Definitions and signs of sine, cosine, tangent of an angle. How to remember the values of cosines and sines of the main points of the number circle Trigonometric circle positive and negative

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; 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

I watched an interesting video about Grandi's row One minus one plus one minus one - Numberphile. Mathematicians lie. They did not perform an equality test in their reasoning.

This resonates with my reasoning about .

Let's take a closer look at the signs that mathematicians are cheating us. At the very beginning of the reasoning, mathematicians say that the sum of the sequence DEPENDS on whether the number of elements in it is even or not. This is an OBJECTIVELY ESTABLISHED FACT. What happens next?

Next, mathematicians subtract the sequence from unity. What does this lead to? This leads to a change in the number of elements in the sequence - an even number changes to an odd number, an odd number changes to an even number. After all, we have added one element equal to one to the sequence. Despite all the external similarity, the sequence before the transformation is not equal to the sequence after the transformation. Even if we are talking about an infinite sequence, we must remember that an infinite sequence with an odd number of elements is not equal to an infinite sequence with an even number of elements.

Putting an equal sign between two sequences different in the number of elements, mathematicians claim that the sum of the sequence DOES NOT DEPEND on the number of elements in the sequence, which contradicts an OBJECTIVELY ESTABLISHED FACT. Further reasoning about the sum of an infinite sequence is false, because it is based on a false equality.

If you see that mathematicians place brackets in the course of proofs, rearrange the elements of a mathematical expression, add or remove something, be very careful, most likely they are trying to deceive you. Like card conjurers, mathematicians divert your attention with various manipulations of the expression in order to eventually give you a false result. If you can’t repeat the card trick without knowing the secret of cheating, then in mathematics everything is much simpler: you don’t even suspect anything about cheating, but repeating all the manipulations with a mathematical expression allows you to convince others of the correctness of the result, just like when have convinced you.

Question from the audience: And infinity (as the number of elements in the sequence S), is it even or odd? How can you change the parity of something that has no parity?

Infinity for mathematicians is like the Kingdom of Heaven for priests - no one has ever been there, but everyone knows exactly how everything works there))) I agree, after death you will be absolutely indifferent whether you lived an even or odd number of days, but ... Adding just one day at the beginning of your life, we will get a completely different person: his last name, first name and patronymic are exactly the same, only the date of birth is completely different - he was born one day before you.

And now to the point))) Suppose a finite sequence that has parity loses this parity when going to infinity. Then any finite segment of an infinite sequence must also lose parity. We do not observe this. The fact that we cannot say for sure whether the number of elements in an infinite sequence is even or odd does not mean at all that the parity has disappeared. Parity, if it exists, cannot disappear into infinity without a trace, as in the sleeve of a card sharper. There is a very good analogy for this case.

Have you ever asked a cuckoo sitting in a clock in which direction the clock hand rotates? For her, the arrow rotates in the opposite direction to what we call "clockwise". It may sound paradoxical, but the direction of rotation depends solely on which side we observe the rotation from. And so, we have one wheel that rotates. We cannot say in which direction the rotation occurs, since we can observe it both from one side of the rotation plane and from the other. We can only testify to the fact that there is rotation. Complete analogy with the parity of an infinite sequence S.

Now let's add a second rotating wheel, the plane of rotation of which is parallel to the plane of rotation of the first rotating wheel. We still can't tell exactly which direction these wheels are spinning, but we can tell with absolute certainty whether both wheels are spinning in the same direction or in opposite directions. Comparing two infinite sequences S and 1-S, I showed with the help of mathematics that these sequences have different parity and putting an equal sign between them is a mistake. Personally, I believe in mathematics, I do not trust mathematicians))) By the way, in order to fully understand the geometry of transformations of infinite sequences, it is necessary to introduce the concept "simultaneity". This will need to be drawn.

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 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 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 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.

In conclusion, I want to show you how mathematicians manipulate
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 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 (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.

In the last lesson, we successfully mastered (or repeated - as anyone likes) the key concepts of all trigonometry. it trigonometric circle , angle on a circle , sine and cosine of this angle and also mastered signs of trigonometric functions in quarters . Learned in detail. On the fingers, one might say.

But this is still not enough. In order to successfully apply all these simple concepts in practice, we need another useful skill. Namely, the correct working with corners in trigonometry. Without this skill in trigonometry - nothing. Even in the most primitive examples. Why? Yes, because the angle is the key acting figure in all trigonometry! No, not trigonometric functions, not sine with cosine, not tangent with cotangent, namely the corner itself. No angle - no trigonometric functions, yes ...

