, together they will turn into one portion of the borscht. Here I suggest you a little distract from the borscht and remember the distant childhood. Remember how we were taught to fold the bunnies and clerk together? It was necessary to find how much animals would succeed. What did they teach us then to do? We were taught to tear off the units of measurements from numbers and add numbers. Yes, one any number can be folded with another any number. This is a direct path to the authis of modern mathematics - we do it is not clear what, it is not clear why and very well understand how this refers to reality, because of the three levels of mathematics differences only one. It will be more correct to learn to move from one units of measurement to others.

And bunnies, and clarops, and animals can be calculated in pieces. One common unit of measurement for different objects allows us to fold them together. This is a children's task option. Let's look at a similar task for adults. What happens if you fold bunnies and money? Here you can offer two solutions.

First option. We define the market value of the bunnies and fold it with the amount of money. We received the total cost of our wealth in the cash equivalent.

Second option. You can add the number of bunnies with the number of cash bills available. We will receive the number of movable property in pieces.

As you can see, the same arrangement law allows you to get different results. It all depends on what exactly we want to know.

But back to our boors. Now we can see what will happen at different values \u200b\u200bof the angle of linear angular functions.

The angle is zero. We have a salad, but there is no water. We can not cook borsch. The amount of boards is also zero. This does not mean that zero borschor is zero water. Zero zero can be at zero salad (straight angle).

For me personally, it is the main mathematical evidence of the fact that. Zero does not change the number when adding. This is because the addition itself is impossible if there is only one term and there is no second term. You can treat it anyhow, but remember - all mathematical operations with zero came up with the mathematics themselves, so throwing your logic and stupidly tool the definitions invented by mathematicians: "The division on zero is impossible", "any number multiplied by zero is zero" , "For a duck point zero" and other nonsense. It is just once to remember that zero is not a number, and you will never have a question, is a zero natural number or not, because such a question is generally deprived of any meaning: how can be considered a number that the number is not. It is like asking what color is invisible color. Add zero to the number is the same as painting paint, which is not. Dry tassel washed and talk to everyone that "we painted." But I was a little distracted.

The angle is greater than zero, but less than forty-five degrees. We have a lot of lettuce, but little water. As a result, we get a thick borsch.

The angle is forty-five degrees. We have in equal amounts water and salad. This is the perfect borsch (and forgive me a cook, it's just a mathematics).

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

Right angle. We have water. Only memories remained from salad, because the angle we continue to measure from the line, which once marked the salad. We can not cook borsch. The amount of borscht is zero. In this case, hold on and drink water while it is)))

Here. Something like this. I can tell here and other stories that will be more than appropriate here.

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

The appearance of mathematics on our planet.

All these stories in the language of mathematics are told using linear angular functions. Some other time I will show you the real place of these functions in the structure of mathematics. In the meantime, back to trigonometry of borscht and consider the projection.

saturday, October 26, 2019

wednesday, August 7, 2019

Completing the conversation about, you need to consider the infinite set. It gave that the concept of "infinity" acts on mathematicians as a boating to the rabbit. Awesome horror before infinity deprives mathematicians of common sense. Here is an example:

The source is located. Alpha denotes valid number. The sign of equality in the above expressions suggests that if to infinity to add a number or infinity, nothing will change, resulting in the same infinity. If you take an infinite set as an example natural numbersThe considered examples can be represented in this form:

For visual proof of their mathematics, many different methods came up with. Personally, I look at all these methods, like on dance of shamans with tambourines. Essentially, they all are reduced to the fact that either part of the numbers are not busy and new guests are settled in them, or to the fact that part of visitors are thrown into the corridor to free the place for guests (very humanly). I outlined my opinion on such solutions in the form of a fantastic story about the blonde. What are my reasoning based on? The resettlement of the endless number of visitors requires infinitely much time. After we freed the first room for the guest, one of the visitors will always follow the corridor from your room to the neighboring century. Of course, the time factor can be stupidly ignored, but it will be not written from the category of "fools." It all depends on what we do: Customize reality for mathematical theories or vice versa.

What is the "endless hotel"? The endless hotel is a hotel where there is always any number of free places, no matter how many rooms are busy. If all rooms in the infinite corridor "for visitors" are occupied, there is another endless corridor with guest numbers. Such corridors will be an infinite set. In this case, the "endless hotel" is an infinite number of floors in an infinite amount of housings on an infinite amount of planets in an infinite number of universes created by an infinite amount of gods. Mathematics are not able to remove from banal household problems: God-Allah-Buddha is always only one, the hotel is one, the corridor is only one. Here are mathematicians and are trying to sweep the ordinal numbers of hotel rooms, convincing us in the fact that you can "shove the unpiered".

