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Trigonometry, as a science, originated in the Ancient East. The first trigonometric ratios were developed by astronomers to create an accurate calendar and orientate by the stars. These calculations related to spherical trigonometry, while in the school course they study the ratio of the sides and angle of a flat triangle.

Trigonometry is a branch of mathematics dealing with the properties of trigonometric functions and the relationship between the sides and angles of triangles.

During the heyday of culture and science in the 1st millennium AD, knowledge spread from the Ancient East to Greece. But the main discoveries of trigonometry are the merit of the men of the Arab Caliphate. In particular, the Turkmen scientist al-Marazvi introduced such functions as tangent and cotangent, compiled the first tables of values ​​for sines, tangents and cotangents. The concept of sine and cosine was introduced by Indian scientists. A lot of attention is devoted to trigonometry in the works of such great figures of antiquity as Euclid, Archimedes and Eratosthenes.

Basic quantities of trigonometry

The basic trigonometric functions of a numerical argument are sine, cosine, tangent, and cotangent. Each of them has its own graph: sine, cosine, tangent and cotangent.

The formulas for calculating the values ​​of these quantities are based on the Pythagorean theorem. It is better known to schoolchildren in the formulation: “Pythagorean pants, equal in all directions,” since the proof is given on the example of an isosceles right triangle.

Sine, cosine and other dependencies establish a relationship between acute angles and sides of any right triangle. We give formulas for calculating these quantities for angle A and trace the relationship of trigonometric functions:

As you can see, tg and ctg are inverse functions. If we represent the leg a as the product of sin A and the hypotenuse c, and the leg b as cos A * c, then we get the following formulas for the tangent and cotangent:

trigonometric circle

Graphically, the ratio of the mentioned quantities can be represented as follows:

The circle, in this case, represents all possible values ​​of the angle α - from 0° to 360°. As can be seen from the figure, each function takes a negative or positive value depending on the angle. For example, sin α will be with a “+” sign if α belongs to the I and II quarters of the circle, that is, it is in the range from 0 ° to 180 °. With α from 180° to 360° (III and IV quarters), sin α can only be a negative value.

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

The values ​​of α equal to 30°, 45°, 60°, 90°, 180° and so on are called special cases. The values ​​of trigonometric functions for them are calculated and presented in the form of special tables.

These angles were not chosen by chance. The designation π in the tables is for radians. Rad is the angle at which the length of a circular arc corresponds to its radius. This value was introduced in order to establish a universal relationship; when calculating in radians, the actual length of the radius in cm does not matter.

The angles in the tables for trigonometric functions correspond to radian values:

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

Properties of trigonometric functions: sine and cosine

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

Consider a comparative table of properties for a sine wave and a cosine wave:

sinusoidcosine wave
y = sin xy = cos x
ODZ [-1; 1]ODZ [-1; 1]
sin x = 0, for x = πk, where k ϵ Zcos x = 0, for x = π/2 + πk, where k ϵ Z
sin x = 1, for x = π/2 + 2πk, where k ϵ Zcos x = 1, for x = 2πk, where k ϵ Z
sin x = - 1, at x = 3π/2 + 2πk, where k ϵ Zcos x = - 1, for x = π + 2πk, where k ϵ Z
sin (-x) = - sin x, i.e. odd functioncos (-x) = cos x, i.e. the function is even
the function is periodic, the smallest period is 2π
sin x › 0, with x belonging to quarters I and II or from 0° to 180° (2πk, π + 2πk)cos x › 0, with x belonging to quarters I and IV or from 270° to 90° (- π/2 + 2πk, π/2 + 2πk)
sin x ‹ 0, with x belonging to quarters III and IV or from 180° to 360° (π + 2πk, 2π + 2πk)cos x ‹ 0, with x belonging to quarters II and III or from 90° to 270° (π/2 + 2πk, 3π/2 + 2πk)
increases on the interval [- π/2 + 2πk, π/2 + 2πk]increases on the interval [-π + 2πk, 2πk]
decreases on the intervals [ π/2 + 2πk, 3π/2 + 2πk]decreases in intervals
derivative (sin x)' = cos xderivative (cos x)’ = - sin x

Determining whether a function is even or not is very simple. It is enough to imagine a trigonometric circle with signs of trigonometric quantities and mentally “fold” the graph relative to the OX axis. If the signs are the same, the function is even; otherwise, it is odd.

The introduction of radians and the enumeration of the main properties of the sinusoid and cosine wave allow us to bring the following pattern:

It is very easy to verify the correctness of the formula. For example, for x = π/2, the sine is equal to 1, as is the cosine of x = 0. Checking can be done by looking at tables or by tracing function curves for given values.

Properties of tangentoid and cotangentoid

The graphs of the tangent and cotangent functions differ significantly from the sinusoid and cosine wave. The values ​​tg and ctg are inverse to each other.

