Today we will consider problems B8 with trigonometry in its classical sense, where ordinary right triangles. Therefore, there will be no trigonometric circles and negative angles today - only ordinary sines and cosines.

Such tasks account for approximately 30% of the total. Remember: if the angle π is mentioned at least once in problem B8, it is solved in completely different ways. We will definitely review them in the near future. And now the main definition of the lesson:

A triangle is a figure on a plane, consisting of three points and segments that connect them. In fact, this is a closed broken line of three links. The points are called the vertices of the triangle, and the segments are called the sides. It is important to note that the vertices must not lie on the same straight line, otherwise the triangle degenerates into a segment.

Quite often, a triangle is called not only the broken line itself, but also the part of the plane that is bounded by this broken line. Thus, the area of ​​a triangle can be determined.

Two triangles are called equal if one can be obtained from the other by one or more plane movements: translation, rotation, or symmetry. In addition, there is the concept of similar triangles: their angles are equal, and the corresponding sides are proportional ...

This is triangle ABC. Moreover, it is a right triangle: in it ∠C = 90°. These are the ones most often encountered in problem B8.

All you need to know to solve problem B8 is a few simple facts from geometry and trigonometry, as well as general scheme decision in which these facts are used. Then it remains just to "fill your hand."

Let's start with the facts. They are divided into three groups:

1. Definitions and consequences from them;
2. Basic identities;
3. Symmetries in a triangle.

It cannot be said that any of these groups is more important, more difficult or easier. But the information they contain allows us to decide any task B8. Therefore, you need to know everything. So let's go!

Group 1: definitions and consequences from them

Consider triangle ABC , where ∠C is a straight line. First, definitions:

The sine of an angle is the ratio of the opposite leg to the hypotenuse.

The cosine of an angle is the ratio of the adjacent leg to the hypotenuse.

The tangent of an angle is the ratio of the opposite leg to the adjacent leg.

One angle or segment can be included in different right triangles. Moreover, very often the same segment is a leg in one triangle and a hypotenuse in another. But more on this later, but for now we will work with the usual angle A. Then:

1. sin A = BC : AB ;
2. cos A = AC : AB ;
3. tan A = BC : AC .

The main consequences of the definition:

1. sin A = cos B ; cos A = sin B - the most commonly used corollaries
2. tg A \u003d sin A : cos A - connects the tangent, sine and cosine of one angle
3. If ∠A + ∠B = 180°, i.e. angles are adjacent, then: sin A \u003d sin B; cos A = -cos B .

Believe it or not, these facts are enough to solve about a third of all B8 trigonometric problems.

Group 2: basic identities

The first and most important identity is the Pythagorean theorem: the square of the hypotenuse is equal to the sum squares of legs. As applied to the triangle ABC, discussed above, this theorem can be written as follows:

AC 2 + BC 2 = AB 2

And immediately - a small remark that will save the reader from many mistakes. When you solve a problem, always (hey, always!) write down the Pythagorean theorem in this form. Do not try to immediately express the legs, as is usually required. You may save a couple of lines of calculations, but it was on this “saving” that more points were lost than anywhere else in geometry.

The second identity is from trigonometry. As follows:

sin 2 A + cos 2 A = 1

That's what it's called: the basic trigonometric identity. It can be used to express cosine in terms of sine and vice versa.

Group 3: Symmetries in a triangle

What is written below applies only to isosceles triangles. If this does not appear in the problem, then the facts from the first two groups are enough to solve.

So, consider an isosceles triangle ABC, where AC = BC. Draw the height CH to the base. We get the following facts:

1. ∠A = ∠B . As a consequence, sin A = sin B ; cos A = cos B ; tg A = tg B .
2. CH is not only the height, but also the bisector, i.e. ∠ACH = ∠BCH . Similarly, are equal and trigonometric functions these corners.
3. Also CH is the median, so AH = BH = 0.5 AB .

Now that all the facts have been considered, let's proceed directly to the solution methods.

