How to find the smallest positive angle. Positive and negative angles in trigonometry. Protection of personal information

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 as an example 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 recorded the actions in algebraic system notation and in the system of notation adopted in set theory, with a detailed enumeration of the elements of the set. 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 to us mental capacity(or vice versa, deprive us of free thought).

Sunday, August 4, 2019

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

We read: "... rich theoretical background Mathematics of Babylon 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 for set theory, mathematicians have come up with own language and own designations. 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 it is not complete solution Problems. 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 indefinitely 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.

Wednesday, July 4, 2018

I already told you that, with the help of which shamans try to sort "" realities. How do they do it? How does the formation of the set actually take place?

Let's take a closer look at the definition of a set: "a collection of different elements, conceived as a single whole." Now feel the difference between the two phrases: "thinkable as a whole" and "thinkable as a whole." The first phrase is the end result, the multitude. The second phrase is a preliminary preparation for the formation of the set. At this stage, reality is divided into separate elements ("whole") from which a multitude ("single whole") will then be formed. At the same time, the factor that allows you to combine the "whole" into a "single whole" is carefully monitored, otherwise the shamans will not succeed. After all, shamans know in advance exactly what set they want to demonstrate to us.

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 can 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.

Saturday, June 30, 2018

If mathematicians cannot reduce a concept to other concepts, then they do not understand anything in mathematics. I answer: how do the elements of one set differ from the elements of another set? The answer is very simple: numbers and units of measure.

It is today that everything that we do not take belongs to some set (as mathematicians assure us). By the way, did you see in the mirror on your forehead a list of those sets to which you belong? And I have not seen such a list. I will say more - not a single thing in reality has a tag with a list of sets to which this thing belongs. Sets are all inventions of shamans. How do they do it? Let's look a little deeper into history and see how the elements of the set looked before the mathematicians-shamans pulled them apart into their sets.

A long time ago, when no one had heard of mathematics yet, and only trees and Saturn had rings, huge herds of wild elements of sets roamed the physical fields (after all, shamans had not yet invented mathematical fields). They looked like this.

Yes, do not be surprised, from the point of view of mathematics, all elements of sets are most similar to sea urchins- from one point, like needles, units of measurements stick out in all directions. For those who, I remind you that any unit of measurement can be geometrically represented as a segment of arbitrary length, and a number as a point. Geometrically, any quantity can be represented as a bundle of segments sticking out in different sides from one point. This point is the zero point. I will not draw this work of geometric art (no inspiration), but you can easily imagine it.

What units of measurement form an element of the set? Any that describe this element from different points of view. These are the ancient units of measurement used by our ancestors and which everyone has long forgotten about. These are the modern units of measurement that we use now. These are units of measurement unknown to us, which our descendants will come up with and which they will use to describe reality.

We figured out the geometry - the proposed model of the elements of the set has a clear geometric representation. And what about physics? Units of measurement - this is the direct connection between mathematics and physics. If shamans do not recognize units of measurement as a full-fledged element of mathematical theories, this is their problem. I personally can’t imagine a real science of mathematics without units of measurement. That is why, at the very beginning of the story about set theory, I spoke of it as the Stone Age.

But let's move on to the most interesting - to the algebra of elements of sets. Algebraically, any element of the set is a product (the result of multiplication) of different quantities. It looks like this.

I deliberately did not use the conventions adopted in set theory, since we are considering an element of a set in its natural habitat before the advent of set theory. Each pair of letters in brackets denotes a separate value, consisting of the number indicated by the letter " n" and units of measurement, indicated by the letter " a". Indices near the letters indicate that the numbers and units of measurement are different. One element of the set can consist of an infinite number of values \u200b\u200b(as long as we and our descendants have enough imagination). Each bracket is geometrically represented by a separate segment. In the example with the sea urchin one bracket is one needle.

How do shamans form sets from different elements? In fact, by units of measurement or by numbers. Understanding nothing in mathematics, they take different sea urchins and carefully examine them in search of that single needle by which they form a set. If there is such a needle, then this element belongs to the set; if there is no such needle, this element is not from this set. Shamans tell us fables about mental processes and a single whole.

As you may have guessed, the same element can belong to a variety of sets. Next, I will show you how sets, subsets and other shamanistic nonsense are formed. As you can see, "the set cannot have two identical elements", but if there are identical elements in the set, such a set is called a "multiset". Reasonable beings will never understand such logic of absurdity. This is the level of talking parrots and trained monkeys, in which the mind is absent from the word "completely." Mathematicians act as ordinary trainers, preaching their absurd ideas to us.