  1. Y = tgx.
  2. The tangent tends to the values ​​of y at x = π/2 + πk, but never reaches them.
  3. The smallest positive period of the tangentoid is π.
  4. Tg (- x) \u003d - tg x, i.e., the function is odd.
  5. Tg x = 0, for x = πk.
  6. The function is increasing.
  7. Tg x › 0, for x ϵ (πk, π/2 + πk).
  8. Tg x ‹ 0, for x ϵ (— π/2 + πk, πk).
  9. Derivative (tg x)' = 1/cos 2 ⁡x .

Consider the graphical representation of the cotangentoid below in the text.

The main properties of the cotangentoid:

  1. Y = ctgx.
  2. Unlike the sine and cosine functions, in the tangentoid Y can take on the values ​​of the set of all real numbers.
  3. The cotangentoid tends to the values ​​of y at x = πk, but never reaches them.
  4. The smallest positive period of the cotangentoid is π.
  5. Ctg (- x) \u003d - ctg x, i.e., the function is odd.
  6. Ctg x = 0, for x = π/2 + πk.
  7. The function is decreasing.
  8. Ctg x › 0, for x ϵ (πk, π/2 + πk).
  9. Ctg x ‹ 0, for x ϵ (π/2 + πk, πk).
  10. Derivative (ctg x)' = - 1/sin 2 ⁡x Fix

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. All. 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; subtraction 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 will happen for different values ​​of the angle of the linear angle 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 denotes a 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 an infinite set of natural numbers as an example, then the considered examples can be represented as follows:

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 down the operations in algebraic notation and in 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 another infinite set is added to one 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 a 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 will not go far to confirm my words - it has a language and conventions that are different from the language and conventions of 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 A consisting of four people. This set is formed on the basis of "people" Let's designate the elements of this set through the letter A, 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 A 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 managed to come to a common opinion about the essence of paradoxes ... mathematical analysis, set theory, new physical and philosophical approaches were involved in the study of the issue; 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 rests 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 (naturally, 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.

If you are already familiar with trigonometric circle , and you just want to refresh individual elements in your memory, or you are completely impatient, then here it is, :

Here we will analyze everything in detail step by step.

The trigonometric circle is not a luxury, but a necessity

Trigonometry many are associated with an impassable thicket. Suddenly, so many values ​​​​of trigonometric functions pile up, so many formulas ... But it’s, after all, it didn’t work out at first, and ... off and on ... sheer misunderstanding ...

It is very important not to wave your hand at values ​​of trigonometric functions, - they say, you can always look at the spur with a table of values.

If you constantly look at the table with the values ​​​​of trigonometric formulas, let's get rid of this habit!

Will save us! You will work with it several times, and then it will pop up in your head on its own. Why is it better than a table? Yes, in the table you will find a limited number of values, but on the circle - EVERYTHING!

For example, say, looking at standard table of values ​​of trigonometric formulas , which is the sine of, say, 300 degrees, or -45.


No way? .. you can, of course, connect reduction formulas... And looking at the trigonometric circle, you can easily answer such questions. And you will soon know how!

And when solving trigonometric equations and inequalities without a trigonometric circle - nowhere at all.

Introduction to the trigonometric circle

Let's go in order.

First, write down the following series of numbers:

And now this:

And finally this one:

Of course, it is clear that, in fact, in the first place is, in the second place is, and in the last -. That is, we will be more interested in the chain .

But how beautiful it turned out! In which case, we will restore this “wonderful ladder”.

And why do we need it?

This chain is the main values ​​​​of sine and cosine in the first quarter.

Let's draw a circle of unit radius in a rectangular coordinate system (that is, we take any radius along the length, and declare its length to be unit).

From the “0-Start” beam, we set aside in the direction of the arrow (see Fig.) corners.

We get the corresponding points on the circle. So, if we project the points onto each of the axes, then we will get exactly the values ​​from the above chain.

Why is that, you ask?

Let's not take everything apart. Consider principle, which will allow you to cope with other, similar situations.

Triangle AOB is a right triangle with . And we know that opposite the angle at lies a leg twice as small as the hypotenuse (our hypotenuse = the radius of the circle, that is, 1).

Hence, AB= (and hence OM=). And by the Pythagorean theorem

I hope something is clear now.

So point B will correspond to the value, and point M will correspond to the value

Similarly with the rest of the values ​​of the first quarter.

As you understand, the axis familiar to us (ox) will be cosine axis, and the axis (oy) - sinus axis . Later.

To the left of zero on the cosine axis (below zero on the sine axis) there will, of course, be negative values.

So, here it is, the ALL-POWERFUL, without which nowhere in trigonometry.

But how to use the trigonometric circle, we'll talk in.