Properties of sine, cosine, tangent and cotangent of an angle. Trigonometric circle

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, it can be thought of as a rectangle, with one side representing lettuce and the other side representing water. The sum of these two sides will indicate 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 from a mathematical point of view? How can the sum of two line segments become trigonometry? To understand this, we need linear angular functions.

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

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

Is it possible to do without linear angular functions? It’s possible, because mathematicians still manage without them. The trick of mathematicians is that they always tell us only about those problems that they themselves know how to solve, and never talk about those problems that they cannot solve. Look. If we know the result of addition and one term, we use subtraction to find the other term. All. We don’t know other problems and we don’t know how to solve them. What should we do if we only know the result of the addition and do not know both terms? In this case, the result of the addition must be decomposed into two terms using linear angular functions. Next, we ourselves choose what one term can be, and linear angular functions show what the second term should be so that the result of the addition is exactly what we need. There can be an infinite number of such pairs of terms. IN Everyday life We can do just fine without decomposing the sum; subtraction is enough for us. But when scientific research laws of nature, decomposing a sum into its components can be very useful.

Another law of addition that mathematicians don't like to talk about (another of their tricks) requires that the terms have the same units of measurement. For salad, water, and borscht, these could be units of weight, volume, value, or unit of measurement.

The figure shows two levels of difference for mathematical . 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 field of units of measurement, which are shown in square brackets and indicated by the letter U. This is what physicists do. We can understand the third level - differences in the area of ​​​​the objects being described. Different objects can have the same number of identical units of measurement. How important this is, we can see in the example of borscht trigonometry. If we add subscripts to the same designation of units of measurement of different objects, we can say exactly which mathematical quantity describes a specific object and how it changes over time or due to our actions. Letter W I will designate water with a letter S I'll designate the salad with a letter B- borsch. This is what linear angular functions for borscht will look like.

If we take some part of the water and some part of the salad, together they will turn into one portion 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 there would be. What were we taught to do then? We were taught to separate units of measurement from numbers and add numbers. Yes, any one number can be added to any other number. This is a direct path to the autism of modern mathematics - we do it incomprehensibly what, incomprehensibly why, and very poorly understand how this relates to reality, because of the three levels of difference, mathematicians operate with only one. It would be more correct to learn how to move from one unit of measurement to another.

Bunnies, 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 amount of money. We got the total value of our wealth in monetary terms.

Second option. You can add the number of bunnies to the number of banknotes we have. We will receive 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 let's get back to our borscht. Now we can see what will happen when different meanings angle of linear angular functions.

The angle is zero. We have salad, but no water. We can't cook borscht. The amount of borscht is also zero. This does not mean at all that zero borscht is equal to zero water. There can be zero borscht with 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 happens because addition itself is impossible if there is only one term and the second term is missing. You can feel about this as you like, but remember - all mathematical operations with zero were invented by mathematicians themselves, so throw away your logic and stupidly cram the definitions invented by mathematicians: “division by zero is impossible”, “any number multiplied by zero equals zero” , “beyond the puncture point zero” and other nonsense. It is enough to remember once that zero is not a number, and you will never again have a question whether zero is a natural number or not, because such a question loses all meaning: how can something that is not a number be considered a number? It's like asking what color an invisible color should be classified as. Adding a zero to a number is the same as painting with paint that is not there. We waved a dry brush and told everyone that “we 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 not enough water. As a result, we will get thick borscht.

The angle is forty-five degrees. We have equal quantities of water and salad. This is the perfect borscht (forgive me, chefs, 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 salad. You will get liquid borscht.

Right angle. We have water. All that remains of the salad are memories, as we continue to measure the angle from the line that once marked the salad. We can't cook borscht. The amount of borscht is zero. In this case, hold on and drink water while you have it)))

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

Two friends had their shares in a common business. After killing 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 borscht trigonometry and consider projections.

Saturday, October 26, 2019

I watched an interesting video about Grundy series One minus one plus one minus one - Numberphile. Mathematicians lie. They did not perform an equality check during their reasoning.

This echoes my thoughts about .

Let's take a closer look at the signs that mathematicians are deceiving us. At the very beginning of the argument, mathematicians say that the sum of a sequence DEPENDS on whether it has an even number of elements 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 of the sequence - an even number changes to an odd number, an odd number changes to an even number. After all, we 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 talk about an infinite sequence, we must remember that the demon final sequence with an odd number of elements is not equal to an infinite sequence with an even number of elements.

By putting an equal sign between two sequences with different numbers 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, since it is based on a false equality.

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

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

Infinity is for mathematicians, like the Kingdom of Heaven is 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 into 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.