General scheme for solving problem B8

Geometry differs from algebra in that it does not have simple and universal algorithms. Each task has to be solved from scratch - and this is its complexity. However, general recommendations can still be given.

To begin with, the unknown side (if any) should be denoted by X . Then we apply the solution scheme, which consists of three points:

1. If there is an isosceles triangle in the problem, apply to it all possible facts from the third group. Find equal angles and express their trigonometric functions. In addition, an isosceles triangle is rarely a right triangle. Therefore, look for right-angled triangles in the problem - they are definitely there.
2. Apply the facts from the first group to the right triangle. The end goal is to get an equation with respect to the variable X . Find X - solve the problem.
3. If the facts from the first group were not enough, we apply the facts from the second group. And again looking for X .

Examples of problem solving

Now let's try using the knowledge gained to solve the most common problems B8. Do not be surprised that with such an arsenal, the decision text will not be much longer than the original condition. And it pleases:)

A task. In triangle ABC, angle C is 90°, AB = 5, BC = 3. Find cos A .

By definition (Group 1), cos A = AC : AB . The hypotenuse AB is known to us, but the leg AC will have to be looked for. Let's denote it AC = x .

Let's move on to group 2. Triangle ABC is a right triangle. According to 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 find the cosine:

cos A = AC: AB = 4: 5 = 0.8.

A task. In triangle ABC, angle B is 90°, cos A = 4/5, BC = 3. BH is the height. Find AH.

Denote the required side AH = x and consider the triangle ABH . It is rectangular, and ∠AHB = 90° by convention. Therefore cos A = AH : AB = x : AB = 4/5. This is a proportion, it can be rewritten like this: 5 x = 4 AB. Obviously, we will find x if we know AB.

Consider triangle ABC. It is also rectangular, with cos A = AB : AC . Neither AB nor AC are known to us, so we pass to the second group of facts. We write down the main trigonometric identity:

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.

Since the trigonometric functions of an acute angle are positive, we get sin A = 3/5. On the other hand, sin A = BC : AC = 3: AC . We get the proportion:

3:AC=3:5;
3 AC = 3 5;
AC = 5.

So, AC = 5. Then AB = AC cos A = 5 4/5 = 4. Finally, we find AH = x:

5 x = 4 4;
x = 16/5 = 3.2.

A task. In triangle ABC AB = BC , AC = 5, cos C = 0.8. Find the height CH .

Denote the required height CH = x . Before us is an isosceles triangle ABC, in which AB \u003d BC. Therefore, from the third group of facts we have:

∠A = ∠C ⇒ cos A = cos C = 0.8

Consider the triangle ACH . It is rectangular (∠H = 90°) with AC = 5 and cos A = 0.8. By definition, cos A = AH : AC = AH : 5. We get the proportion:

AH:5=8:10;
10 AH = 5 8;
AH = 40: 10 = 4.

It remains to use the second group of facts, namely the Pythagorean theorem for the triangle ACH :

AH 2 + CH 2 = AC 2;
4 2 + x 2 = 5 2 ;
x 2 \u003d 25 - 16 \u003d 9;
x=3.

A task. IN right triangle ABC ∠B = 90°, AB = 32, AC = 40. Find the sine of angle CAD.

Since we know the hypotenuse AC = 40 and the leg AB = 32, we can find the cosine of the angle A : cos A = AB : AC = 32: 40 = 0.8. It was a fact from the first group.

Knowing the cosine, you can find the sine through the basic trigonometric identity (a fact from the second 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 finding the sine, the fact that the trigonometric functions of an acute angle are positive was again used. It remains to note that the angles BAC and CAD are adjacent. From the first group of facts we have:

∠BAC + ∠CAD = 180°;
sin CAD = sin BAC = sin A = 0.6.

A task. In triangle ABC AC = BC = 5, AB = 8, CH is the height. Find tg A .

Triangle ABC is isosceles, CH is height, so note that AH = BH = 0.5 AB = 0.5 8 = 4. This is a fact from the third group.

