Today we look at the problem of B8 with trigonometry at the її classical razuminn, de vichayutsya zvichaynі. straight-cut tricots. Therefore, there won’t be any trigonometric kіl and negative cutіv this year - there will be no more than an equal sine and cosine.
Such a goal is to become approximately 30% of the total amount. Remember: if the task is B8, at least once you guess kut π, you won't be violating in other ways. We are obov'azkovo razglyana їh the next hour. And now - a smut for the lesson:
Trikutnik - a figure on a flat surface, which is formed from three points and winds, which are connected. In fact, the entire lane is closed with three lanes. Krapki are called tops of trikutnik, and vіdrіzki - sides. It is important to respect that the peaks are not guilty lie on the same straight line, otherwise the tricot turns around the tops.
It is often called trikutnik not only the laman itself, but also a part of the area, as it is surrounded by the laman. In this rank, you can designate the trikutnik area.
Two trikutniks are called equal, because one can be taken from the other way of one or more of the area: zsuva, turn or symmetry. In addition, it is necessary to understand similar tricks: they are equal, and the other sides are proportional.
Tse trikutnik ABC. Moreover, it is a straight-cut tricot: in Newmou ∠C = 90°. The most common ones are used in problem B8.
Everything you need to know to solve the problem B8 - a bunch of simple facts from geometry and trigonometry, as well as a big scheme of decoupling, in which facts are victorious. Let's get rid of just "fill your hand."
Let's start from the facts. The stench is divided into three groups:
- Appreciation of that heritage from them;
- Basic identity;
- Symmetry at trikutnik.
It is impossible to say which of these groups is important, which is simple. But the information that is hidden in them, allow you to read be-yaké zavdannya B8. So you need to know everything. So let's go!
Group 1: their legacy
Let's look at the tricot ABC, where ∠C is a straight line. For the cob - vyznachennya:
Sinus kuta - tse extension of the protilegus leg to hypotension.
The cosine of the kuta is the value of the adjoining leg to the hypotension.
The tangent of the kuta is the extension of the protractile leg to the snug one.
One kut or vіdrіzok can go up to rіznyh straight-cut trikutnikіv. More than that, more often than not, the very same crotch with a leg in one tricot and hypotenuse in the other. Ale about tsedali, but for the time being pracyuvatimemo zі svechaynim kut A. Todi:
- sinA=BC:AB;
- cosA=AC:AB;
- tgA=BC:AC.
Main findings from the appointment:
- sin A = cos B; cos A = sin B
- tg A \u003d sin A: cos A - call the tangent, sine and cosine of one kut
- Yakscho ∠A + ∠B = 180°, tobto. cut the sum, then: sin A \u003d sin B; cos A = -cos B.
If you want - wilt, if you want - no, but there are enough facts to solve about a third of all trigonometric tasks B8.
Group 2: basic identity
The first and most common identity is the Pythagorean theorem: the square of the hypotenuse is equal to the sum of the squares of the categories. A hundred and fifty tricutnik ABC, looked at above, this theorem can be written as follows:
AC 2 + BC 2 = AB 2
I immediately - little respect, as if on the coast of a chitach in the presence of rich pardons. If you fail the task, zavzhd (feel, zavzhd!) Write down the Pythagorean theorem yourself in this way. Do not try to hang the leg straight, as it is necessary. It is possible that you save a couple of rows of calculations, but on your own “economy” more points were spent, lower be it in geometry.
Another identity is from trigonometry. Looking at the approaching rank:
sin 2 A + cos 2 A = 1
This is how it is called: basic trigonometric totality. With yoga, it is possible to help through the sine viraziti cosine i navpaki.
Group 3: Symmetries in tricoutnik
Those that are written below are worth less than equal-femoral trikutniks. If the task is not such a figure, then there are enough facts from the first two groups to figure it out.
Also, let's look at the equilateral tricot ABC de AC \u003d BC. Draw to the base the height of CH. We take into account the following facts:
- ∠A = ∠B. As a last resort, sin A = sin B; cos A = cos B; tan A = tan B.
- CH - yak height, and th bisector, tobto. ∠ACH = ∠BCH. Similarly, equal and trigonometric functions of these cuts.
- Also CH - tse median, to that AH = BH = 0.5 AB.
Now, if all the facts are considered, let's move on without a trace to the methods of solving.
Heading scheme for decomposing tasks B8
Geometry looks like algebra, since there are no simple and universal algorithms in it. The skin must be brought up from scratch - and tse folding. However, it is still possible to give some protegeal recommendations.
For the cob of the cob, designate an unknown side (such as є) for X. Let's create a solution scheme, which consists of three points:
- As a task, it’s a tricouter, to zastosuvat up to the new all possible facts from the third group. Find the equals of cuts and analyze their trigonometric functions. In addition, a ribbed tricot is rarely straight-cut. That's why joke the straight-cut tricutniks - they stink there obov'yazkovo є.
- Zastosuvati up to a rectilinear tricot in fact from the first group. Kіntseva meta - otrimati ryvnyannia schodo zminnoї X . We know X - we untie the task.