Now let’s get to the point))) Let’s say that a finite sequence that has parity loses this parity when going to infinity. Then any finite segment of an infinite sequence must lose parity. We don't see this. The fact that we cannot say for sure whether an infinite sequence has an even or odd number of elements does not mean that parity has disappeared. Parity, if it exists, cannot disappear without a trace into infinity, like in a sharpie’s sleeve. There is a very good analogy for this case.

Have you ever asked the cuckoo sitting in the clock in which direction the clock hand rotates? For her, the arrow rotates in the opposite direction to what we call “clockwise”. As paradoxical as it may sound, 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 plane of rotation 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 say for sure in which direction these wheels rotate, but we can absolutely tell whether both wheels rotate in the same direction or in the opposite direction. Comparing two infinite sequences S And 1-S, I showed with the help of mathematics that these sequences have different parities and putting an equal sign between them is a mistake. Personally, I trust mathematics, I don’t trust mathematicians))) By the way, 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. The point is that the concept of “infinity” affects mathematicians like a boa constrictor affects a rabbit. The trembling horror of infinity deprives mathematicians common sense. Here's an example:

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

To clearly prove that they were right, mathematicians came up with many different methods. Personally, I look at all these methods as shamans dancing with tambourines. Essentially, they all boil down to the fact that either some of the rooms are unoccupied and new guests are moving in, or that some of the visitors are thrown out into the corridor to make room for guests (very humanly). I presented my view on such decisions in the form of a fantasy story about the Blonde. What is my reasoning based on? Relocating an infinite number of visitors takes an infinite amount of time. After we have vacated the first room for a guest, 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 be in the category of “no law is written for fools.” It all depends on what we are doing: adjusting reality to mathematical theories or vice versa.

What is an “endless hotel”? An infinite hotel is a hotel that always has any number of empty beds, regardless of how many rooms are occupied. If all the rooms in the endless "visitor" corridor are occupied, there is another endless corridor with "guest" rooms. There will be an infinite number of such corridors. Moreover, 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 are not able to distance themselves from banal everyday problems: there is always only one God-Allah-Buddha, there is only one hotel, there is only one corridor. So mathematicians are trying to juggle the serial numbers of hotel rooms, convincing us that it is possible to “shove in the impossible.”

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 are there - one or many? There is no correct answer to this question, since we invented numbers ourselves; numbers do not exist in Nature. Yes, Nature is great at counting, but for this she uses other mathematical tools that are not familiar to us. I’ll tell you what Nature thinks another time. Since we invented numbers, we ourselves will decide how many sets of natural numbers there are. Let's consider both options, as befits real scientists.

Option one. “Let us be given” one single set of natural numbers, which lies serenely on the shelf. We take this set from the shelf. That's it, there are no other natural numbers left on the shelf and 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 one from the set we have already taken and return it to the shelf. After that, we can take one from the shelf and add it to what we have left. As a result, we will again get an infinite set of natural numbers. You can write down all our manipulations like this:

I wrote down the actions in algebraic notation and in set theory notation, with a detailed listing 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 unit is added.

Option two. We have many different infinite sets of natural numbers on our shelf. I emphasize - DIFFERENT, despite the fact that they are practically indistinguishable. Let's 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. This is 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 you add another infinite set 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 is for measuring. Now imagine that you added one centimeter to the ruler. This will be a different line, not equal to the original one.

You can accept or not accept my reasoning - it is your own business. But if someday you come across math problems, think about whether you are following the path of false reasoning trodden by generations of mathematicians. After all, studying mathematics, first of all, forms a stable stereotype of thinking in us, and only then adds to our mental abilities (or, conversely, deprives us of free-thinking).

pozg.ru

Sunday, August 4, 2019

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

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

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

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

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

Saturday, August 3, 2019

How to divide a set into subsets? To do this, you need to enter a new unit of measurement that is present in some of the elements of the selected set. Let's look at an example.

May we have plenty A consisting of four people. This set is formed on the basis of “people.” Let us denote the elements of this set by the letter A, the subscript with a number will indicate the serial number of each person in this set. Let's introduce a new unit of measurement "gender" and denote it by the letter b. Since sexual characteristics are inherent in all people, we multiply each element of the set A based on gender b. Notice that our set of “people” has now become a set of “people with gender characteristics.” After this we can divide the sexual characteristics into male bm and women's bw sexual characteristics. Now we can apply a mathematical filter: we select one of these sexual characteristics, no matter which one - male or female. If a person has it, then we multiply it by one, if there is no such sign, we multiply it by zero. And then we use regular school mathematics. Look what happened.

After multiplication, reduction and rearrangement, we ended up with two subsets: the subset of men Bm and a subset of women Bw. Mathematicians reason in approximately the same way when they apply set theory in practice. But they don’t tell us the details, but give us the finished result - “a lot of people consist of a subset of men and a subset of women.” Naturally, you may have a question: how correctly has the mathematics been applied in the transformations outlined above? I dare to assure you that, in essence, the transformations were done correctly; it is enough to know the mathematical basis of arithmetic, Boolean algebra and other branches of mathematics. What it is? Some other time I will tell you about this.