Now consider the triangle ACH : it has ∠AHC = 90°. You can express the tangent: tg A \u003d CH: AH. But AH = 4, so it remains to find the side CH , which we denote CH = x . By the Pythagorean theorem (a fact from group 2) we have:

AH 2 + CH 2 = AC 2;
4 2 + x 2 = 5 2 ;
x 2 \u003d 25 - 16 \u003d 9;
x=3.

Now everything is ready to find the tangent: tg A = CH : AH = 3: 4 = 0.75.

A task. In triangle ABC AC = BC, AB = 6, cos A = 3/5. Find the height AH.

Denote the required height AH = x . Again triangle ABC is isosceles, so note that ∠A = ∠B , hence cos B = cos A = 3/5. This is a fact from the third group.

Consider triangle ABH . By assumption, it is rectangular (∠AHB = 90°), and the hypotenuse AB = 6 and cos B = 3/5 are known. But cos B = BH : AB = BH : 6 = 3/5. We got the ratio:

BH:6=3:5;
5 BH = 6 3;
BH = 18/5 = 3.6.

Now let's find AH = x using the Pythagorean theorem for the triangle 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.

Additional Considerations

There are non-standard tasks where the facts and schemes discussed above are useless. Alas, in this case, a truly individual approach is needed. They like to give similar tasks at all kinds of "trial" and "demonstration" exams.

Below are two real tasks that were offered at the trial exam in Moscow. Few coped with them, which indicates the high complexity of these tasks.

A task. In a right triangle ABC, a median and an altitude are drawn from the angle C = 90°. It is known that ∠A = 23°. Find ∠MCH .

Note that the median CM is drawn to the hypotenuse AB , so M is the center of the circumscribed circle, i.e. AM = BM = CM = R, where R is the radius of the circumscribed circle. Hence triangle ACM is isosceles, and ∠ACM = ∠CAM = 23°.

Now consider triangles ABC and CBH. By assumption, both triangles are right triangles. Moreover, ∠B is general. Therefore, triangles ABC and CBH are similar in two angles.

In similar triangles, the corresponding elements are proportional. In particular:

BCH = BAC = 23°

Finally, consider ∠C . It is direct, and furthermore, ∠C = ∠ACM + ∠MCH + ∠BCH . In this equality, ∠MCH is the desired one, and ∠ACM and ∠BCH are known and equal to 23°. We have:

90° = 23° + MCH + 23°;
MCH = 90° - 23° - 23° = 44°.

A task. The perimeter of the rectangle is 34 and the area is 60. Find the diagonal of this rectangle.

Let's denote the sides of the rectangle: AB = x, BC = y. Let's express the perimeter:

P ABCD \u003d 2 (AB + BC) \u003d 2 (x + y) \u003d 34;
x + y = 17.

Similarly, we express the area: S ABCD = AB BC = x y = 60.

Now consider triangle ABC. It is rectangular, so we write down the Pythagorean theorem:

AB 2 + BC 2 = AC 2 ;
AC 2 = x 2 + y 2 .

Note that the formula for the square of the difference implies the equality:

x 2 + y 2 \u003d (x + y) 2 - 2 x y \u003d 17 2 - 2 60 \u003d 289 - 120 \u003d 169

So AC 2 = 169, hence AC = 13.

Simply put, these are vegetables cooked in water according to a special recipe. I will consider two initial components (vegetable salad and water) and the finished result - borscht. Geometrically, this can be represented as a rectangle in which one side denotes lettuce, the other side denotes water. The sum of these two sides will denote borscht. The diagonal and area of ​​such a "borscht" rectangle are purely mathematical concepts and are never used in borscht recipes.

How do lettuce and water turn into borscht in terms of mathematics? How can the sum of two segments turn into trigonometry? To understand this, we need linear angle functions.

You won't find anything about linear angle functions in math textbooks. But without them there can be no mathematics. The laws of mathematics, like the laws of nature, work whether we know they exist or not.

