Trigonometric circle Definition of values. How to remember the values \u200b\u200bof cosine and sinuses of the main points of the numerical circle
Example 1.
Find the radical mode of an angle of equal a) 40 °, b) 120 °, c) 105 °
a) 40 ° \u003d 40 · π / 180 \u003d 2π / 9
b) 120 ° \u003d 120 · π / 180 \u003d 2π / 3
c) 105 ° \u003d 105 · π / 180 \u003d 7π / 12
Example 2.
Find a degree to the angle of the angle of pronounced in radians a) π / 6, b) π / 9, c) 2 · π / 3
a) π / 6 \u003d 180 ° / 6 \u003d 30 °
b) π / 9 \u003d 180 ° / 9 \u003d 20 °
c) 2π / 3 \u003d 2 × 180 ° / 6 \u003d 120 °
Definition of sinus, cosine, tangent and catangens
The sine of an acute angle T of a rectangular triangle is equal to the attitude of the opposite category to the hypotenuse (Fig. 1):
The cosine of an acute angle T of a rectangular triangle is equal to the ratio of the adjacent catech for hypotenuse (Fig. 1):
These definitions belong to the rectangular triangle and are special cases of the definitions that are presented in this section.
Position the same right triangle In a numeric circle (Fig. 2).
We see that catat b. equal to a certain amount y. on the Y axis (ordinate axes), catat but equal to a certain amount x. on the X axis (abscissa axis). A hypotenuse from equal to the circle radius (R).
Thus, our formulas acquire a different appearance.
Since B \u003d y., a \u003d. x., C \u003d R, then:
y X.
SIN T \u003d -, COS T \u003d -.
R R.
By the way, then a different kind is found, naturally, the formulas of Tangent and Kotangens.
Since TG T \u003d B / A, CTG T \u003d A / B, then the true equations are true:
tG T \u003d. y./x.,
cTG \u003d. x./y..
But back to sinus and cosine. We are dealing with a numeric circle in which the radius is 1. So it turns out:
y.
SIN T \u003d - \u003d y.,
1
x.
cOS T \u003d - \u003d x..
1
So we come to the third, simpler type of trigonometric formulas.
These formulas are applicable not only to acute, but also to any other corner (stupid or deployed).
Definitions and formulas COS T, SIN T, TG T, CTG T.
From the formulas of Tangent and Kotangen follows another formula:
Equations of the numerical circle.
Sinus, Kosinus, Tangent and Kotnence signs in the fourth circumference:
| 1st quarter | 2nd quarter | 3rd quarter | 4th quarter | |
| COS T. | + | – | – | + |
| SIN T. | + | + | – | – |
| TG T, CTG T | + | – | + | – |
Cosine and sine main points of the numerical circumference:
How to remember the values \u200b\u200bof cosine and sinuses of the main points of the numerical circle.
First of all, you need to know that in each pair numbers the cosine values \u200b\u200bare first, the sinus values \u200b\u200bare second.
1) Note: With all the many points of the numerical circle, we only have a case with five numbers (in the module):
1 √2 √3
0; -; --; --; 1.
2 2 2
Make this "Opening" - and you will remove the psychological fear of the abundance of numbers: they are actually just five.
2) Let's start with integers 0 and 1. They are only on the axes of coordinates.
It is not necessary to learn by heart, where, for example, the cosine in the module has a unit, and where 0.
At the ends of the axis kosineov (axis h.), of course, cosines are equal to module 1, and sines are equal to 0.
At the ends of the axis sinusov (axis w.) sinuses are equal to module 1, and cosinees are equal to 0.
Now about signs. No zero sign has no sign. As for 1 - here you just need to remember the simplest thing: From the course of the 7th grade you know that on the axis h. Right from the center coordinate plane - Positive numbers, left - negative; on axis w. Up from the center go positive numbers, down - negative. And then you will not be mistaken with a sign 1.
3) We now turn to fractional values.