The logic of your reasoning, I will demonstrate you on the example of an infinite set of natural numbers. First you need to answer a very simple question: how many sets of natural numbers exist - one or much? There is no correct answer to this question, because numbers came up with themselves, there are no numbers in nature. Yes, nature knows how to count perfectly, but for this it uses other mathematical tools that are not familiar to us. How nature believes, I will tell you another time. Since the numbers came up with us, we ourselves decide how many sets of natural numbers exist. Consider both options, as is submitted by this scientist.

Option first. "Let us give" one-sole set of natural numbers, which serene lies on the shelf. Take it from the shellf this is a lot. Everything, other natural numbers on the shelf there is no left and take them nowhere. We can not add a unit to this set, as we already have it. And if you really want? No problem. We can take a unit of the many have already taken and bring it back to the shelf. After that, we can take a unit from the shelter and add it to what we have left. As a result, we again get an infinite set of natural numbers. Write all our manipulations like this:

I recorded the actions in the algebraic system of designations and in the system of designations adopted in the theory of sets, with a detailed listing of sets of sets. The lower index indicates that the many natural numbers we have the only one. It turns out that the set of natural numbers will remain unchanged only if it is subtracted from it a unit and add the same unit.

Option second. We have a lot of different infinite sets of natural numbers on our shelf. I emphasize - different, despite the fact that they are practically not distinguishes. Take one of these sets. Then, from another set of natural numbers, we take a unit and add a set of already taken by us. We can even fold two sets of natural numbers. That's what we do:

The lower indexes "one" and "two" indicate that these elements belonged to different sets. Yes, if you add an unit to an infinite set, the result is also an infinite set, but it will not be the same as the initial set. If one infinite set is added to one infinite set, the result is a new infinite set consisting of elements of the first two sets.

The set of natural numbers is used for the account just as a ruler for measurements. Now imagine that you added one centimeter to the ruler. This will already be another line, not equal to the original one.

You can accept or not accept my reasoning is your personal matter. But if you ever come across mathematical problems, think about whether you are walking along the trail of false reasoning, trotted generations of mathematicians. After all, classes in mathematics, first of all, form a steady stereotype of thinking, and only then add mental abilities to us (or vice versa, deprive us of freightness).

pozg.ru.

sunday, August 4, 2019

Updated postscript to the article about and saw this wonderful text in Wikipedia:

Read: "... rich theoretical Foundation Mathematics of Babylon did not have a holistic nature and was reduced to the set of scattered techniques devoid common system and evidence. "

Wow! What are we smart and how well we can see the shortcomings of others. And we slightly look at modern mathematics in the same context? Slightly paraphrasing the given text, I personally managed the following:

The rich theoretical basis of modern mathematics is not a holistic nature and comes down to the set of scattered sections devoid of a common system and evidence base.

For confirmation of your words, I will not walk far - it has a language and conditional designations other than the language and the symbols of many other sections of mathematics. The same names in different sections of mathematics can have a different meaning. The most obvious Lumps of modern mathematics, I want to devote a whole cycle of publications. See you soon.

saturday, August 3, 2019

How to divide the set on subsets? To do this, enter a new unit of measure, which is present from the part of the elements of the selected set. Consider an example.

Let we have many BUTconsisting of four people. This set is formed on the basis of "people" we denote the elements of this set through the letter butThe lower index with the number will indicate the sequence number of each person in this set. We introduce a new unit of measurement "penis" and denote its letter b.. Since sexual signs are inherent in all people, multiply every element of the set BUT on sexual sign b.. Please note that now our many people have become many "people with sexual signs." After that, we can split genital signs for men bM. and women bW Sexual signs. Now we can apply a mathematical filter: we choose one of these sexual signs, which is indifferent to what is male or female. If he is present in humans, then you multiply it on one, if there is no such a sign - you multiply it on zero. And then apply the usual school mathematics. See what happened.

After multiplication, abbreviations and regrouping, we received two subsets: a subset of men BM. and a subset of women BW. Approximately the same mathematicians reason when they use the theory of sets in practice. But in the details they do not devote us to us, but give out the finished result - "a lot of people consists of a subset of men and a subset of women." Naturally, you may have a question how correctly mathematics are applied in the above transformations? I dare to assure you, essentially the transformations done everything correctly, it is enough to know the mathematical justification of arithmetic, boolean algebra and other sections of mathematics. What it is? Anyone else's time I will tell you about it.

As for examples, it is possible to combine two sets into one premise, pose a unit of measurement present at the elements of these two sets.