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

Is it possible to do without linear angular functions? You can, because mathematicians still manage without them. The trick of mathematicians lies in the fact that they always tell us only about those problems that they themselves can solve, and never tell us about those problems that they cannot solve. See. If we know the result of the addition and one term, we use subtraction to find the other term. Everything. We do not know other problems and we are not able to solve them. What to do if we know only the result of the addition and do not know both terms? In this case, the result of addition must be decomposed into two terms using linear angular functions. Further, we ourselves choose what one term can be, and the linear angular functions show what the second term should be in order for the result of the addition to be exactly what we need. There can be an infinite number of such pairs of terms. IN Everyday life we do very well without decomposing the sum, subtracting is enough for us. But in scientific studies of the laws of nature, the expansion of the sum into terms can be very useful.

Another law of addition that mathematicians don't like to talk about (another trick of theirs) requires the terms to have the same unit of measure. For lettuce, water, and borscht, these may be units of weight, volume, cost, or unit of measure.

The figure shows two levels of difference for math. The first level is the differences in the field of numbers, which are indicated a, b, c. This is what mathematicians do. The second level is the differences in the area of ​​units of measurement, which are shown in square brackets and are indicated by the letter U. This is what physicists do. We can understand the third level - the differences in the scope of the described objects. Different objects can have the same number of the same units of measure. How important this is, we can see on the example of borscht trigonometry. If we add subscripts to the same notation for the units of measurement of different objects, we can say exactly what mathematical quantity describes a particular object and how it changes over time or in connection with our actions. letter W I will mark the water with the letter S I will mark the salad with the letter B- Borsch. Here's what the linear angle functions for borscht would look like.

If we take some part of the water and some part of the salad, together they will turn into one serving of borscht. Here I suggest you take a little break from borscht and remember your distant childhood. Remember how we were taught to put bunnies and ducks together? It was necessary to find how many animals will turn out. What then were we taught to do? We were taught to separate units from numbers and add numbers. Yes, any number can be added to any other number. This is a direct path to the autism of modern mathematics - we do not understand what, it is not clear why, and we understand very poorly how this relates to reality, because of the three levels of difference, mathematicians operate on only one. It will be more correct to learn how to move from one unit of measurement to another.

And bunnies, and ducks, and little animals can be counted in pieces. One common unit of measurement for different objects allows us to add them together. This is a children's version of the problem. Let's look at a similar problem for adults. What do you get when you add bunnies and money? There are two possible solutions here.

First option. We determine the market value of the bunnies and add it to the available cash. We got the total value of our wealth in terms of money.

Second option. You can add the number of bunnies to the number of banknotes we have. We will get the amount of movable property in pieces.

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

But back to our borscht. Now we can see what happens when different meanings angle of linear angular functions.

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

For me personally, this is the main mathematical proof of the fact that . Zero does not change the number when added. This is because addition itself is impossible if there is only one term and the second term is missing. You can relate to this as you like, but remember - all mathematical operations with zero were invented by mathematicians themselves, so discard your logic and stupidly cram the definitions invented by mathematicians: "division by zero is impossible", "any number multiplied by zero equals zero" , "behind the point zero" and other nonsense. It is enough to remember once that zero is not a number, and you will never have a question whether zero is a natural number or not, because such a question generally loses all meaning: how can one consider a number that which is not a number. It's like asking what color to attribute an invisible color to. Adding zero to a number is like painting with paint that doesn't exist. They waved a dry brush and tell everyone that "we have painted." But I digress a little.

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 borscht.

The angle is forty-five degrees. We have equal amounts of water and lettuce. This is the perfect borscht (may the cooks forgive me, it's just math).

The angle is greater than forty-five degrees but less than ninety degrees. We have a lot of water and little lettuce. Get liquid borscht.

Right angle. We have water. Only memories remain of the lettuce, as we continue to measure the angle from the line that once marked the lettuce. We can't cook borscht. The amount of borscht is zero. In that case, hold on and drink water while it's available)))

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

The two friends had their shares in the common business. After the murder of one of them, everything went to the other.

The emergence of mathematics on our planet.

All these stories are told in the language of mathematics 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, let's return to the trigonometry of borscht and consider projections.