In all denominators, fractions are the same number 2. I won't be wrong to write in the denominator.
In the middle of the quarters, the cosine and sinus have an absolutely identical value of module: √2 / 2. In which case, they are familiar with plus or minus - see Table above. But you can hardly need such a table: you know this from the same course grade 7.
All coming to the axis h. Points have absolutely identical module cosine and sine values: (√3 / 2; 1/2).
The values \u200b\u200bof all closest to the axis w. Points are also absolutely identical in the module - and in them the same numbers, only they "changed" in some places: (1/2; √3 / 2).
Now about signs - here your interesting alternation (although with signs, we believe, you should easily understand and so).
If in the first quarter the values \u200b\u200band cosine, and sinus with a plus sign, then in a diametrically opposite (third) they are with a minus sign.
If in the second quarter with a minus sign only cosines, then in a diametrically opposite (fourth) - only sinuses.
It remains only to recall that in each combination of cosine and sinus values, the first number is the cosine value, the second number is the value of the sine.
Pay attention to another regularity: sinus and cosine of all diametrically opposite circumference points are absolutely equal to the module. Take, for example, opposite points π / 3 and 4π / 3:
cOS π / 3 \u003d 1/2, SIN π / 3 \u003d √3 / 2
COS 4π / 3 \u003d -1/2, SIN 4π / 3 \u003d -√3 / 2
The values \u200b\u200bof cosine and sinuses of two opposite points are different from the sign. But here there is its own pattern: sines and cosines of diametrically opposite points always have opposite signs.
It's important to know:
The values \u200b\u200bof cosine and sinuses of the numeric circle points are sequentially increasing or declining in a strictly defined order: from the smallest value to the largest and vice versa (see section "Ascending and descending trigonometric functions"- However, it is easy to see this, only just looking at the numerical circle above).
In descending order, such an alternation of values \u200b\u200bis obtained:
√3 √2 1 1 √2 √3
1; --; --; -; 0; – -; – --; – --; –1
2 2 2 2 2 2
They increase strictly in reverse order.
Realizing this simple patternYou will learn to quite easily determine the values \u200b\u200bof the sine and cosine.
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.
Allow you to establish a number of characteristic results - properties of sinus, cosine, tangent and catangens. In this article, we will consider three basic properties. The first of them indicates the signs of sinus, cosine, tangent and catangens angle α depending on whether the angle of which coordinate quarter is α. Further, we consider the property of the frequency, which establishes the invariaries of sine, cosine, tangent and catangent of the angle α when changing this angle to a whole number of revolutions. The third property expresses the relationship between the values \u200b\u200bof sine, cosine, tangent and catangent of opposite angles α and -α.
If you are interested in the properties of the functions of sinus, cosine, tangent and catangent, they can be studied in the appropriate section of the article.
Navigating page.
Sinus, Kosinus, Tangens and Cotangens signs on quarters
Below at this point will be encountered the phrase "Corner I, II, III and IV coordinate quarters". Explain what is the corners.
Take a single circle, we note on it the starting point A (1, 0), and turn it around the point O to the angle α, and we will assume that we will fall into point a 1 (x, y).
They say that the angle α is an angle I, II, III, IV coordinate quarterif the point A 1 lies in I, II, III, IV quarter, respectively; If the angle α is such that the point A 1 lies on any of the coordinate direct Ox or Oy, then this angle does not belong to any of the four quarters.
For clarity, we give a graphic illustration. The drawings below show the angles of rotation 30, -210, 585 and -45 degrees, which are angles I, II, III and IV coordinate quarters, respectively.
Corners 0, ± 90, ± 180, ± 270, ± 360, ... Degrees do not belong to any of the coordinate quarters.
Now we will understand which signs are the values \u200b\u200bof sinus, cosine, tangent and catangent angle of rotation α depending on whether the angle of which quarter is α.
For sinus and cosine it is simple.