- Even though the facts from the first group were insufficient, the facts from the other group were stagnant. I'm joking again X.
Apply the solution of tasks
And now we’ll try for help to gain knowledge of the broadest task B8. Do not be surprised that with such an arsenal, the text of the decision will not appear rich in mind. I'm glad :)
Manager. For tricot ABC cut C is 90 °, AB = 5, BC = 3. Find cos A.
For appointments (group 1) cos A = AC : AB . The hypotenuse AB is visible to us, and the axis of the legs AC is brought to shukati. Significantly yoga AC = x.
Let's move on to group 2. Tricut ABC is straight. For the Pythagorean theorem:
AC 2 + BC 2 = AB 2;
x 2 + 3 2 = 5 2;
x 2 \u003d 25 - 9 \u003d 16;
x = 4.
Now you can know the cosine:
cos A = AC: AB = 4: 5 = 0.8.
Manager. In trikutnik ABC cut B is 90 °, cos A = 4/5, BC = 3. BH - height. Find AH.
Significantly joking bіk AH = x і look at trikunik ABH. Він is straight-cut, moreover, ∠AHB = 90° behind the head. To that cos A = AH: AB = x: AB = 4/5. This proportion, її can be rewritten as follows: 5 x = 4 AB. Obviously, we know x, so we know AB.
Let's take a look at the tricot ABC. Vin is also rectilinear, moreover, cos A = AB: AC. We do not know neither AB nor AC, so let's move on to another group of facts. Let us write down the main trigonometric toto- nism:
sin 2 A + cos 2 A = 1;
sin 2 A \u003d 1 - cos 2 A \u003d 1 - (4/5) 2 \u003d 1 - 16/25 \u003d 9/25.
The scales of the trigonometric function of the acute kuta are positive, it is necessary to have sin A = 3/5. From the lower side, sin A = BC: AC = 3: AC. We take the proportion:
3:AC=3:5;
3 AC = 3 5;
AC = 5.
Also, AC = 5. Then AB = AC cos A = 5 4/5 = 4. We know AH = x:
5 x = 4 4;
x = 16/5 = 3.2.
Manager. In trikutniku ABC AB = BC, AC = 5, cos C = 0.8. Find the height of CH.
Significantly for the Shukan, the height is CH = x. In front of us is a rіnofemoral tricot ABC, in yakomu AB \u003d BC. Also, from the third group of facts, maybe:
∠A = ∠C ⇒ cos A = cos C = 0.8
Let's take a look at the tricot ACH. Vin is straight-cut (∠H = 90°), and AC = 5 and cos A = 0.8. By appointment, cos A \u003d AH: AC \u003d AH: 5. We take the proportion:
AH:5=8:10;
10 AH = 5 8;
AH = 40: 10 = 4.
I lost my speed with another group of facts, and the Pythagorean theorem itself for the tricutnik ACH:
AH 2 + CH 2 = AC 2;
4 2 + x 2 = 5 2;
x 2 \u003d 25 - 16 \u003d 9;
x = 3.
Manager. For a straight knit ABC ∠B = 90°, AB = 32, AC = 40. Find the sine of the cut CAD .
Oskіlki us v_doma hypotenuse AC = 40 and leg AB = 32, you can know the cosine of the cut A: cos A = AB: AC = 32: 40 = 0.8. This is a fact from the first group.
Knowing the cosine, you can know the sine through the basic trigonometric identity (a fact from another group):
sin 2 A + cos 2 A = 1;
sin 2 A \u003d 1 - cos 2 A \u003d 1 - 0.8 2 \u003d 0.36;
sin A = 0.6.
When the sine is significant, the fact that the trigonometric functions of the acute kuta are positive became clear again. Lost respect, scho kuti BAC and CAD sum_zhnі. From the first group of facts may:
∠BAC + ∠CAD = 180°;
sin CAD = sin BAC = sin A = 0.6.
Manager. Trikutniku has ABC AC = BC = 5, AB = 8, CH is height. Find tg A.
Trikutnik ABC - equal thighs, CH - height, it is respectable that AH = BH = 0.5 AB = 0.5 8 = 4. This fact is from the third group.
Now let's look at the tricot ACH: Newmu has ∠AHC = 90°. You can use the tangent: tg A = CH: AH. Ale AH = 4, then it is left to know the side of CH, since CH = x is significant. According to the Pythagorean theorem (fact from group 2) we can:
AH 2 + CH 2 = AC 2;
4 2 + x 2 = 5 2;
x 2 \u003d 25 - 16 \u003d 9;
x = 3.
Now everyone is ready to know the tangent: tg A \u003d CH: AH \u003d 3: 4 \u003d 0.75.
Manager. Tricutnik ABC AC = BC, AB = 6, cos A = 3/5. Find the altitude of AH.
Significantly Shukan Visot AH = x. I know the tricoutnik ABC - equal thighs, it is respectful that ∠A = ∠B, also, cos B = cos A = 3/5. This fact is from the third group.