How to work with corners on a circle? To do this, we need to ironically learn two points.

1) How Are the angles on a circle counted?

2) What are they counted (measured)?

The answer to the first question is the topic of today's lesson. We will deal with the first question in detail right here and now. The answer to the second question will not be given here. Because it's quite developed. Like the second question itself, it’s very slippery, yes.) I won’t go into details for now. This is the topic of the next separate lesson.

Shall we start?

How are angles calculated on a circle? Positive and negative angles.

Those who read the title of the paragraph may already have their hair on end. How so?! Negative corners? Is this even possible?

to the negative numbers we have already got used to it. We can represent them on the numerical axis: positive to the right of zero, negative to the left of zero. Yes, and we look at the thermometer outside the window periodically. Especially in winter, in frost.) And the money on the phone is in the "minus" (i.e. duty) sometimes go away. It's all familiar.

But what about the corners? It turns out that negative angles in mathematics also happen! It all depends on how to count this very angle ... no, not on a number line, but on a number circle! I mean, in a circle. Circle - here it is, an analogue of the number line in trigonometry!

So, How are the angles on a circle calculated? There is nothing to be done, we will have to draw this very circle first.

I'll draw this beautiful picture:

It is very similar to the pictures from the previous lesson. There are axes, there is a circle, there is an angle. But there is also new information.

I also added numbers for 0°, 90°, 180°, 270° and 360° on the axes. Now this is more interesting.) What are these numbers? Correctly! These are the values of the angles measured from our fixed side, which fall on the coordinate axes. We recall that the fixed side of the angle is always firmly attached to the positive semiaxis OX. And any angle in trigonometry is measured from this semiaxis. This basic origin of the angles must be kept in mind ironically. And the axes - they intersect at right angles, right? So we add 90 ° in each quarter.

And more added red arrow. With a plus. The red one is on purpose to catch the eye. And it stuck in my memory well. For this must be remembered reliably.) What does this arrow mean?

So it turns out, if we turn our corner plus arrow(counterclockwise, in the course of the numbering of quarters), then the angle will be considered positive! The figure shows an angle of +45° as an example. By the way, please note that the axial angles 0°, 90°, 180°, 270° and 360° are also rewound precisely in plus! By the red arrow.

Now let's look at another picture:

Almost everything is the same here. Only the angles on the axes are numbered reversed. Clockwise. And they have a minus sign.) blue arrow. Also with a minus. This arrow is the direction of the negative reading of the angles on the circle. She shows us that if we postpone our corner clockwise, then angle will be considered negative. For example, I showed an angle of -45°.

By the way, please note that the numbering of quarters never changes! It doesn't matter if we wind corners in plus or minus. Always strictly counterclockwise.)

Remember:

1. The beginning of the counting of angles is from the positive semiaxis ОХ. By the hour - "minus", against the clock - "plus".

2. The numbering of the quarters is always counterclockwise, regardless of the direction of the calculation of the angles.

By the way, signing the angles on the axes 0°, 90°, 180°, 270°, 360°, each time drawing a circle, is not a requirement at all. This is purely for understanding the essence. But these numbers must be present in your head when solving any problem in trigonometry. Why? Yes, because this elementary knowledge gives answers to many other questions in all trigonometry! The most important question is in which quarter does the angle we are interested in fall? Believe it or not, the correct answer to this question solves the lion's share of all other problems with trigonometry. We will deal with this important lesson (the distribution of angles in quarters) in the same lesson, but a little later.

The values of the angles lying on the coordinate axes (0°, 90°, 180°, 270° and 360°) must be remembered! Remember firmly, to automatism. And both in plus and minus.

But from this moment the first surprises begin. And along with them tricky questions addressed to me, yes ...) And what will happen if the negative angle on the circle match the positive? It turns out that the same point on a circle can be denoted as a positive angle, and a negative one ???

Quite right! So it is.) For example, a positive angle of +270° occupies on a circle the same position , which is the negative angle -90°. Or, for example, a positive angle of +45° on a circle will take the same position , which is the negative angle -315°.

We look at the next picture and see everything:

Similarly, a positive angle of +150° will go where a negative angle of -210°, a positive angle of +230° will go to the same place as a negative angle of -130°. And so on…

And now what i can do? How exactly to count the angles, if it is possible this way and that? How right?