By definition, the sine of the angle α is the ordinate point A 1. Obviously, in the I and II coordinate quarters, it is positive, and in the III and IV of the quarters - is negative. Thus, the sinus of the angle α has a sign plus in the I and II of the quarters, and the minus sign is in the III and VI of the quarters.
In turn, the cosine of the angle α is the abscissa point A 1. In I and IV quarters, it is positive, and in the II and III of the quarters - negative. Consequently, the cosine values \u200b\u200bof the angle α in I and IV quarters are positive, and in the II and III of the quarters are negative.
To determine the signs on the quarteens and catangens and Kotanznes, it is necessary to recall their definitions: Tangent is the ratio of the order of the point A 1 to the abscissa, and the Cotanence is the abscissa ratio of the point A 1 to the ordinate. Then because the rules of division of numbers With the same and different signs it follows that tangent and catangenes have a plus sign when the inscissions and order signs of the point A 1 are the same, and have a minus sign - when the inscissa signs and the ordinate points are different. Consequently, Tangent and Cotangenes angle have a sign + in I and III coordinate quarters, and a minus sign - in the II and IV quarters.
Indeed, for example, in the first quarter and the abscissa x, and the ordinate y point A 1 are positive, then the private X / Y, and private Y / X - positively, therefore, Tangent and Kotannce have signs +. And in the second quarter, the abscissa X is negative, and the ordinate y is positive, therefore x / y, and y / x are negative, from where Tangent and Kotangenes have a minus sign.
Go to the next property of sinus, cosine, tangent and catangens.
Property of periodicity
Now we will analyze, perhaps the most obvious property of sinus, cosine, tangent and catangent angle. It consists in the following: when the angle changes to an integer number of complete revolutions, the value of sinus, cosine, tangent and the catangent of this angle do not change.
This is understandable: when the angle is changed for a whole number of revolutions, we will always fall into a point A 1 on a single circle, consequently, the values \u200b\u200bof the sine, cosine, tangent and the catangent remain unchanged, since the coordinates of the point A 1 are unchanged.
With the help of the formulas under consideration, the property of sinus, cosine, tangent and catangens can be written as: sin (α + 2 · · z) \u003d sinα, cos (α + 2 · π · z) \u003d cosα, tg (α + 2 · π · z) \u003d TGα, CTG (α + 2 · · · z) \u003d Ctgα, where α is an angle of rotation in radians, z - any, the absolute value of which indicates the number of complete revolutions, to which the angle α changes, and the number Z sign indicates the direction turn.
If the angle of rotation α is defined in degrees, the specified formulas will be thrown in the form of sin (α + 360 ° · z) \u003d sinα, cos (α + 360 ° · z) \u003d cosα, tg (α + 360 ° · z) \u003d TGα , CTG (α + 360 ° · z) \u003d CTGα.
We give examples of using this property. For example, 
, as 
, but 
. Here is another example: or.
This property, together with the formulas of bringing, is very often used when calculating the values \u200b\u200bof sinus, cosine, tangent and catangen "large" corners.
The considered properties of sinus, cosine, tangent and Kotangenes sometimes refer to the property of frequency.
Properties of sinus, cosine, tangents and catangers of opposite angles
Let 1 be the point obtained as a result of the rotation of the initial point A (1, 0) around the point O to the angle α, and the point A 2 is the result of the rotation of the point A to the angle -α, the opposite corner α.
The property of sinuses, cosinees, tangents and catangents of opposite angles is based on a sufficiently obvious fact: the points mentioned above and 1 or 2 either coincide (at), or are located symmetrically relative to the OX axis. That is, if point A 1 has coordinates (x, y), then the point A 2 will have coordinates (x, -y). Hence the definitions of sinus, cosine, tangent and catangent write equality and.
Comparing them, come to the ratios between sinus, cosine, tangents and catangents of the opposite angles α and -α species.
This is the considered property in the formula formula.
We give examples of using this property. For example, equality and 
.