Let's take a look at the tricot ABH. Behind the head vein is straight-cut (∠AHB = 90°), moreover, in the house the hypotenuse AB = 6 і cos B = 3/5. Ale cos B = BH: AB = BH: 6 = 3/5. Take the proportion:
BH:6=3:5;
5 BH = 6 3;
BH = 18/5 = 3.6.
Now we know AH = x by the Pythagorean theorem for the tricot ABH :
AH 2 + BH 2 = AB 2;
x 2 + 3.6 2 \u003d 6 2;
x 2 \u003d 36 - 12.96 \u003d 23.04;
x = 4.8.
Dodatkovі mirkuvannya
Buvayut non-standard zavdannya, de looking more facts and schemes of marni. It's a pity that such a time needs a true individual case. Like zavdannya to love to give at all "trial" and "demonstration" trials.
Below are two real missions, which were shown on a trial EDI near Moscow. Alone ran into them, to tell about the high folding of their zavdan.
Manager. For a straight-cut tricot ABC іz kuta C = 90°, a median and height were carried out. Apparently, A = 23°. Find ∠MCH.
Respectfully, the median CM is drawn to the hypotension AB to that M is the center of the described stake, that is. AM = BM = CM = R, where R is the radius of the described stake. Also, tricot ACM is equi-femoral and ∠ACM = ∠CAM = 23°.
Now let's take a look at the ABC and CBH tricots. For the mind, offending trikutniks are straight. In addition, ∠B is hot. Also, tricots ABC and CBH are similar to two cuts.
Similar tricots have proportional elements. Zokrema:
BCH=BAC=23°
Let's take a look at ∠C . Vin is direct, i, moreover, ∠C = ∠ACM + ∠MCH + ∠BCH . In ts_y evenness ∠MCH - hooting, and ∠ACM and ∠BCH in vіdomі ta equal 23 °. Maemo:
90° = 23° + MCH + 23°;
MCH = 90° - 23° - 23° = 44°.
Manager. The perimeter of a rectangle is 34, and the area is 60. Find the diagonal of the rectangle.
Significantly, the sides of a rectangle are: AB = x, BC = y. Virazimo perimeter:
P ABCD = 2 (AB + BC) = 2 (x + y) = 34;
x+y=17.
Similarly, we can see the area: S ABCD \u003d AB BC \u003d x y \u003d 60.
Now let's take a look at the tricot ABC. Vin is straight-cut, so we write down the Pythagorean theorem:
AB 2 + BC 2 = AC 2;
AC 2 = x 2 + y 2.
Respectfully, that from the formula of the square of the difference, the equality is clear:
x 2 + y 2 \u003d (x + y) 2 - 2 x y \u003d 17 2 - 2 60 \u003d 289 - 120 \u003d 169
Also, AC2=169, stars AC=13.
Trikutnik maє miraculous power - tse zhorstka to post, tobto. with a post-yniy dozhiny side, it is not possible to change the shape of the tricot. Tsya power trikutnik rob yoga indispensable in technіtsі and everyday life. Structural elements in the shape of the knitwear take their shape, for example, elements in the shape of a square or a parallelogram. In addition, the tricot is the simplest bagatok, and whether it’s a bagatok, you can imagine it in a look at a set of trikutniks.
The main power and trikutnik formulas
Designation: A, B, C - kuti trikutnika,
a, b, c - opposite sides,
R - radius of the described stake,
r - radius of the inscribed stake,
p - napіvperimeter, (a + b + c) / 2,
S - trikutnik area.
The sides of the tricoutnik are tied with offensive irregularities
a ≤ b + c
b ≤ a + c
c ≤ a + b
In one of them, the tricutnik is called a virogenim in one of them. They gave a glimpse of non-virogeneous vibrations.
Trikutnik can be unequivocally (to the point of zsuvu and turn) to be assigned to the next three main elements:
a, b, c - on three sides;
a, b, C - from both sides and kutu between them;
a, B, C - to the side and two lie down to it.
The sum of kutiv be-a kind of trikutnik is post_yna
A + B + C = 180°
1. Rectangular tricot. Designation of trigonometric functions.
We can look at a straight-cut tricutnik, showing a little one.Kut B = 90° (straight).
Sine function: sin(A) = a/b.
Cosine function: cos(A) = c/b.
Tangent function: tg(A) = a/c.
Cotangent function: ctg(A) = c/a.
2. Rectangular tricot. Trigonometric formulas.
a = b * sin(A)c = b * cos(A)
a = c * tg(A)
Div. also:
- The Pythagorean theorem is a sample of simple proofs of the theorem.
3. Rectangular tricot. Pythagorean theorem.
b2 = a2 + c2For the help of the Pythagorean theorem, you can induce a direct kut, as if by hand there are no suitable tools, for example, cosince. For the help of two lines or two cloaks of a sack, we’ll take a cathet with a length of 3 and 4. We’ll destroy it or we’ll tear it up, the docks of a high blood pressure will not become equal 5 (3 2 + 4 2 \u003d 5 2).
On the page of the Pythagorean theorem, a few simple proofs of the theorem are listed.