Saturday, October 26, 2019

Wednesday, August 7, 2019

Concluding the conversation about , we need to consider an infinite set. Gave in that the concept of "infinity" acts on mathematicians, like a boa constrictor on a rabbit. The quivering horror of infinity deprives mathematicians of common sense. Here is an example:

The original source is located. Alpha stands for real number. The equal sign in the above expressions indicates that if you add a number or infinity to infinity, nothing will change, the result will be the same infinity. If we take as an example an infinite set natural numbers, the considered examples can be presented in the following form:

To visually prove their case, mathematicians have come up with many different methods. Personally, I look at all these methods as the dances of shamans with tambourines. In essence, they all come down to the fact that either some of the rooms are not occupied and new guests are settled in them, or that some of the visitors are thrown out into the corridor to make room for the guests (very humanly). I presented my view on such decisions in the form of a fantastic story about the Blonde. What is my reasoning based on? Moving an infinite number of visitors takes an infinite amount of time. After we have vacated the first guest room, one of the visitors will always walk along the corridor from his room to the next one until the end of time. Of course, the time factor can be stupidly ignored, but this will already be from the category of "the law is not written for fools." It all depends on what we are doing: adjusting reality to mathematical theories or vice versa.

What is an "infinite hotel"? An infinity inn is an inn that always has any number of vacancies, no matter how many rooms are occupied. If all the rooms in the endless hallway "for visitors" are occupied, there is another endless hallway with rooms for "guests". There will be an infinite number of such corridors. At the same time, the "infinite hotel" has an infinite number of floors in an infinite number of buildings on an infinite number of planets in an infinite number of universes created by an infinite number of Gods. Mathematicians, on the other hand, are not able to move away from banal everyday problems: God-Allah-Buddha is always only one, the hotel is one, the corridor is only one. So mathematicians are trying to juggle the serial numbers of hotel rooms, convincing us that it is possible to "shove the unpushed".

I will demonstrate the logic of my reasoning to you using 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 many? There is no correct answer to this question, since we ourselves invented numbers, there are no numbers in Nature. Yes, Nature knows how to count perfectly, but for this she uses other mathematical tools that are not familiar to us. As Nature thinks, I will tell you another time. Since we invented the numbers, we ourselves will decide how many sets of natural numbers exist. Consider both options, as befits a real scientist.

Option one. "Let us be given" a single set of natural numbers, which lies serenely on a shelf. We take this set from the shelf. That's it, there are no other natural numbers left on the shelf and there is nowhere to take them. We cannot add one to this set, since we already have it. What if you really want to? No problem. We can take a unit from the set we have already taken and return it to the shelf. After that, we can take a unit from the shelf and add it to what we have left. As a result, we again get an infinite set of natural numbers. You can write all our manipulations like this:

I have written the operations in algebraic notation and set theory notation, listing the elements of the set in detail. The subscript indicates that we have one and only set of natural numbers. It turns out that the set of natural numbers will remain unchanged only if one is subtracted from it and the same one is added.

Option two. We have many different infinite sets of natural numbers on the shelf. I emphasize - DIFFERENT, despite the fact that they are practically indistinguishable. We take one of these sets. Then we take one from another set of natural numbers and add it to the set we have already taken. We can even add two sets of natural numbers. Here's what we get:

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

The set of natural numbers is used for counting in the same way as a ruler for measurements. Now imagine that you have added one centimeter to the ruler. This will already be a different line, not equal to the original.

You can accept or not accept my reasoning - this is your own business. But if you ever run into mathematical problems, consider whether you are on the path of false reasoning, trodden by generations of mathematicians. After all, mathematics classes, first of all, form a stable stereotype of thinking in us, and only then they add mental abilities to us (or vice versa, they deprive us of free thinking).

pozg.ru

Sunday, August 4, 2019

I was writing a postscript to an article about and saw this wonderful text on Wikipedia:

We read: "... the rich theoretical basis of Babylonian mathematics did not have a holistic character and was reduced to a set of disparate techniques, devoid of common system and evidence base.