It remains only to notice that the property of sinuses, cosinees, tangents and catangents of opposite angles, as well as the previous property, is often used when calculating the values \u200b\u200bof sinus, cosine, tangent and catangent, and allows you to completely get away from negative angles.
Bibliography.
- Algebra: Studies. For 9 cl. environments Shk. /u. N. Makarychev, N. G. Mindyuk, K. I. Neshkov, S. B. Suvorov; Ed. S. A. Telikovsky. - M.: Education, 1990.- 272 C.: Il.- ISBN 5-09-002727-7
- Algebra and starting analysis: studies. For 10-11 cl. general education. institutions / A. N. Kolmogorov, A. M. Abramov, Yu. P. Dudnitsyn, etc.; Ed. A. N. Kolmogorova.- 14th ed. - M.: Enlightenment, 2004.- 384 C.: Il.- ISBN 5-09-013651-3.
- Bashmakov M. I. Algebra and start analysis: studies. For 10-11 cl. environments shk. - 3rd ed. - M.: Enlightenment, 1993. - 351 C.: Il. - ISBN 5-09-004617-4.
- Gusev V. A., Mordkovich A. G. Mathematics (benefit for applicants in technical schools): studies. benefit. - m.; Higher. Shk., 1984.-351 p., Il.
Type of lesson: systematization of knowledge and intermediate control.
Equipment: Trigonometric circle, tests, cards with tasks.
Objectives lesson: Systematize the studied theoretical material on the definitions of sinus, cosine, the tangent of the corner; Check the degree of learning of knowledge on this topic and application in practice.
Tasks:
- To summarize and consolidate the concepts of sinus, cosine and tangent angle.
- To form a comprehensive view of trigonometric functions.
- Promote students' desire and the need to study trigonometric material; Briefing the culture of communication, the ability to work in groups and needs in self-education.
"Who does a smallest and thinks himself, that
It becomes more reliable, stronger, smarter.
(V.Shukshin)
DURING THE CLASSES
I. Organizational moment
The class is represented by three groups. In each group consultant.
Teacher reports the topic, objectives and objectives of the lesson.
II. Actualization of Knowledge (Front Work with Class)
1) Work in groups on tasks:
1. Formulate the definition of sin angle.
- What signs do sin α in each coordinate quarter?
- Under what values \u200b\u200bit makes sense, expression sin α, and what values \u200b\u200bcan it take?
2. The second group of the same questions for COS α.
3. The third group of answers is preparing on the same TG α and CTG α issues.
At this time, three students operate on their own on cards on cards (representatives of different groups).
Card number 1.
Practical work.
Using a single circle, calculate the values \u200b\u200bof SIN α, COS α and TG α for an angle of 50, 210 and - 210.
Card number 2.
Determine the sign of expression: TG 275; COS 370; SIN 790; TG 4.1 and SIN 2.
Card number 3.
1) Calculate:
2) Compare: COS 60 and COS 2 30 - SIN 2 30
2) orally:
a) proposed a number of numbers: 1; 1.2; 3; 0 ,, - 1. Among them are too much. What kind of SIN α or COS α can express these numbers (can SIN α or COS α take these values).
b) does the meaning of the expression: COS (-); SIN 2; TG 3: CTG (- 5); ; CTG0;
CTG (- π). Why?
c) there is a smallest and the greatest value SIN or COS, TG, CTG.
D) Is it true?
1) α \u003d 1000 is an angle of II of the II quarter;
2) α \u003d - 330 is an angle of IV quarter.
e) the numbers correspond to the same point on a single circle.
3) work at the board
№ 567 (2; 4) - find the value of the expression
№ 583 (1-3) Determine the sign of expression
Homework:table in notebook. № 567 (1, 3) № 578
III. Assimplement of additional knowledge. Trigonometry in palm
Teacher: It turns out that the values \u200b\u200bof sinuses and cosine angles "are" on your palm. Stretch your hand (any) and dig as stronger fingers (as on the poster). One student is invited. We measure the corners between our fingers.