"The power of a straight-cut tricoutnik" - Proof. The sum of two good cuts of a straight-cut tricot is 90 °. First dominion. Let's look at the straight-cut ABC tricot, yakomu? A-straight, B \u003d 30 ° i mean? W = 60°. Another power. Other power Other power Third power Zavdannya. Looks like a straight-cut ABC tricot, in which the AC leg has more half of the PS hypotenuse.
"Trigonometry" - the basic formulas of flat trigonometry. Cotangent - ratio of cosine to sine (tobto value, wrapped to tangent). Trigonometry. For the hospitality of the kutіv novі vznachennya spіvpadat іz kolishni. Trikutnik area: Cosine - extension of the adjacent leg to hypotenuse. Menelaus of Alexandria (100 AD) Having written the "Sphere" in three books.
"Zavdannya on a straight-cut tricutnik" - The Pythagoreans were engaged in the proof of the sign of equivalence of tricutniks. In Egypt, Thales got stuck on a rich rock, cultivating science in Thebes and Memphis. Biography Thales. Not far away stands the great temple of Apollo with marmur altars and statues. Miletus is the fatherland of Thales. Far off the road, Milesian merchants-sailors were breaking.
"Straight-cut paralepiped" - The faces of the parallelepiped, which do not have parallel vertices, are called protile. A parallelepiped is a hexahedron, all of its faces (substantiated) are parallelograms. Volume of a rectangular parallelepiped. The word was used by the old Greek confessions of Euclid and Heron. Dovzhina height height. The parallelepiped, the mustache of a square, is called a cube.
"Trigonometriya 10 class" - V_dpovid_. 1st variant (2nd variant) Calculate: Work with tests. Mathematical dictation. Historical proof. The robot beat the board. "The transformation of trigonometric virases". So that it would be easier for everyone to live, so that it would be viable, so that it could. Proof of the sameness.
"Volume of a right-angled parallelepiped" - How do the ribs line up with the rib AE? Vіdrіzok. Reminder for knowing the area of the surface of a rectangular parallelepiped. Rivni. Square. 5. The cube has equal edges. Razvyazannya tasks. Mathematics Grade 5 Cube. Dovzhini, width and height. (Flat, volume). Yakі peaks lie down to the foundation? 4. Paralepiped has 8 ribs.
To say simply, tse veggies, cooked by the water for a special recipe. I will look at two ingredients (vegetable salad and water) and the finished result is borscht. Geometrically, it is possible like a rectangle, in which one side means lettuce, the other side means water. The sum of the two sides is significant borscht. The diagonal and area of such a "borscht" straight cut is simply mathematical concepts and in no way vikoristovuyutsya in recipes for preparing borscht.
Like lettuce and water turn into borscht at the sight of mathematics? How can the sum of two breezes transform into trigonometry? To be clear, we need linear edge functions.
You don’t know anything about linear kutov functions from the assistants of mathematics. Aje can't do mathematics without them. The laws of mathematics, like the laws of nature, are practiced independently, in addition, we know about their foundations.
Linear kutov_ functions - tse laws of folding. Marvel at how algebra transforms into geometry, and geometry transforms into trigonometry.
What can you do without linear hood functions? You can, even mathematicians do without them. The trick of mathematics lies in the fact that the stench always tell us only about those tasks, like the stench can vire, and in no way tell about those tasks, like the stink does not viirish. Marvel. As we know the result of the folding of that one supplement, for the sake of another supplement, we win the prize. Mustache. We don’t know any other tasks and we can’t believe it. Why work in that mood, how can we only see the result of the additional payment and not feel the insult of the additional payment? In this case, the result of the addition should be divided into two additions for the help of linear kutovyh functions. Let us already choose for ourselves, as if we could only add one more, and show the linear kutov functions, as we can have other add-ons, so that the result of the add-on is the same as we need. There can be no such pairs of dodanks. In everyday life, we miraculously manage without spreading the bag, we have enough knowledge. And the axis, with scientific achievements of the laws of nature, laying out sums for dodanki can be needed.
Another law of folding, about which mathematicians do not like to talk (another one of their cunning), vimaga, so that the additions are small, however, alone in the world. For a salad, drive that borscht, you can be alone in the world, obsyagu, vartost, or alone in the world.
The little one shows two equal numbers for mathematics. First rіven - tse vіdminnostі in the field of numbers, yakі znachenі a, b, c. Tse those who are engaged in mathematicians. Other rіven - tse vіdmіnnostі in the area of \u200b\u200bone vimir, as indicated by the square arms, it is marked with a letter U. Physicists are engaged in this. We can understand the third row - the diversity of the area of describing objects. Different objects can mother the same number of the same loneliness in the world. Naskіlki tse important, we can give borscht a trigonometry butt. As far as we add the lower index to the same value of one in the world of different objects, we can say exactly how the mathematical value describes a particular object and how it changes with the hour or at the link with the actions. letter W I will sign water, letter S lettuce lettuce B- Borsch. The axis yak vyglyadatimut linear kutovі functions for borscht.