Once upon a time, the engineers who built the bridge were in a boat under the bridge during the tests of the bridge. If the bridge collapsed, the mediocre engineer died under the rubble of his creation. If the bridge could withstand the load, the talented engineer built other bridges.

No matter how mathematicians hide behind the phrase "mind me, I'm in the house", or rather "mathematics studies abstract concepts", there is one umbilical cord that inextricably connects them with reality. This umbilical cord is money. Let us apply mathematical set theory to mathematicians themselves.

We studied mathematics very well and now we are sitting at the cash desk, paying salaries. Here a mathematician comes to us for his money. We count the entire amount to him and lay it out on our table into different piles, in which we put bills of the same denomination. Then we take one bill from each pile and give the mathematician his "mathematical salary set". We explain the mathematics that he will receive the rest of the bills only when he proves that the set without identical elements is not equal to the set with identical elements. This is where the fun begins.

First of all, the deputies' logic will work: "you can apply it to others, but not to me!" Further, assurances will begin that there are different banknote numbers on banknotes of the same denomination, which means that they cannot be considered identical elements. Well, we count the salary in coins - there are no numbers on the coins. Here the mathematician will frantically recall physics: different coins have different amounts of dirt, the crystal structure and arrangement of atoms for each coin is unique ...

And now I have the most interesting question: where is the boundary beyond which elements of a multiset turn into elements of a set and vice versa? Such a line does not exist - everything is decided by shamans, science here is not even close.

Look here. We select football stadiums with the same field area. The area of the fields is the same, which means we have a multiset. But if we consider the names of the same stadiums, we get a lot, because the names are different. As you can see, the same set of elements is both a set and a multiset at the same time. How right? And here the mathematician-shaman-shuller takes out a trump ace from his sleeve and begins to tell us either about a set or a multiset. In any case, he will convince us that he is right.

To understand how modern shamans operate with set theory, tying it to reality, it is enough to answer one question: how do the elements of one set differ from the elements of another set? I will show you, without any "conceivable as not a single whole" or "not conceivable as a single whole."

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:

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.

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).

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 the mathematics of Babylon did not have a holistic character and was reduced to a set of disparate techniques, devoid of a common system and evidence base."

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.

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.

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:

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:

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.

Wednesday, July 4, 2018

I already told you that, with the help of which shamans try to sort "" realities. How do they do it? How does the formation of the set actually take place?

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.

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.

Saturday, June 30, 2018

Yes, do not be surprised, from the point of view of mathematics, all elements of the sets are most similar to sea urchins - from one point, like needles, units of measurement stick out in all directions. For those who, I remind you that any unit of measurement can be geometrically represented as a segment of arbitrary length, and a number as a point. Geometrically, any quantity can be represented as a bundle of segments sticking out in different directions from one point. This point is the zero point. I will not draw this work of geometric art (no inspiration), but you can easily imagine it.

In the last lesson, we successfully mastered (or repeated - as anyone likes) the key concepts of all trigonometry. This is 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. For a successful practical application all these simple concepts 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! Not no trigonometric functions, not a sine with a cosine, not a tangent with a 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 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! Most main question – 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, positive angle+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°. Etc…

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 largest negative injection" 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 in which angles are calculated can enormously influence 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 negative side- 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! 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 successful solution any assignments in mathematics. And trigonometry as well.

So with general rule counting the angles on the circle figured out. 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 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 problems.) 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 necessary information. Like how aid when 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! 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 the 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. 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 ambiguously. 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? Perfectly! 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 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!

Injection: °π rad =

Convert to: radians degrees 0 - 360° 0 - 2π positive negative Calculate

When lines intersect, there are four different areas with respect to the point of intersection.
These new areas are called corners.

The picture shows 4 different angles formed by the intersection of lines AB and CD

Usually, angles are measured in degrees, which is denoted as °. When an object makes a full circle, that is, moves from point D through B, C, A, and then back to D, it is said to have rotated 360 degrees (360°). So a degree is $\frac(1)(360)$ of a circle.

Angles greater than 360 degrees

We talked about how when an object makes a full circle around a point, then it goes 360°, however, when an object makes more than one circle, then it makes an angle more than 360 degrees. This is a common occurrence in Everyday life. The wheel goes through many circles when the car is moving, i.e. it forms an angle greater than 360°.