Answer: anyway correct! Mathematics does not prohibit any of the two directions for counting angles. And the choice of a specific direction depends solely on the task. If the task does not say anything in plain text about the sign of the angle (such as "determine the greatest negative corner" etc.), then we work with the most convenient angles for us.

Of course, for example, in such cool topics as trigonometric equations and inequalities, the direction of the calculation of angles can have a huge impact on the answer. And in the relevant topics, we will consider these pitfalls.

Remember:

Any point on the circle can be denoted by both positive and negative angles. Anyone! What we want.

Now let's think about this. We found out that the angle of 45° is exactly the same as the angle of -315°? How did I find out about these same 315° ? Can't you guess? Yes! Through a full turn.) In 360 °. We have a 45° angle. How much is missing before a full turn? Subtract 45° from 360° - here we get 315° . We wind in the negative direction - and we get an angle of -315 °. Still unclear? Then look at the picture above again.

And this should always be done when translating positive angles into negative ones (and vice versa) - draw a circle, note about a given angle, we consider how many degrees are missing before a full turn, and we wind the resulting difference in the opposite direction. And that's it.)

What else is interesting about the corners that occupy the same position on the circle, what do you think? And the fact that such corners exactly the same sine, cosine, tangent and cotangent! Is always!

For example:

Sin45° = sin(-315°)

Cos120° = cos(-240°)

Tg249° = tg(-111°)

Ctg333° = ctg(-27°)

And now this is extremely important! What for? Yes, all for the same!) To simplify expressions. For simplification of expressions is a key procedure for a successful solution any assignments in mathematics. And trigonometry as well.

So, we figured out the general rule for counting angles on a circle. Well, if we here hinted at full turns, about quarters, then it would be time to twist and draw these very corners. Shall we draw?)

Let's start with positive corners. They will be easier to draw.

Draw angles within one revolution (between 0° and 360°).

Let's draw, for example, an angle of 60°. Everything is simple here, no frills. We draw coordinate axes, a circle. You can directly by hand, without any compass and ruler. We draw schematically A: We don't have drafting with you. There is no need to comply with GOSTs, they will not be punished.)

You can (for yourself) mark the values of the angles on the axes and indicate the arrow in the direction against the clock. After all, we are going to save money as a plus?) You can not do this, but you need to keep everything in your head.

And now we draw the second (movable) side of the corner. What quarter? In the first, of course! For 60 degrees is strictly between 0° and 90°. So we draw in the first quarter. at an angle about 60 degrees to the fixed side. How to count about 60 degrees without a protractor? Easily! 60° is two thirds of a right angle! We mentally divide the first quarter of the circle into three parts, we take two-thirds for ourselves. And we draw ... How much we actually get there (if we attach a protractor and measure it) - 55 degrees or 64 - it doesn’t matter! It is important that still somewhere about 60°.

We get an image:

That's all. And no tools were needed. We develop an eye! It will come in handy in geometry tasks.) This unsightly drawing can be indispensable when you need to scratch a circle and an angle in haste, without really thinking about beauty. But at the same time scribble right, without errors, with all the necessary information. For example, as an aid in solving trigonometric equations and inequalities.

Now let's draw an angle, for example, 265°. Guess where it might be? Well, it's clear that not in the first quarter and not even in the second: they end at 90 and 180 degrees. You can think that 265° is 180° plus another 85°. That is, to the negative semiaxis OX (where 180 °) must be added about 85°. Or, even easier, to guess that 265 ° does not reach the negative semi-axis OY (where 270 °) of some unfortunate 5 °. In a word, in the third quarter there will be this corner. Very close to the negative axis OY, to 270 degrees, but still in the third!

Draw:

Again, absolute precision is not required here. Let in reality this angle turned out to be, say, 263 degrees. But the most important question (what quarter?) we answered correctly. Why is this the most important question? Yes, because any work with an angle in trigonometry (whether we draw this angle or not) begins with the answer to this very question! Is always. If you ignore this question or try to answer it mentally, then mistakes are almost inevitable, yes ... Do you need it?

Remember:

Any work with an angle (including drawing this very angle on a circle) always begins with determining the quarter in which this angle falls.

Now, I hope you will draw the angles correctly, for example, 182°, 88°, 280°. AT correct quarters. In the third, first and fourth, if anything ...)

The fourth quarter ends at a 360° angle. This is one full turn. Pepper is clear that this angle occupies the same position on the circle as 0 ° (ie, the reference point). But the corners don't end there, yeah...

What to do with angles greater than 360°?