For example, we take a part of the water and a part of the salad, at once the stench is transformed into one portion of borscht. Here I will preach to you the trochs of vodvoliktisya in the borscht and guess the childishness in the distance. Remember, how did they teach us to stack bunnies at once and that pumpkin? It was necessary to know, skilki of the whole look like weide. What did they teach us to work for? We were taught to learn the unicity of the world of numbers and add up the numbers. So, whether or not a number can be added to another number. This is a direct path to autism of modern mathematics - mi robimo nezrazumilo, nezrazumіlo navіscho and even nastily sensibly, as if reality is troubling, even with three rіvnіv vіdmіnnosti mathematicians operate with more than one. It would be better to learn how to go from one alone to the other.
І bunnies, і kachechok, і zvіryat can be porahuvat in pieces. One solemn unity of peace for various objects allows us to put them together. Tse childish variant of the task. Look at the similar task for the grown-ups. What do you see, how to fold the bunnies that pennies? Here you can suggest two solutions.
First option. Significantly the market price of the bunnies and fold it with an obvious penny sum. We took away the total wealth of our wealth from a penny equivalent.
Another option. You can put a lot of rabbits together with a lot of penny bills that we have. We take away a small amount of dry lane from pieces.
Like Bachite, that same law of folding allows you to take different results. All lay in the form of what we want to know.
Ale, let's turn to our borscht. Now we can wonder what to consider for the different values of the cut of the linear cut functions.
Kut is equal to zero. We may have a salad, but we don’t have water. We can't cook borscht. The quantity of borscht is also equal to zero. Tse zovsim does not mean that zero borscht is equal to zero water. Zero borscht can be buti with zero salad (straight kut).
Especially for me, the main mathematical proof of the fact that . Zero does not change the number of days before the date. It’s worth it to that which is impossible to add itself, for example, there is only one addendum and the other daily addendum. You can put it all right, but remember - all the mathematical operations with zero were invented by the mathematicians themselves, to that, give your logic and stupidly cramming the meaning, invented by mathematicians: , dorivnyuє zero", "behind the point zero" is the other way around. To remember once, that zero is not a number, and you already don’t win any food, that is zero by a natural number of chi, so that such food is taken care of by any sense: how can you take in a number those who are not a number. It's all the same, what to feed, to what color you can see an invisible color. Add zero to - tse those same, scho farbuvati farboi, as if you don’t know. They waved a dry penzlik and we say to everyone that "we were farmed." Ale, I was a little excited.
Kut greater for zero, ale less than forty-five degrees. We may have a lot of salad, but a little water. As a result, we take thick borscht.
Kut dorivnyuє forty-five degrees. We can have water and salad in equal quantities. This is the ideal borscht (keep it up for me to cook, it’s just math).
Kut more than forty-five degrees, ale less than ninety degrees. We have a lot of water and a little salad. Viide is a rare borscht.
Straight cut. We have water. In the salad, we lost more than hope, the shards of the kut mi continue to die in the line, as if it meant salad. We can't cook borscht. The quantity of borscht is equal to zero. At such a time, try and drink water, while it’s out)))
Axis. Like so. I can tell other stories here, as they will be more ancient.
Two friends small their shares in a shared business. After driving in one of them, everything went to the other.
Appearance of mathematics on the planet.
All of the history of my mathematics is told for the help of linear kutov functions. As another time, I will show you the real scope of these functions in the structure of mathematics. In the meantime, let's turn to trigonometry, wrestling with that clear projection.
Saturday, July 26, 2019
Wednesday, 7 September 2019
Concluding the rozmov about, it is necessary to look at the faceless. It gave to the fact that the understanding of "inconsistency" on mathematicians is like a boa constrictor on a rabbit. The trembling gasp in front of the inconsistency helps mathematicians to a healthy mind. Axis butt:
Pershodzherelo to know. Alpha means real number. The sign of equivalence in pointing virazes is about those who can add a number to indistinctness, or inconsistency, nothing changes, as a result, such inconsistency itself will appear. If I take impersonal natural numbers in the form of a butt, then looking at the butts can be represented in this way:
For a scientific proof of their correctness, mathematicians used a wide variety of methods. I am especially amazed at all methods, like at the dance of shamans with tambourines. In fact, all the smells are brought to the point that either a part of the rooms is not occupied and new guests are settled in them, or a part of them is left at the corridor to call the place for the guests (or call it in a human way). My glance at similar solutions, I clave at the form of a fantastic explanation about the Blonde. Why are my mirrorings grounded? The resettlement of an inexhaustible number of people will require a lot of time. After that, as we have opened the first room for the guest, one of the guards will walk along the corridor from his room to the end of the century. Obviously, the factor can be stupidly ignored for a while, but it will still be in the category of "the law of no scriptures for fools." To deposit everything according to what we are borrowing: we imagine reality under mathematical theories chi navpaki.