As you can see, units of measurement and ordinary mathematics turn the theory of sets into the relic of the past. A sign of the fact that with the theory of sets is not all right, it is that for the theory of mathematics sets, their own language and their own designations came up. Mathematics were accepted as shamans once come. Only shamans know how "correctly" apply their "knowledge." These "knowledge" they teach us.

In conclusion, I want to show you how mathematics manipulate with.

monday, January 7, 2019

In the fifth century BC, the ancient Greek philosopher Zenon Elayky formulated his famous apiorials, the most famous of which is Achilles and Turtle Aritia. This is how it sounds:

Suppose Achilles runs ten times faster than the turtle, and is behind it at a distance of a thousand steps. For the time, for which Achilles is running through this distance, a hundred steps will crash in the same side. When Achilles runs a hundred steps, the turtle will crawl about ten steps, and so on. The process will continue to infinity, Achilles will never catch up to the turtle.

This reasoning has become a logical shock for all subsequent generations. Aristotle, Diogen, Kant, Hegel, Hilbert ... All of them somehow considered the Apriology of Zenon. Shock turned out to be so strong that " ... Discussions continue and at present, to come to the general opinion on the essence of paradoxes to the scientific community has not yet been possible ... A mathematical analysis, the theory of sets, new physical and philosophical approaches was involved in the study of the issue; None of them became a generally accepted issue of the issue ..."[Wikipedia," Yenon Apriya "]. Everyone understands that they are blocked, but no one understands what deception is.

From the point of view of mathematics, Zeno in his Aproria clearly demonstrated the transition from the value to. This transition implies application instead of constant. As far as I understand, the mathematical apparatus of the use of variables of units of measurement is either yet not yet developed, or it was not applied to the Aporition of Zenon. The use of our ordinary logic leads us to a trap. We, by inertia of thinking, use permanent time measurement units to the inverter. From a physical point of view, it looks like a slowdown in time to its complete stop at the moment when Achilles is stuffed with a turtle. If time stops, Achilles can no longer overtake the turtle.

If you turn the logic usually, everything becomes in place. Achilles runs at a constant speed. Each subsequent segment of its path is ten times shorter than the previous one. Accordingly, the time spent on its overcoming, ten times less than the previous one. If you apply the concept of "infinity" in this situation, it will correctly say "Achilles infinitely will quickly catch up the turtle."

How to avoid this logical trap? Stay in permanent time measurement units and do not move to reverse values. In the language of Zenon, it looks like this:

For that time, for which Achilles runs a thousand steps, a hundred steps will crack the turtle to the same side. For the next time interval, equal to the first, Achilles will run another thousand steps, and the turtle will crack a hundred steps. Now Achilles is an eight hundred steps ahead of the turtle.

This approach adequately describes reality without any logical paradoxes. But it is not complete solution Problems. On the Zenonian Agrac of Achilles and Turtle is very similar to the statement of Einstein on the irresistibility of the speed of light. We still have to study this problem, rethink and solve. And the decision you need to seek not in infinitely large numbers, and in units of measurement.

Another interesting Yenon Aproria tells about the flying arrows:

The flying arrow is still, since at every moment she rests, and since it rests at every moment of time, it always rests.

In this manor, the logical paradox is very simple - it is enough to clarify that at each moment the flying arrow is resting at different points of space, which, in fact, is the movement. Here you need to note another moment. According to one photo of the car on the road, it is impossible to determine the fact of its movement, nor the distance to it. To determine the fact of the car's motion, you need two photos made from one point at different points in time, but it is impossible to determine the distance. To determine the distance to the car, two photos made from different points of space at one point in time, but it is impossible to determine the fact of movement (naturally, additional data is still needed for calculations, trigonometry to help you). What I want to pay special attention is that two points in time and two points in space are different things that should not be confused, because they provide different opportunities for research.
I'll show the process on the example. We select "Red solid to the pillow" - this is our "whole". At the same time, we see that these things are with a bow, and there is without a bow. After that, we select part of the "whole" and form a lot of "with a bow." So the shamans make their feed, tie their theory of sets to reality.

Now let's make a little dirty. Take a "hard in a pary with a bow" and unite these "whole" in color sign, swing red elements. We got a lot of "red". Now the question is on the backbone: the obtained sets "with a bow" and "red" are the same set or two different sets? Only shamans know the answer. More precisely, they themselves know nothing, but they will say, so it will be.

This simple example shows that the theory of sets is completely useless when it comes to reality. What's the secret? We formed a lot of "red solid in a pary with a bow." The formation occurred in four different units of measurement: color (red), strength (solid), roughness (in a pull), decorations (with a bow). Only the set of units of measurement allows adequately to describe the real objects in the language of mathematics. That's what it looks like.