In order to find out the number of cycles (circles passed) during the rotation of an object, we count the number of times that 360 must be added to itself to get a number equal to or less than a given angle. In the same way, we find a number, which we multiply by 360 to get a number that is smaller but closest to the given angle.

Example 2
1. Find the number of circles described by the object forming the angle
a) 380°
b) 770°
c) 1000°
Decision
a) 380 = (1 × 360) + 20
The object described one circle and 20°
Since the circle $20^(\circ) = \frac(20)(360) = \frac(1)(18)$
The object described $1\frac(1)(18)$ circles.

B) 2 × 360 = 720
770 = (2 × 360) + 50
The object described two circles and 50°
$50^(\circ) = \frac(50)(360) = \frac(5)(36)$ circles
The object described $2\frac(5)(36)$ circles
c)2 × 360 = 720
1000 = (2 × 360) + 280
$280^(\circ) = \frac(260)(360) = \frac(7)(9)$ circles
The object described $2\frac(7)(9)$ circles

When an object rotates clockwise, it forms a negative angle of rotation, and when it rotates counterclockwise, it forms a positive angle. Up to this point, we have considered only positive angles.

In diagram form, a negative angle can be drawn as shown below.

The figure below shows the sign of the angle, which is measured from a common straight line, 0 axis (abscissa - x axis)

This means that if there is a negative angle, we can get the corresponding positive angle.
For example, the bottom of a vertical line is 270°. When measured in the negative direction, we get -90 °. We simply subtract 270 from 360. Given a negative angle, we add 360 to get the corresponding positive angle.
When the angle is -360°, it means that the object has made more than one clockwise circle.

Example 3
1. Find the corresponding positive angle
a) -35°
b) -60°
c) -180°
d) - 670°

2. Find the corresponding negative angle 80°, 167°, 330° and 1300°.
Decision
1. To find the corresponding positive angle, we add 360 to the angle value.
a) -35°= 360 + (-35) = 360 - 35 = 325°
b) -60°= 360 + (-60) = 360 - 60 = 300°
c) -180°= 360 + (-180) = 360 - 180 = 180°
d) -670°= 360 + (-670) = -310
This means one circle clockwise (360)
360 + (-310) = 50°
The angle is 360 + 50 = 410°

2. To get the corresponding negative angle, we subtract 360 from the angle value.
80° = 80 - 360 = - 280°
167° = 167 - 360 = -193°
330° = 330 - 360 = -30°
1300° = 1300 - 360 = 940 (one lap completed)
940 - 360 = 580 (second round completed)
580 - 360 = 220 (third round passed)
220 - 360 = -140°
The angle is -360 - 360 - 360 - 140 = -1220°
So 1300° = -1220°

Radian

A radian is the angle from the center of a circle that encloses an arc whose length is equal to the radius of the given circle. This is a unit of measure for an angular quantity. This angle is approximately equal to 57.3°.
In most cases, this is referred to as glad.
Thus $1 rad \approx 57.3^(\circ)$

Radius=r=OA=OB=AB
Angle BOA is one radian

Since the circumference is given as $2\pi r$, there are $2\pi$ radii in the circle, and hence $2\pi$ radians in the whole circle.

Radians are usually expressed in terms of $\pi$ to avoid decimal parts in calculations. In most books, the acronym glad (rad) does not occur, but the reader should know that, when we are talking about an angle, then it is specified through $\pi$, and the units of measurement automatically become radians.

$360^(\circ) = 2\pi\rad$
$180^(\circ) = \pi\rad$,
$90^(\circ) = \frac(\pi)(2) rad$,
$30^(\circ) = \frac(30)(180)\pi = \frac(\pi)(6) rad$,
$45^(\circ) = \frac(45)(180)\pi = \frac(\pi)(4) rad$,
$60^(\circ) = \frac(60)(180)\pi = \frac(\pi)(3) rad$
$270^(\circ) = \frac(270)(180)\pi = \frac(27)(18)\pi = 1\frac(1)(2)\pi\ rad$

Example 4
1. Convert 240°, 45°, 270°, 750° and 390° to radians via $\pi$.
Decision
Multiply the angles by $\frac(\pi)(180)$.
$240^(\circ) = 240 \times \frac(\pi)(180) = \frac(4)(3)\pi=1\frac(1)(3)\pi$
$120^(\circ) = 120 \times \frac(\pi)(180) = \frac(2\pi)(3)$
$270^(\circ) = 270 \times \frac(1)(180)\pi = \frac(3)(2)\pi=1\frac(1)(2)\pi$
$750^(\circ) = 750 \times \frac(1)(180)\pi = \frac(25)(6)\pi=4\frac(1)(6)\pi$
$390^(\circ) = 390 \times \frac(1)(180)\pi = \frac(13)(6)\pi=2\frac(1)(6)\pi$