What is a "non-skinny hotel"? Neskinchenniy hotel - tse hotel, de zavzhd є whether there are a number of free places, regardless of how many rooms are occupied. As well as all rooms in the non-limited corridor for occupants, there is another non-limited corridor with rooms for guests. There will be no such corridors. At the same time, the "innumerable hotel" has an infinite number of surfaces, an infinite number of corps on an infinite number of planets, an infinite number of all-worlds, created by an infinite number of Gods. Well, mathematicians are not able to stand aside from banal post-butt problems: God-Allah-Buddha - there is only one leader, the hotel - one wine, the corridor - only one. The axis of mathematics and help to sort out the ordinal numbers of hotel rooms, reconsidering us from what we can "get in the wrong".
I will show you the logic of my reflections on the example of the infinite multiplier of natural numbers. More often than not, it is necessary to ask for a simple question: how many multiplies of natural numbers do you need - one chi is rich? There is no correct type of nutrition, the shards of the number were invented by ourselves, there are no numbers in Nature. So, Nature is good in goodness, but for whom won't victorious other mathematical tools that are not for us. As nature cares, I will tell you one more time. The shards of the number were invented by us, we ourselves virishuvatememo, the scalings of the multiplications of natural numbers are used. Let's look at insulting options, how to lie with the right scholars.
First option. "Let us be given" one-one impersonal natural numbers, like lying on the floor without a turbo. We take the police for the faceless. All other natural numbers were not left out on the field and they were taken nowhere. We can’t add one to the next multiplier, the shards are already out. And what else do you want? No problem. We can take one with the multiplier we have already taken and turn it on the floor. If so, we can take a single piece from the police and add it to what is left. As a result, we again take away the impersonal natural numbers. You can record all our manipulations like this:
I have written down dії in the system of algebra value and in the system of value, adopted in the theory of multipliers, with detailed remapping of the elements of the multiplier. The lower index indicates those that we have a lot of natural numbers in one and the same. It appears that the impersonal natural numbers will be left with the inevitable only in that fall, as if they saw one and add another one.
The option is different. We have a lot of different, inexhaustible multiplications of natural numbers lying on the floor. Naked - RIZNIKH, do not marvel at those who practically do not stink. Let's take one of these multiples. Let's take one from the other impersonal natural numbers and add to the multiplier we have already taken. We can add two multipliers of natural numbers. The axis of what is in us weide:
The lower indices "one" and "two" point to those that these elements belonged to different multiples. So, if you add one to an inexhaustible multiplier, as a result, you will see an inexhaustible multiplicity, but it won’t be like that, like a multiplier of cobs. If you add up to one infinite multiplier, add another infinite multiplier, as a result, you will create a new infinite multiplier, which is formed from the elements of the first two multipliers.
A lot of natural numbers are victorious for the rahunka just like a line for vimiryuvan. Now show that you added one centimeter to the line. Tse will be another line, as it is not a good one.
You can accept or not accept my mirkuvannya - your special officer is on the right. But if you are stuck with mathematical problems, think about why you don’t walk with the stitch of pardons, trodden by generations of mathematicians. Even if we’re busy with mathematics, let’s try to form a stable stereotype of thought in us, and then we’ll give us romatic vibes (or, on the contrary, free-thinking will allow us).
pozg.ru
week, 4 serp 2019
Having added the postscript to the article about that, having read this wonderful text from Wikipedia:
It reads: "... the rich theoretical basis of mathematics to Babylon, in the presence of a solid character, was reduced to a set of different approaches, facilitating the total system and the evidence base."
Wow! As if we are reasonable, we can well bachiti a few others. And why should we marvel at modern mathematics in such a way? Slightly paraphrasing the pointing text, especially for me it was like this:
The rich theoretical basis of modern mathematics is not of a solid nature and can be reduced to a set of different divisions, adding to the general system and the evidence base.
For confirmation of my words, I will not go far - I can say that smart words, I can see that smart words of riches of other branches of mathematics. One and the same name among different branches of mathematics can be the mother of different sense. I would like to dedicate a whole cycle of publications to the most obvious mistakes of modern mathematics. See you soon.
Saturday, 03 September 2019
How to subdue the impersonal to the submultiple? For whom it is necessary to introduce a new unity of the world, which is a part of the element in the combined multiplier. Let's look at an example.
Let us have an impersonal BUT, What is made up of some people. Formed qiu multiplier for the sign "people" a, the lower index with a number indicates the ordinal number of the skin person in this plural. We introduce a new unit for the "status sign" and significantly її letter b. Shards of state are signs of power in all people, many times the skin element is many BUT on the sign b. Reveal respect, that now our faceless "people" have changed into faceless "people with statuary signs." If so, we can divide state marks on people bm that woman bw article signs. Now we can set up a mathematical filter: we choose one of these statutory signs, which one is a human or a woman. If there is a presence in people, then we multiply її by one, if there are no such signs - we multiply її by zero. And then zastosovuєmo zvichaynu school mathematics. Wonder what happened.