The letter "A" with different indices indicates different units of measurement. In brackets allocated units of measurement on which the "whole" is highlighted at the preliminary step. Behind the brackets made a unit of measurement, which is formed by a set. The latter line shows the final result - the element of the set. As you can see, if you use units of measurement to form a set, then the result does not depend on the order of our actions. And this is already mathematics, not dance of shamans with tambourines. Shamans can be "intuitive" to come to the same result by arguing it "apparent", because the units of measurement are not included in their "scientific" arsenal.

Using units of measurement, it is very easy to divide one or combine several sets into one alarm. Let's look at the algebra of this process more carefully.

Trigonometry, like science, originated in the ancient East. The first trigonometric ratios were derived by astronomers to create an accurate calendar and focus on the stars. These calculations refer to spherical trigonometry, while in school course We study the ratio of the parties and the angle of the flat triangle.

Trigonometry is a section of mathematics engaged in the properties of trigonometric functions and dependence between the parties and the corners of the triangles.

During the heyday of the culture and science of the first millennium, our era of knowledge has spread from the ancient East to Greece. But the main discoveries of trigonometry is the merit of husbands of the Arab Caliphate. In particular, the Turkmen scientist al-Marazvi entered the functions such as Tangent and Kotangent, compiled first tables for sinus values, tangents and catangents. The concept of sine and cosine is introduced by Indian scientists. Trigonometry is devoted to a lot of attention in the writings of such great leaders of antiquity, like Euclidea, Archimedes and Eratosthene.

The main values \u200b\u200bof trigonometry

The main trigonometric functions of the numerical argument are sinus, cosine, tangent and catangent. Each of them has its own schedule: sinusoid, cosineida, tangensoid and catangensoid.

The basis of the formulas for calculating the values \u200b\u200bof the specified quantities is the Pythagoreo theorem. She is more famous for schoolchildren in the wording: "Pythagoras pants, in all directions are equal," since the proof is provided by the example of an equally rectangular triangle.

Sinus, cosine and other dependences establish a link between sharp corners and sides of any rectangular triangle. We give formulas for calculating these values \u200b\u200bfor angle A and trace the relationship of trigonometric functions:

As can be seen, TG and CTG are inverse functions. If you submit catat A as a piece of SIN A and hypotenuses with, and roll b in the form of COS A * C, we will receive the following formulas for Tangent and Kotangent:

Trigonometric circle

The graphically, the ratio of said values \u200b\u200bcan be represented as follows:

Circle, in this case, is all possible angle α - from 0 ° to 360 °. As can be seen from the figure, each function takes a negative or positive value depending on the corner value. For example, SIN α will be with the "+" sign, if α belongs to the I and II of the quarter of the circle, that is, it is between 0 ° to 180 °. With α from 180 ° to 360 ° (III and IV quarters), SIN α can only be a negative value.

Let's try to build trigonometric tables for specific angles and find out the value of values.

The values \u200b\u200bα are 30 °, 45 °, 60 °, 90 °, 180 °, and so on - are called special cases. The values \u200b\u200bof trigonometric functions are calculated for them and are presented in the form of special tables.

These angles are not chosen by any accident. The designation π in the tables stands for radians. Rad is an angle at which the length of the circumference arc corresponds to its radius. This value was introduced in order to establish a universal dependence, when calculating radians, the actual radius length in cm does not matter.

Corners in tables for trigonometric functions correspond to radian values:

So, it is not difficult to guess that 2π is a complete circle or 360 °.

Properties of trigonometric functions: sinus and cosine

In order to consider and compare the main properties of sinus and cosine, tangent and catangens, it is necessary to draw their functions. This can be done in the form of a curve located in a two-dimensional coordinate system.

Consider a comparative table of properties for sinusoids and cosineids:

Sinusoid	Kosinusoid
y \u003d sin x	y \u003d COS X
Odz [-1; one]	Odz [-1; one]
sin x \u003d 0, at x \u003d πk, where k ε z	cos x \u003d 0, at x \u003d π / 2 + πk, where k ε z
sin x \u003d 1, at x \u003d π / 2 + 2πk, where k ε z	cos x \u003d 1, at x \u003d 2πk, where k ε z
sin x \u003d - 1, at x \u003d 3π / 2 + 2πk, where k ε z	cos x \u003d - 1, at x \u003d π + 2πk, where k ε z
sIN (-X) \u003d - SIN X, i.e. function is odd	cOS (-X) \u003d COS X, i.e. function is even
	function periodic, the smallest period - 2π
sIN X\u003e 0, with x-owned I and II quarters or from 0 ° to 180 ° (2πk, π + 2πk)	cOS X\u003e 0, with x-X-owned I and IV quarters or from 270 ° to 90 ° (- π / 2 + 2πk, π / 2 + 2πk)
sIN X \u003c0, with x-X-owned III and IV quarters or from 180 ° to 360 ° (π + 2πk, 2π + 2πk)	cos x \u003c0, with x-x and third quarters or from 90 ° to 270 ° (π / 2 + 2πk, 3π / 2 + 2πk)
increases on the interval [- π / 2 + 2πk, π / 2 + 2πk]	increases on the interval [-π + 2πk, 2πk]
decreases at intervals [π / 2 + 2πk, 3π / 2 + 2πk]	decreases at intervals
derivative (SIN X) '\u003d COS X	derivative (COS X) '\u003d - SIN X