2. Convert the following angles to degrees.
a) $\frac(5)(4)\pi$
b) $3.12\pi$
c) 2.4 radians
Decision
$180^(\circ) = \pi$
a) $\frac(5)(4) \pi = \frac(5)(4) \times 180 = 225^(\circ)$
b) $3.12\pi = 3.12 \times 180 = 561.6^(\circ)$
c) 1 rad = 57.3°
$2.4 = \frac(2.4 \times 57.3)(1) = 137.52$

Negative angles and angles greater than $2\pi$ radians

To convert a negative angle to a positive one, we add it to $2\pi$.
To convert a positive angle to a negative one, we subtract $2\pi$ from it.

Example 5
1. Convert $-\frac(3)(4)\pi$ and $-\frac(5)(7)\pi$ to positive angles in radians.

Decision
Add to the corner $2\pi$
$-\frac(3)(4)\pi = -\frac(3)(4)\pi + 2\pi = \frac(5)(4)\pi = 1\frac(1)(4)\ pi$

$-\frac(5)(7)\pi = -\frac(5)(7)\pi + 2\pi = \frac(9)(7)\pi = 1\frac(2)(7)\ pi$

When an object rotates through an angle greater than $2\pi$;, it makes more than one circle.
In order to determine the number of revolutions (circles or cycles) in such an angle, we find such a number, multiplying by $2\pi$, the result is equal to or less than, but as close as possible to the given number.

Example 6
1. Find the number of circles passed by the object at given angles
a) $-10\pi$
b) $9\pi$
c) $\frac(7)(2)\pi$

Decision
a) $-10\pi = 5(-2\pi)$;
$-2\pi$ implies one cycle in the clockwise direction, it means that
the object made 5 cycles clockwise.

b) $9\pi = 4(2\pi) + \pi$, $\pi =$ half cycle
the object made four and a half cycles counterclockwise

c) $\frac(7)(2)\pi=3.5\pi=2\pi+1.5\pi$, $1.5\pi$ is equal to three quarters of the cycle $(\frac(1.5\pi)(2\pi)= \frac(3)(4))$
the object has traveled one and three-quarters of a cycle counterclockwise

Counting angles on a trigonometric circle.

Attention!
There are additional
material in Special Section 555.
For those who strongly "not very..."
And for those who "very much...")

It is almost the same as in the previous lesson. There are axes, a circle, an angle, everything is chin-china. Added numbers of quarters (in the corners of a large square) - from the first to the fourth. And then suddenly who does not know? As you can see, the quarters (they are also called the beautiful word "quadrants") are numbered counterclockwise. Added angle values on axes. Everything is clear, no frills.

And added a green arrow. With a plus. What does she mean? Let me remind you that the fixed side of the corner always nailed to the positive axis OH. So, if we twist the moving side of the corner plus arrow, i.e. in ascending quarter numbers, the angle will be considered positive. For example, the picture shows a positive angle of +60°.

If we postpone the corners in the opposite direction, clockwise, angle will be considered negative. Hover over the picture (or touch the picture on the tablet), you will see a blue arrow with a minus. This is the direction of the negative reading of the angles. A negative angle (-60°) is shown as an example. And you will also see how the numbers on the axes have changed ... I also translated them into negative angles. The numbering of the quadrants does not change.

Here, usually, the first misunderstandings begin. How so!? And if the negative angle on the circle coincides with the positive!? And in general, it turns out that the same position of the movable side (or a point on the numerical circle) can be called both a negative angle and a positive one!?

Yes. Exactly. Let's say a positive angle of 90 degrees takes on a circle exactly the same position as a negative angle of minus 270 degrees. A positive angle, for example +110° degrees, takes exactly the same position as the negative angle is -250°.

No problem. Everything is correct.) The choice of a positive or negative calculation of the angle depends on the condition of the assignment. If the condition says nothing plain text about the sign of the angle, (like "determine the smallest positive angle", etc.), then we work with values that are convenient for us.

The exception (and how without them?!) are trigonometric inequalities, but there we will master this chip.