After multiplying, quickly and regrouping, we took away two submultiples: a multiplicity of people bm and a lot of women bw. Approximately this is how mathematicians mumble themselves, if they put the theory of multipliers into practice. But in the details, the stench does not attach us, but you see the finished result - "impersonal people are made up of more people and more women." Zvichayno, can you blame the nutrition, how much mathematics is correctly zastosovannya in more advanced transformations? I can commend you, really, everything is done correctly, to inform the nobility of the mathematical priming of arithmetic, Boolean algebra and other branches of mathematics. What is it? As if another time, I will tell you about it.
If there are hundreds of supermultiples, then it is possible to combine two multipliers in one supermultiple, having selected one in the world, but the elements have two multiplies.
Like a bachite, alone in the world, that natural mathematics transforms the theory of multipliers into a relic of the past. I will acquaint those who, for the theory of multipliers, are not all the same, those who, for the theory of multipliers, mathematicians foresaw the language of language and knowledge of power. Mathematicians blamed it as if shamans were robbed. Only shamans know how to “correctly” zastosovuvat their “knowledge”. Tsim "knowing" stink to teach us.
Finally, I want to show you how mathematicians manipulate z.
Monday, Sep 7, 2019
In the fifth century before ours, the ancient Greek philosopher Zenon of Eleisky formulated his famous aporia, which he found to be the aporia "Achiles and the tortoise". Axis yak won sound:
Admissible, Achilles lives ten times closer, lower than a tortoise, and stays behind her for a thousand rocks. For that hour, for a kind of Achilles, to pass through the distance, a turtle in the same bіk propovs a hundred rokіv. If Achilles lives a hundred miles, the tortoise prophesies ten more miles, and so on. The process continues to inexorable, Achilles so no way can the tortoise be cured.
The change of the world has become a logical shock for all the coming generations. Aristotle, Diogenes, Kant, Hegel, Hilbert... Everyone else looked at Zeno's aporia differently. The shock was strong on the floor, sho " ... the discussions are going on and in the given hour, to think about the reality of the paradoxes in the science of science has not yet come a long way ... mathematical analysis, the theory of multiplicity, new physical and philosophical approaches were carried out until the end; zhoden і from them without becoming the most famous cherishes of nutrition.[Wikipedia, "Aporia of Zeno"]. Everyone knows what to fool them, but no one knows what deception is.
From the point of view of mathematics, Zeno, in his aporia, clearly demonstrated the transition from the value to . Tsey transition may be on the uvazi zastosuvannya zamіst postіynyh. Naskіlki razumіu, mathematicheskij apparatus zastosuvannya zmіnnyh odinіru or more raspravleniya, or yogo zastosuvanya until Zeno's aporia. Zastosuvannya our supreme logic to bring us to pasture. Mi, for the inertia of the mind, zastosovuєmo postiyni odinі vіru an hour before the turned value. From the physical point of view, it looks like an hour before the last tooth at the moment when Achilles is equal to the tortoise. As the time goes by, Ahiles can no longer overtake the tortoise.
If we turn the logic around to us, then everything falls into place. Achilles lives from fast swedishness. The skin of the stepping yogo path is ten times shorter than the front. Obviously, the hour, which is stained on the yogo hem, is ten times less than the front. If you want to understand the "inconsistency" in this situation, then you correctly say "Achilles is inexcusably fast on the tortoise."
How to uniqnut tsієї logical pasta? Get lost in fasting loneliness at the end of the day and move on to deadly values. My Zeno looks like this:
For that hour, for a kind of Achilles, to pass a thousand miles, the turtle at that very bek propoved a hundred miles. For the next hour, which is better than the first, Achilles will live another thousand miles, and the tortoise will prophesy a hundred miles. Now Achilles is on the vіsіmsot krokіv vperedzhaє tortoise.
Tsey pіdhіd adequately signifies reality without everyday logical paradoxes. But it's not the top of the problem. Einstein's assertion about the inexhaustibility of the swidkost of light is even similar to Zenon's aporia "Achilles and the tortoise". We still need to live up to the problem, rethink and virishiti. The first decision is necessary to shukati not in infinitely great numbers, but in the loneliness of the world.
Insha tsikava aporiya Zeno opovіda about the arrow, scho to fly.
An arrow to fly is unruly, to that which at the skin moment of the hour it rests, and the shards of it rest at the skin of the hour, then it will rest forever.
In this aporia, the logical paradox is even more simple - to ask for clarification that at the skin moment it’s time to shoot, to fly, to rest in different points of the open space, to, in the air, and є with your hands. Here the next point is to be noted. According to one photograph of a car on a road, it is impossible to tell the fact of yogo rush, no way to see it. To determine the fact of the collapse of the car, two photographs are needed, broken from the same point in different moments and hours, but it is not possible to determine the difference. For the purpose of getting to the car, you will need two photographs, broken from different points of space at one moment of the hour, but you can’t determine the fact of the collapse (naturally, you will need additional data for the investigation, trigonometry will help you). What I want to pay special respect to, then the price of those that are two points in the hour and two points in the space - the price of speech, if not a trick of the rogue, even if the stench gives a difference of opportunity for follow-up.