A triangle is taken, where there is an angle of 30, 45 and 60 90 and apply the top of the angle to the moon bug on the palm. The Moon's Buds is at the intersection of the member's continuation and thumb. One side we combine with the little finger, and the other side with one of the other fingers.
It turns out between the little finger and the thumb 90, between the little finger and the nameless - 30, between the little finger and the average - 45, between the little finger - 60. And this is all people without exception.
mysietician number 0 - corresponds to 0,
Unnamed number 1 - corresponds to 30,
Middle No. 2 - corresponds to 45,
Indicating number 3 - corresponds to 60,
Big number 4 - corresponds to 90.
Thus, we have 4 fingers on our hand and remember the formula:
№ finger |
Angle |
Value |
|
||
|
||
|
This is just a mnemonic rule. In general, the value of SIN α or COS α should be known by heart, but sometimes this rule will help in a difficult moment.
Comeume the rule for COS (angles unchanged, and the countdown from the thumb). Physical pause associated with SIN α or COS α signs.
IV. Checking learning Zun.
Independent work with feedback
Each student receives a test (4 options) and a list with answers to all the same.
Test
Option 1
1) With what angle turning the radius takes the same position as when turning at an angle 50.
2) Find the value of the expression: 4COS 60 - 3SIN 90.
3) Which of the numbers is less than zero: SIN 140, COS 140, SIN 50, TG 50.
Option 2.
1) With what angle turning the radius also takes the position as when it is rotated at an angle 10.
2) Find the value of the expression: 4COS 90 - 6Sin \u200b\u200b30.
3) Which of the numbers is greater than zero: SIN 340, COS 340, SIN 240, TG (- 240).
Option 3.
1) Find the value of the expression: 2CTG 45 - 3COS 90.
2) Which of the numbers is less than zero: SIN 40, COS (- 10), TG 210, SIN 140.
3) the angle of which quarter is the angle α if sin α\u003e 0, cos α< 0.
Option 4.
1) Find the value of the expression: TG 60 - 6CTG 90.
2) Which of the numbers is less than zero: sin (- 10), COS 140, TG 250, COS 250.
3) the angle of which quarter is the angle α if CTG α< 0, cos α> 0.
BUT |
B. |
IN |
G. |
D. |
E. |
J. |
Z. |
AND |
L. |
M. |
|
N. |
ABOUT |
P |
R |
FROM |
T. |
W. |
F. |
H. |
Sh |
YU |
I |
(word - trigonometry key)
V. Information from the history of trigonometry
Teacher: Trigonometry is a fairly important section of mathematics for a person's life. Modern view Trigonometry gave the largest mathematician 18 century Leonard Eileler - Swiss by origin long years Worked in Russia and who was a member of the St. Petersburg Academy of Sciences. It introduced the known definitions of trigonometric functions formulated and proved well-known formulas, we will learn them later. The life of Euler is very interesting and I advise you to get acquainted with it on the book of Yakovlev "Leonard Euler".
(Message guys on this topic)
Vi. Summing up the lesson
Game "Cross - Noliki"
Two students are involved in the most active. They are supported by groups. The task solution is recorded in the notebook.
Tasks
1) find an error
a) sin 225 \u003d - 1.1 V) SIN 115< О
b) cos 1000 \u003d 2 g) COS (- 115)\u003e 0
2) Express the angle in degrees
3) Express the angle 300 in the radians
4) What the largest and smallest value may have an expression: 1+ SIN α;
5) Determine the sign of expression: SIN 260, COS 300.
6) In which quarter of the numeric circle is the point
7) Determine the signs of expressions: COS 0.3π, SIN 195, CTG 1, TG 390
8) Calculate:
9) Compare: SIN 2 and SIN 350
VII. Reflection lesson
Teacher: Where can we meet with trigonometry?
On what classes in grade 9, and now you apply the concepts of SIN α, COS α; TG α; CTG α and for what purpose?