And now a question for you. How do I know that the position of the 110° angle is the same as the position of the -250° angle?
I will hint that this is due to the full turnover. In 360°... Not clear? Then we draw a circle. We draw on paper. Marking the corner about 110°. And believe how much remains until a full turn. Just 250° remains...

Got it? And now - attention! If the angles 110° and -250° occupy the circle same position, then what? Yes, the fact that the angles are 110 ° and -250 ° exactly the same sine, cosine, tangent and cotangent!
Those. sin110° = sin(-250°), ctg110° = ctg(-250°) and so on. Now this is really important! And in itself - there are a lot of tasks where it is necessary to simplify expressions, and as a basis for the subsequent development of reduction formulas and other intricacies of trigonometry.

Of course, I took 110 ° and -250 ° at random, purely for example. All these equalities work for any angles occupying the same position on the circle. 60° and -300°, -75° and 285°, and so on. I note right away that the corners in these couples - various. But they have trigonometric functions - the same.

I think you understand what negative angles are. It's quite simple. Counter-clockwise is a positive count. Along the way, it's negative. Consider angle positive or negative depends on us. From our desire. Well, and more from the task, of course ... I hope you understand how to move in trigonometric functions from negative to positive angles and vice versa. Draw a circle, an approximate angle, and see how much is missing before a full turn, i.e. up to 360°.

Angles greater than 360°.

Let's deal with angles that are greater than 360 °. And such things happen? There are, of course. How to draw them on a circle? Not a problem! Suppose we need to understand in which quarter an angle of 1000 ° will fall? Easily! We make one full turn counterclockwise (the angle was given to us positive!). Rewind 360°. Well, let's move on! Another turn - it has already turned out 720 °. How much is left? 280°. It is not enough for a full turn ... But the angle is more than 270 ° - and this is the border between the third and fourth quarter. So our angle of 1000° falls into the fourth quarter. Everything.

As you can see, it's quite simple. Let me remind you once again that the angle of 1000° and the angle of 280°, which we obtained by discarding the "extra" full turns, are, strictly speaking, various corners. But the trigonometric functions of these angles exactly the same! Those. sin1000° = sin280°, cos1000° = cos280° etc. If I were a sine, I wouldn't notice the difference between these two angles...

Why is all this necessary? Why do we need to translate angles from one to another? Yes, all for the same.) In order to simplify expressions. Simplification of expressions, in fact, is the main task of school mathematics. Well, along the way, the head is training.)

Well, shall we practice?)

We answer questions. Simple at first.

1. In which quarter does the angle -325° fall?

2. In which quarter does the angle 3000° fall?

3. In which quarter does the angle -3000° fall?

There is a problem? Or insecurity? We go to Section 555, Practical work with a trigonometric circle. There, in the first lesson of this very " practical work..." everything is detailed ... In such questions of uncertainty shouldn't!

4. What is the sign of sin555°?

5. What is the sign of tg555°?

Determined? Fine! Doubt? It is necessary to Section 555 ... By the way, there you will learn how to draw tangent and cotangent on trigonometric circle. A very useful thing.

And now the smarter questions.

6. Bring the expression sin777° to the sine of the smallest positive angle.

7. Bring the expression cos777° to the cosine of the largest negative angle.

8. Convert the expression cos(-777°) to the cosine of the smallest positive angle.

9. Bring the expression sin777° to the sine of the largest negative angle.

What, questions 6-9 puzzled? Get used to it, there are not such formulations on the exam ... So be it, I will translate it. Only for you!

The words "reduce the expression to ..." mean to transform the expression so that its value hasn't changed a appearance changed in accordance with the task. So, in tasks 6 and 9, we should get a sine, inside which is the smallest positive angle. Everything else doesn't matter.

I will give the answers in order (in violation of our rules). But what to do, there are only two signs, and only four quarters ... You will not scatter in options.

6. sin57°.

7.cos(-57°).

8.cos57°.

9.-sin(-57°)

I suppose that the answers to questions 6-9 confused some people. Especially -sin(-57°), right?) Indeed, in the elementary rules for counting angles there is room for errors ... That is why I had to make a lesson: "How to determine the signs of functions and give angles on a trigonometric circle?" In Section 555. There tasks 4 - 9 are sorted out. Well sorted, with all the pitfalls. And they are here.)

In the next lesson, we will deal with the mysterious radians and the number "Pi". Learn how to easily and correctly convert degrees to radians and vice versa. And we will be surprised to find that this elementary information on the site enough already to solve some non-standard trigonometry puzzles!