Wow! How smart we are and how well we can see the shortcomings of others. Is it weak for us to look at modern mathematics in the same context? Slightly paraphrasing the above text, personally I got the following:

The rich theoretical basis of modern mathematics does not have a holistic character and is reduced to a set of disparate sections, devoid of a common system and evidence base.

I won’t go far to confirm my words - it has a language and symbols that are different from the language and symbols many other branches of mathematics. The same names in different branches of mathematics can have different meanings. I want to devote a whole cycle of publications to the most obvious blunders of modern mathematics. See you soon.

Saturday, August 3, 2019

How to divide a set into subsets? To do this, you must enter a new unit of measure, which is present in some of the elements of the selected set. Consider an example.

May we have many BUT consisting of four people. This set is formed on the basis of "people" Let's designate the elements of this set through the letter but, the subscript with a number will indicate the ordinal number of each person in this set. Let's introduce a new unit of measurement "sexual characteristic" and denote it by the letter b. Since sexual characteristics are inherent in all people, we multiply each element of the set BUT on gender b. Notice that our "people" set has now become the "people with gender" set. After that, we can divide the sexual characteristics into male bm and women's bw gender characteristics. Now we can apply a mathematical filter: we select one of these sexual characteristics, it does not matter which one is male or female. If it is present in a person, then we multiply it by one, if there is no such sign, we multiply it by zero. And then we apply the usual school mathematics. See what happened.

After multiplication, reductions and rearrangements, we got two subsets: the male subset bm and a subset of women bw. Approximately the same way mathematicians reason when they apply set theory in practice. But they do not let us in on the details, but give us 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 applied mathematics in the above transformations? I dare to assure you that in fact the transformations are done correctly, it is enough to know the mathematical justification of arithmetic, Boolean algebra and other sections of mathematics. What it is? Some other time I will tell you about it.

As for supersets, it is possible to combine two sets into one superset by choosing a unit of measurement that is present in the elements of these two sets.

As you can see, units of measurement and common math make set theory a thing of the past. A sign that all is not well with set theory is that mathematicians have come up with their own language and notation for set theory. The mathematicians did what the shamans once did. Only shamans know how to "correctly" apply their "knowledge". This "knowledge" they teach us.

Finally, I want to show you how mathematicians manipulate .

Monday, January 7, 2019

In the fifth century BC, the ancient Greek philosopher Zeno of Elea formulated his famous aporias, the most famous of which is the aporia "Achilles and the tortoise". Here's how it sounds:

Let's say Achilles runs ten times faster than the tortoise and is a thousand paces behind it. During the time during which Achilles runs this distance, the tortoise crawls a hundred steps in the same direction. When Achilles has run a hundred steps, the tortoise will crawl another ten steps, and so on. The process will continue indefinitely, Achilles will never catch up with the tortoise.

This reasoning became a logical shock for all subsequent generations. Aristotle, Diogenes, Kant, Hegel, Gilbert... All of them, in one way or another, considered Zeno's aporias. The shock was so strong that " ... discussions continue at the present time, the scientific community has not yet been able to come to a common opinion about the essence of paradoxes ... mathematical analysis, set theory, new physical and philosophical approaches; none of them became a universally accepted solution to the problem ..."[Wikipedia," Zeno's Aporias "]. Everyone understands that they are being fooled, but no one understands what the deception is.

From the point of view of mathematics, Zeno in his aporia clearly demonstrated the transition from the value to. This transition implies applying instead of constants. As far as I understand, the mathematical apparatus for applying variable units of measurement has either not yet been developed, or it has not been applied to Zeno's aporia. The application of our usual logic leads us into a trap. We, by the inertia of thinking, apply constant units of time to the reciprocal. From a physical point of view, it looks like time slowing down to a complete stop at the moment when Achilles catches up with the tortoise. If time stops, Achilles can no longer overtake the tortoise.

If we turn the logic we are used to, everything falls into 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 overcoming it is ten times less than the previous one. If we apply the concept of "infinity" in this situation, then it would be correct to say "Achilles will infinitely quickly overtake the tortoise."