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 when scientific research The laws of nature 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 when different values 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
Viewed an interesting video about row Grande One minus one plus one minus one - Numberphile . Mathematics lie. They did not verify equality during their reasoning.
This echoes my arguments about.
Let's look at the signs of deceiving us with mathematicians. At the very beginning of reasoning, mathematics say that the sum of the sequence depends on the even number of elements in it or not. This is an objectively established fact. What happens next?
Further mathematics from the unit deduct the sequence. What does this lead to? This leads to a change in the number of sequence elements - even quantity changes to an odd, odd changes to even. After all, we added to a sequence one element equal to one. Despite all the external similarity, the sequence before the conversion is not equal to the sequence after the transformation. Even if we argue about the infinite sequence, it is necessary to remember that the infinite sequence with an odd number of elements is not equal to an infinite sequence with an even number of elements.
By signing the equality between two different elements by sequences, mathematics argue that the sequence sum does not depend on the number of elements in the sequence, which contradicts the objectively established fact. Further reasoning about the sum of the infinite sequence is false, since they are based on false equality.
If you see that mathematics in the course of evidence set brackets, the elements of the mathematical expression are rearranged by places, something is added or removed, be very attentive, most likely you are trying to deceive you. Like card magicians, mathematics with various manipulations with an expression distract your attention to sustate the false result as a result. If the card focus you can not repeat, not knowing the secret of deception, then in mathematics everything is much simpler: you don't even suspect anything about the deception, but the repetition of all manipulations with the mathematical expression allows you to convince others in the correctness of the result, just like when Well, convinced you.
Question from the hall: and infinity (as the number of elements in the sequence s), is it even or odd? How can the parity be changed that parity does not have?
Infinity for mathematicians, as the kingdom of heaven for Popov - no one has ever been there, but everyone knows exactly how everything is arranged there))) I agree, after death you will be absolutely indifferent, even or an odd number of days you lived, but ... adding Only one day at the beginning of your life, we will get a completely different person: the last name, the name and patronymic of him is exactly the same, only the date of birth is completely different - he was born in one day before you.
And now essentially))) Suppose the final sequence that has parity loses this parity when moving to infinity. Then, any finite segment of the infinite sequence should lose parity. We do not observe this. The fact that we cannot say for sure, an even or odd number of elements in an infinite sequence, does not mean that the parity disappeared. Can not parity if it is, disappeared without a trace in infinity, as in the sleeve of Shulera. For this case there is a very good analogy.
You never asked the cuckoo sitting in the clock, in what direction the arrow of the clock rotates? For her, the arrow rotates in the opposite direction of the one we call "clockwise". As it does not paradoxically sound, but the direction of rotation depends solely on which side we observe the rotation. And so, we have one wheel that rotates. We cannot say, in which direction is rotation, since we can observe it both on the one hand the plane of rotation and the other. We can only witness the fact that the rotation is. Complete analogy with the parity of the infinite sequence S..
Now add the second rotating wheel, the plane of rotation of which is parallel to the plane of rotation of the first rotating wheel. We still can't say for sure, in which direction these wheels rotate, but we can absolutely just say, both wheels are rotated in one direction or in opposite. Comparing two endless sequences S. and 1-S.I, with the help of mathematics, showed that these sequences have different parity and put the sign of equality between them - this is an error. I personally believe mathematics, I do not trust mathematicians))) By the way, for a complete understanding of the geometry of transformations of infinite sequences, it is necessary to introduce the concept "simultaneity". It will need to draw it.
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 a 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 as an example, take an infinite set of natural numbers, then the 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 under 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 actions in algebraic system Designations and in the system of designations adopted in the theory of sets, with a detailed listing of elements of the set. 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:
We read: "... A rich theoretical basis of 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, is that for the theory of mathematics, they invented own language and your own designations. 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
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, the scientific community has not yet been possible ... to research the issue was involved mathematical analysis, set theory, new physical and philosophical approaches; 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 burst 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.