Determine whether the function is even or not very simple. It is enough to present a trigonometric circle with signs of trigonometric values \u200b\u200band mentally "folded" the schedule relative to the OX axis. If the signs coincide, the function is even, otherwise - an odd.

Introduction of radians and transfer of the main properties of sinusoids and cosineids allow you to bring the following regularity:

Make sure the formula is very simple. For example, for x \u003d π / 2 sinus is 1, as well as cosine x \u003d 0. You can check for the tables or tracing the functions of functions for the specified values.

Properties of Tangensoids and Kotangensoids

The graphs of the functions of Tangent and Kotangent differ significantly from sinusoids and cosineids. The values \u200b\u200bof TG and CTG are back to each other.

Y \u003d TG x.
TangentSoid tends to values \u200b\u200by at x \u003d π / 2 + πk, but never reaches them.
The lowest positive period of tangensoid is equal to π.
TG (- x) \u003d - TG x, i.e., the function is odd.
TG x \u003d 0, at x \u003d πk.
The function is increasing.
TG x\u003e 0, at x ε (πk, π / 2 + πk).
TG x \u003c0, at x ε (- π / 2 + πk, πk).
Derivative (TG x) '\u003d 1 / COS 2 \u2061X.

Consider the graphic image of the catangensoids below the text.

The main properties of Kotangensoids:

Y \u003d CTG x.
Unlike the functions of sinus and cosine, in TangentSoid Y, it can take the values \u200b\u200bof many valid numbers.
Kothangensoid tends to values \u200b\u200by at x \u003d πk, but never reaches them.
The smallest positive period of the catangensoid is equal to π.
CTG (- x) \u003d - CTG X, i.e. function is odd.
CTG x \u003d 0, at x \u003d π / 2 + πk.
The function is descending.
CTG x\u003e 0, at x ε (πk, π / 2 + πk).
CTG x \u003c0, at x ε (π / 2 + πk, πk).
Derivative (CTG x) '\u003d - 1 / sin 2 \u2061x fix

Trigonometric circle. Single circle. Numerical circle. What it is?

Attention!
This topic has additional
Materials in a special section 555.
For those who are strongly "not very ..."
And for those who are "very ...")

Very often, Terms trigonometric Circle, Single Circle, Numerical Circle Poor understood by students people. And completely in vain. These concepts are a powerful and universal assistant in all sections of trigonometry. In fact, this is a legal crib! Drew a trigonometric circle - and immediately saw the answers! Mustache? So let us ask, sin such a thing will not use. Moreover, it is completely simple.

For successful work with a trigonometric circle, you need to know only three things.

If you like this site ...

By the way, I have another couple of interesting sites for you.)

It can be accessed in solving examples and find out your level. Testing with instant check. Learn - with interest!)

You can get acquainted with features and derivatives.

How are the salad and water turn into borsch in terms of mathematics? How can the sum of two segments be transformed into trigonometry? To understand this, we need linear angular functions.

Linear angular functions are the laws of addition. See how the algebra turns into geometry, and the geometry turns into trigonometry.

Is it possible to do without linear angular functions? It is possible, because mathematics still do without them. The trick of mathematicians is that they always tell us only about those challenges that they themselves can decide, and never tell about those tasks that they do not know how to decide. See. If we know the result of the addition and one term, to search for another complimentary, we use subtraction. Everything. We do not know other tasks and do not know how to solve. What to do in the event that only we are known for the result of addition and are not known both terms? In this case, the result of addition must be decomposed into two terms with linear angular functions. Then we already choose, how can one term may be, and linear angular functions show what the second term should be, so that the result of the addition was exactly what we need. Such pairs of terms can be an infinite set. In everyday life, we wake up without decomposition of the amount, we have enough subtraction. But in the scientific research of the laws of nature, the decomposition of the amount on the components can be very useful.