How to avoid this logical trap? Remain in constant units of time and do not switch to reciprocal values. In Zeno's language, it looks like this:

In the time it takes Achilles to run a thousand steps, the tortoise crawls a hundred steps in the same direction. During the next time interval, equal to the first, Achilles will run another thousand steps, and the tortoise will crawl one hundred steps. Now Achilles is eight hundred paces ahead of the tortoise.

This approach adequately describes reality without any logical paradoxes. But this is not a complete solution to the problem. Einstein's statement about the insurmountability of the speed of light is very similar to Zeno's aporia "Achilles and the tortoise". We have yet to study, rethink and solve this problem. And the solution must be sought not in infinitely large numbers, but in units of measurement.

Another interesting aporia of Zeno tells of a flying arrow:

A flying arrow is motionless, since at each moment of time it is at rest, and since it is at rest at every moment of time, it is always at rest.

In this aporia, the logical paradox is overcome very simply - it is enough to clarify that at each moment of time the flying arrow is at rest at different points in space, which, in fact, is movement. There is another point to be noted here. From one photograph of a car on the road, it is impossible to determine either the fact of its movement or the distance to it. To determine the fact of the movement of the car, two photographs taken from the same point at different points in time are needed, but they cannot be used to determine the distance. To determine the distance to the car, you need two photographs taken from different points in space at the same time, but you cannot determine the fact of movement from them (of course, you still need additional data for calculations, trigonometry will help you). What I want to point out in particular is that two points in time and two points in space are two different things that should not be confused as they provide different opportunities for exploration.
I will show the process with an example. We select "red solid in a pimple" - this is our "whole". At the same time, we see that these things are with a bow, and there are without a bow. After that, we select a part of the "whole" and form a set "with a bow". This is how shamans feed themselves by tying their set theory to reality.

Now let's do a little trick. Let's take "solid in a pimple with a bow" and unite these "whole" by color, selecting red elements. We got a lot of "red". Now a tricky question: are the received sets "with a bow" and "red" the same set or two different sets? Only shamans know the answer. More precisely, they themselves do not know anything, but as they say, so be it.

This simple example shows that set theory is completely useless when it comes to reality. What's the secret? We formed a set of "red solid pimply with a bow". The formation took place according to four different units of measurement: color (red), strength (solid), roughness (in a bump), decorations (with a bow). Only a set of units of measurement makes it possible to adequately describe real objects in the language of mathematics. Here's what it looks like.

The letter "a" with different indices denotes different units of measurement. In parentheses, units of measurement are highlighted, according to which the "whole" is allocated at the preliminary stage. The unit of measurement, according to which the set is formed, is taken out of brackets. The last line shows the final result - an element of the set. As you can see, if we use units to form a set, then the result does not depend on the order of our actions. And this is mathematics, and not the dances of shamans with tambourines. Shamans can “intuitively” come to the same result, arguing it with “obviousness”, because units of measurement are not included in their “scientific” arsenal.

With the help of units of measurement, it is very easy to break one or combine several sets into one superset. Let's take a closer look at the algebra of this process.

Trigonometric relations (functions) in a right triangle

The aspect ratio of a triangle is the basis of trigonometry and geometry. Most problems come down to using the properties of triangles and circles, as well as lines. Consider what trigonometric relations are in simple terms.

Trigonometric ratios in a right triangle are the ratios of the lengths of its sides. Moreover, such a ratio is always the same with respect to the angle that lies between the sides, the ratio between which must be calculated.

The figure shows right triangle ABC.
Consider the trigonometric ratios of its sides with respect to angle A (in the figure, it is also denoted by the Greek letter α).

Consider that side AB of a triangle is its hypotenuse. Side AC is the leg, adjacent to the angle α, and side BC is the leg, opposite angle α.

Regarding the angle α in a right triangle, the following relations exist:

Cosine of an angle is the ratio of the leg adjacent to it to the hypotenuse of a given right triangle. (cm. what is cosine and its properties).
In the figure, the cosine of the angle α is the ratio cosα =AC/AB(adjacent leg divided by the hypotenuse).
Note that for angle β, the adjacent leg is already side BC, so cos β = BC / AB. That is, trigonometric ratios are calculated in accordance with the position of the sides of a right triangle relative to the angle.