I will show the process in practice. Vidbiraemo "chervone hard in pukhirtsyu" - Tse our "tsel". When tsimu mi bachimo, sho tsi things є with a bow, but without a bow. After that, we choose a part of the "whole" and form an impersonal "with a bow". This is how shamans get their own food, tying their theory of multiplies to reality.
And now we're sifting a little mess. Let's take "firmly in puffiness with a bow" and unite "tsili" behind the color sign, vibrating the red elements. We took away the faceless "chervonih". Now food for a drink: take away the multipliers "with a bow" and "chervone" - is it one and the same impersonal or two different multipliers? Vidpovid know less shamans. More precisely, the stench themselves do not know anything, but how to say, so be it.
This simple example shows that the theory of multiples is absolutely marvelous, if one talks about reality. What is the secret? We formed the impersonal "chervone hard in puff with a bow." Molding was done for chotirma with different singles of the world: color (chervone), mint (hard), shortness (at puffy), embellish (with a bow). Only the sukupnіst of loneliness in the world allows to adequately describe the real objects of my mathematics.. The axis looks like it.
The letter "a" with different indices means different ones in the world. At the temples, one sees a vimir, which is seen as the "whole" of the front stage. The loneliness of the world is blamed for the temples, which forms the faceless. The remaining row shows the residual result - the element of the multiplier. Like bachite, like zastosovuvat single vimir for molding a lot, then the result will not lie in the order of our deeds. But it’s mathematics, not dances of shamans with tambourines. Shamans can "intuitively" arrive at the same result, arguing for this "obviousness", even if alone in the world do not enter into their "scientific" arsenal.
For help alone, it’s easy for the world to beat one, or combine a sprat of multiplies into one supermultiple. Let's take a closer look at the algebra of which process.
Trigonometric spіvvіdnosnja (functions) in a straight tricutnik
Spivvіdshenie storіn trikutnik є the basis of trigonometry and geometry. The greater number of zavdans will rise to the level of power of trikutniks and kіl, as well as straight ones. Let's look at what is so trigonometrical spіvvіdnoshennia simple mine.
Trigonometric spіvvіdnennia in a straight-cut trikutnik are called spіvvіdshennya dozhin yoga side. In case of such a spіvvіdnoshnya zavzhdny one and the same, according to the vіvіdshennya to the kut, which lie between the parties, the spіvіdnoshennya between them can be counted.
A straight cut ABC is marked on the little one.
Let's look at the trigonometric expressions of the yogo side of the shodo kuta A (there are also signs of the Greek letter α on the little vein).
Let’s take it to heart that the side AB of the tricot is the hypotenuse. Side AC є leg, lie down to kuta α and side BC is a leg, protile kut α.
Shodo kuta α in a straight-cut tricutnik is to understand the offensive:
cosine kuta called the extension of the leg, which adheres to the new one, to the hypotension of this straight-cut tricutnik. (div. what is the cosine and yoga of power).
On the baby with the cosine of the kuta cosα =AC/AB(Diligent leg dility on hypotenuse).
To give respect, that for the kuta β we will lie on the side of the leg є side BC, to that cos β = BC/AB. Tobto trigonometric spіvvіdnoshennia are counted in vіdpovіdno up to the position of the sides of the rectilinear tricutnik shodo kuta.
With this letter, the meanings can be be-yakim. It’s important less mutually roztashuvannya kuta that side of a straight-cut tricutnik.
Sinus kuta called spіvvіdnoshennia protilegnogo to the new leg to the hypotension of the straight-cut tricutnik (div. scho is the sine and yoga of power).
On the baby with a sinus kuta α є spіvvіdnoshennia sinα = BC/AB(The opposite leg of dilation on the hypotenuse).
Since the oscillators for the designation of the sine are important, and the mutual expansion of the sides of the straight-cut tricutnik according to the given kuta, then for the β kuta the sine function will be sin β = AC/AB.
tangent kuta called spіvvіdnoshnja protilazhnogo given kutu leg to the leg of a straight-cut trikutnik (div. scho take the tangent and yogo power).
On the little tangent kuta tgα = BC/AC. (protilezhny kutu leg dility on adjacent leg)
For kuta β, according to the principles of mutual expansion of sides, the tangent of kuta can be calculated as tg β = AC/BC.
cotangent kuta called spіvvіdnoshnja leg, scho leaning on this kutu, on the prolezhny leg of a straight-cut trikutnik. As you can see from the appointment, the cotangent is a function, connected with the spivvіdnosheniya tangent 1/tg α . Tobto, stink mutually.
manager. Know the trigonometric ratio of trikutnik
At the tricotnik ABC kut C is 90 degrees. cos α = 4/5. Find sin α, sin βSolution. 
Oskilki cos α = 4/5, then AC/AB = 4/5. Tobto the parties spіvvіdnosyatsya like 4:5. Significantly, the length of AC is 4x then AB = 5x.
For the Pythagorean theorem:
BC 2 + AC 2 = AB 2
Todi
BC 2 + (4x) 2 = (5x) 2
BC 2 + 16x2 = 25x2
BC 2 = 9x2
BC=3x
Sin α = BC / AB = 3x / 5x = 3/5
sin β = AC / AB