"Do such things exist?"- you ask. There are, how! It happens, for example, an angle of 444 °. And sometimes, say, an angle of 1000 °. There are all sorts of angles.) Just visually, such exotic angles are perceived a little more complicated than the usual angles within one turn. But you also need to be able to draw and calculate such angles, yes.

To correctly draw such angles on a circle, you need to do the same thing - find out in which quarter does the angle of interest fall. Here the ability to accurately determine the quarter is much more important than for angles from 0 ° to 360 °! The very procedure for determining a quarter is complicated by just one step. Which one, you'll soon see.

So, for example, we need to find out in which quarter the angle 444° falls. We start to spin. Where? As a plus, of course! They gave us a positive angle! +444°. We twist, we twist ... We twisted one turn - we reached 360 °.

How much is left to 444°?We count the remaining tail:

444°-360° = 84°.

So 444° is one full turn (360°) plus another 84°. Obviously, this is the first quarter. So, the angle 444° falls in the first quarter. Half done.

It remains now to depict this angle. How? Very simple! We make one full turn along the red (plus) arrow and add another 84 °.

Like this:

Here I did not clutter up the drawing - sign quarters, draw angles on the axes. All this goodness should have been in my head for a long time.)

But I showed with a "snail" or a spiral how exactly the angle of 444 ° is formed from the angles of 360 ° and 84 °. The dotted red line is one full turn. To which 84° are additionally screwed (solid line). By the way, please note that if this very full turn is discarded, then this will not affect the position of our corner in any way!

But this is important! Angle position 444° completely coincides with an angle position of 84°. There are no miracles, it just happens.)

Is it possible to discard not one full turn, but two or more?

Why not? If the corner is hefty, then it’s not just possible, but even necessary! The angle won't change! More precisely, the angle itself will, of course, change in magnitude. But his position on the circle - no way!) That's why they full momentum, that no matter how many copies you add, no matter how much you subtract, you will still hit the same point. Nice, right?

Remember:

If we add (subtract) to the angle any whole number of complete revolutions, the position of the original corner on the circle will NOT change!

For example:

In which quarter does the angle 1000° fall?

No problem! We consider how many full revolutions sit in a thousand degrees. One revolution is 360°, another one is already 720°, the third is 1080°… Stop! Bust! So, in an angle of 1000 ° sits two full turnover. Throw them out of 1000° and calculate the remainder:

1000° - 2 360° = 280°

So the position of the angle 1000° on the circle same, which is the same as the angle of 280°. With whom it is already much more pleasant to work.) And where does this corner fall? It falls into the fourth quarter: 270° (negative semi-axis OY) plus another ten.

Draw:

Here I no longer drew two full turns with a dotted spiral: it turns out to be painfully long. Just drew the rest of the ponytail from zero, discarding all extra turns. It's like they didn't even exist.)

Once again. In a good way, the angles 444° and 84°, as well as 1000° and 280° are different. But for sine, cosine, tangent and cotangent, these angles are the same!

As you can see, in order to work with angles larger than 360°, you need to define how many full revolutions sit in a given large angle. This is the very additional step that must be done beforehand when working with such angles. Nothing complicated, right?

Dropping full turns, of course, is a pleasant experience.) But in practice, when working with absolutely nightmarish angles, difficulties also occur.

For example:

In which quarter does the angle 31240° fall?

And what, we will add 360 degrees many, many times? It is possible, if it does not burn especially. But we can not only add.) We can also divide!

So let's divide our huge angle into 360 degrees!

By this action, we just find out how many full revolutions are hidden in our 31240 degrees. You can share a corner, you can (whisper in your ear :)) on a calculator.)

We get 31240:360 = 86.777777….

The fact that the number turned out to be fractional is not scary. We are only whole I'm interested in turnovers! Therefore, there is no need to divide to the end.)

So, in our shaggy corner sits as many as 86 full revolutions. Horror…

In degrees it will be86 360° = 30960°

Like this. That is how many degrees can be painlessly thrown out of a given angle of 31240 °. Remains:

31240° - 30960° = 280°

Everything! Angle position 31240° fully identified! In the same place as 280°. Those. fourth quarter.) It seems we have already depicted this angle before? When was the 1000° angle drawn?) There we also went 280 degrees. Coincidence.)

So the moral of the story is this:

If we are given a terrible hefty corner, then:

1. Determine how many full revolutions sit in this corner. To do this, divide the original angle by 360 and discard the fractional part.

2. We consider how many degrees are in the received number of revolutions. To do this, multiply the number of revolutions by 360.