Wherein letter designations can be any. Only relative position matters. angle and sides of a right triangle.

The sine of an angle is the ratio of the leg opposite to it to the hypotenuse of a right triangle (see Fig. what is a sine and its properties).
In the figure, the sine of the angle α is the ratio sinα = BC / AB(opposite leg divided by the hypotenuse).
Since the relative position of the sides of a right-angled triangle relative to a given angle is important for determining the sine, then for the angle β the sine function will be sin β = AC / AB.

Tangent of an angle is the ratio of the leg opposite the given angle to the adjacent leg of a right triangle (see Fig. what is tangent and its properties).
In the figure, the tangent of the angle α will be equal to the ratio tgα = BC / AC. (the leg opposite the corner is divided by the adjacent leg)
For the angle β, guided by the principles relative position sides, the tangent of the angle can be calculated as tan β = AC / BC.

cotangent of an angle is the ratio of the leg adjacent to a given angle to the opposite leg of a right triangle. As can be seen from the definition, the cotangent is this function related to the tangent by the ratio 1/tg α . That is, they are mutually inverse.

A task. Find trigonometric relations in a triangle

In triangle ABC, angle C is 90 degrees. cos α = 4/5. Nadite sin α, sin β

Solution.

Since cos α = 4/5, then AC / AB = 4/5. That is, the sides are related as 4:5. Denote the length of AC as 4x, then AB = 5x.

According to the Pythagorean theorem:
BC 2 + AC 2 = AB 2

Then
BC 2 + (4x) 2 = (5x) 2
BC 2 + 16x 2 = 25x 2
BC 2 = 9x 2
BC=3x

Sin α = BC / AB = 3x / 5x = 3/5
sin β = AC / AB, and its value is already known by condition, that is, 4/5

The triangle has remarkable property is a rigid figure, i.e. If the length of the sides is constant, the shape of the triangle cannot be changed. This property of the triangle makes it indispensable in engineering and construction. Structural elements in the form of a triangle retain their shape, unlike, for example, elements in the form of a square or parallelogram. In addition, a triangle is the simplest polygon and any polygon can be represented as a set of triangles.

Basic properties and formulas of a triangle

Designations:
A, B, C are the angles of the triangle,
a, b, c - opposite sides,
R is the radius of the circumscribed circle,
r is the radius of the inscribed circle,
p is the semiperimeter, (a + b + c) / 2,
S is the area of ​​the triangle.

The sides of a triangle are related by the following inequalities
a ≤ b + c
b ≤ a + c
c ≤ a + b
If equality is satisfied in one of them, the triangle is called degenerate. In what follows, the non-degenerate case is assumed everywhere.

A triangle can be unambiguously (up to shift and rotation) determined by the following triplets of basic elements:
a, b, c - on three sides;
a, b, C - on two sides and the angle between them;
a, B, C - along the side and two corners adjacent to it.

The sum of the angles of any triangle is constant
A + B + C = 180°

1. Right triangle. Definition of trigonometric functions.

Consider the right triangle shown in the figure.

Angle B = 90° (straight).
Sin function: sin(A) = a/b .
Cosine function: cos(A) = c/b .
Tangent function: tg(A) = a/c .
Cotangent function: ctg(A) = c/a .

2. Right triangle. trigonometric formulas.

a = b * sin(A)
c = b * cos(A)
a = c * tg(A)

See also:

• The Pythagorean theorem is a few simple proofs of the theorem.

3. Right triangle. Pythagorean theorem.

b2 = a2 + c2
Using the Pythagorean theorem, you can build a right angle if there are no suitable tools at hand, for example, a square. With the help of two rulers or two pieces of rope, we measure the legs with a length of 3 and 4. Then we shift or push them apart until the length of the hypotenuse becomes equal to 5 (3 2 + 4 2 = 5 2).

The Pythagorean Theorem page provides some simple proofs of the theorem.