3. Subtract these revolutions from the original angle and work with the usual angle in the range from 0° to 360°.

How to work with negative angles?

No problem! In the same way as with positive ones, with only one single difference. What? Yes! You need to turn the corners reverse side, minus! clockwise.)

Let's draw, for example, an angle of -200°. At first, everything is as usual for positive angles - axes, a circle. Let's draw a blue arrow with a minus and sign the angles on the axes in a different way. They, of course, will also have to be counted in the negative direction. These will be all the same angles, stepping through 90°, but counted in the opposite direction, minus: 0°, -90°, -180°, -270°, -360°.

The picture will look like this:

When working with negative angles, there is often a feeling of slight bewilderment. How so?! It turns out that the same axis is both, say, +90° and -270°? Nope, something's wrong here...

Yes, everything is clean and transparent! After all, we already know that any point on the circle can be called both a positive angle and a negative one! Absolutely any. Including on some of the coordinate axes. In our case, we need negative calculation of angles. So we snap off all the corners to minus.)

Now drawing the right angle of -200° is no problem. This is -180° and minus another 20°. We start winding from zero to minus: we fly through the fourth quarter, the third is also past, we reach -180 °. Where to wind the remaining twenty? Yes, it's all right there! By the clock.) Total angle -200° falls into second quarter.

Now you understand how important it is to remember the angles on the coordinate axes?

The angles on the coordinate axes (0°, 90°, 180°, 270°, 360°) must be remembered precisely in order to accurately determine the quarter where the angle falls!

And if the angle is large, with several full turns? It's OK! What difference does it make where these full speeds are turned - in plus or minus? A point on a circle will not change its position!

For example:

In which quadrant does the angle -2000° fall?

All the same! To begin with, we consider how many full revolutions sit in this evil corner. In order not to mess up in signs, let's leave the minus alone for now and just divide 2000 by 360. We get 5 with a tail. The tail does not bother us yet, we will count it a little later when we draw the corner. We believe five full revolutions in degrees:

5 360° = 1800°

Voot. That is how many extra degrees you can safely throw out of our corner without harm to health.

We count the remaining tail:

2000° – 1800° = 200°

And now you can also remember about the minus.) Where will we wind the tail 200 °? Downside, of course! We are given a negative angle.)

2000° = -1800° - 200°

So we draw an angle of -200 °, only without extra turns. I just drew it, but, so be it, I'll paint it one more time. By hand.

The pepper is clear that the given angle -2000 °, as well as -200 °, falls into second quarter.

So, we wind ourselves on a circle ... sorry ... on a mustache:

If a very large negative angle is given, then the first part of working with it (finding the number of full revolutions and discarding them) is the same as when working with a positive angle. The minus sign does not play any role at this stage of the solution. The sign is taken into account only at the very end, when working with the angle remaining after the removal of full turns.

As you can see, drawing negative angles on a circle is no more difficult than drawing positive ones.

Everything is the same, only in the other direction! By the hour!

And now - the most interesting! We've covered positive angles, negative angles, large angles, small angles - the full range. We also found out that any point on the circle can be called a positive and negative angle, we discarded full turns ... No thoughts? Should be postponed...

Yes! Whatever point on the circle you take, it will correspond to endless angles! Large and not so, positive and negative - everyone! And the difference between these angles will be whole number of complete turns. Is always! So the trigonometric circle is arranged, yes ...) That is why reverse the task is to find the angle by the known sine / cosine / tangent / cotangent - is solved ambiguous. And much more difficult. In contrast to the direct problem - to find the entire set of its trigonometric functions for a given angle. And in more serious topics of trigonometry ( arches, trigonometric equations and inequalities ) we will encounter this chip constantly. Getting used.)

1. In what quarter does the angle -345° fall?

2. In which quarter does the angle 666° fall?

3. What quarter does the angle 5555° fall into?

4. What quarter does the -3700° angle fall into?

5. What is the signcos999°?

6. What is the signctg999°?

And did it work? Wonderful! There is a problem? Then you.

Answers:

1. 1

2. 4

3. 2

4. 3

5. "+"

6. "-"

This time, the answers are given in order, breaking with tradition. For there are only four quarters, and there are only two signs. You won't run away...)

In the next lesson, we will talk about radians, about the mysterious number "pi", we will learn how to easily and simply convert radians to degrees and vice versa. And we will be surprised to find that even these simple knowledge and skills will already be quite enough for us to successfully solve many non-trivial problems in trigonometry!