As for supersets, you can combine two sets into one superset by selecting the unit of measurement present in the elements of these two sets.

As you can see, units of measurement and ordinary mathematics make set theory a relic 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. Mathematicians acted as shamans once did. Only shamans know how to “correctly” apply their “knowledge.” They teach us this “knowledge”.

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 steps behind it. During the time it takes Achilles to run this distance, the tortoise will crawl a hundred steps in the same direction. When Achilles runs a hundred steps, the tortoise crawls another ten steps, and so on. The process will continue ad infinitum, Achilles will never catch up with the tortoise.

This reasoning became a logical shock for all subsequent generations. Aristotle, Diogenes, Kant, Hegel, Hilbert... They all considered Zeno's aporia in one way or another. The shock was so strong that " ...discussions continue to this day, to reach a common opinion about the essence of paradoxes scientific community so far it has not been possible... mathematical analysis, set theory, new physical and philosophical approaches were involved in the study of the issue; none of them became a generally accepted solution to the problem..."[Wikipedia, "Zeno's Aporia". Everyone understands that they are being fooled, but no one understands what the deception consists of.

From a mathematical point of view, Zeno in his aporia clearly demonstrated the transition from quantity to . This transition implies application instead of permanent ones. As far as I understand, the mathematical apparatus for using variable units of measurement has either not yet been developed, or it has not been applied to Zeno’s aporia. Applying our usual logic leads us into a trap. We, due to the inertia of thinking, apply constant units of time to the reciprocal value. From a physical point of view, this looks like time slowing down until it stops completely at the moment when Achilles catches up with the turtle. If time stops, Achilles can no longer outrun the tortoise.

If we turn our usual logic around, everything falls into place. Achilles runs at a constant speed. Each subsequent segment of his 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 catch up with the turtle infinitely quickly.”

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

In the time it takes Achilles to run a thousand steps, the tortoise will crawl 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 a hundred steps. Now Achilles is eight hundred steps 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 irresistibility of the speed of light is very similar to Zeno’s aporia “Achilles and the Tortoise”. We still have to study, rethink and solve this problem. And the solution must not be sought endlessly large numbers, but in units of measurement.

Another interesting aporia of Zeno tells about a flying arrow:

A flying arrow is motionless, since at every 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 a flying arrow is at rest at different points in space, which, in fact, is motion. Another point needs 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 whether a car is moving, you need two photographs taken from the same point at different points in time, but you cannot determine the distance from them. To determine the distance to the car, you need two photographs taken from different points space at one point in time, but it is impossible to determine the fact of movement from them (naturally, additional data is still needed for calculations, trigonometry will help you). What I want to draw special attention to is that two points in time and two points in space are different things that should not be confused, because they provide different opportunities for research.
I'll show you the process with an example. We select the “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 part of the “whole” and form a set “with a bow”. This is how shamans get their food by tying their set theory to reality.

Now let's do a little trick. Let’s take “solid with a pimple with a bow” and combine these “wholes” according to color, selecting the red elements. We got a lot of "red". Now the final question: are the resulting 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 it will be.

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 with a pimple and a bow." The formation took place in four different units of measurement: color (red), strength (solid), roughness (pimply), decoration (with a bow). Only a set of units of measurement allows us to adequately describe real objects in the language of mathematics. This is what it looks like.

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

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

The sign of the trigonometric function depends solely on the coordinate quarter in which it is located numeric argument. Last time we learned to convert arguments from a radian measure to a degree measure (see lesson “ Radian and degree measure of an angle”), and then determine this same coordinate quarter. Now let's actually determine the sign of sine, cosine and tangent.

The sine of angle α is the ordinate (y coordinate) of a point on a trigonometric circle that occurs when the radius is rotated by angle α.

The cosine of angle α is the abscissa (x coordinate) of a point on a trigonometric circle, which occurs when the radius is rotated by angle α.

The tangent of the angle α is the ratio of sine to cosine. Or, which is the same thing, the ratio of the y coordinate to the x coordinate.

Notation: sin α = y ; cos α = x ; tg α = y : x .

All these definitions are familiar to you from high school algebra. However, we are not interested in the definitions themselves, but in the consequences that arise on the trigonometric circle. Take a look:

Blue color indicates the positive direction of the OY axis (ordinate axis), red indicates the positive direction of the OX axis (abscissa axis). On this "radar" the signs of trigonometric functions become obvious. In particular:

1. sin α > 0 if angle α lies in the I or II coordinate quadrant. This is because, by definition, sine is an ordinate (y coordinate). And the y coordinate will be positive precisely in the I and II coordinate quarters;
2. cos α > 0, if angle α lies in the 1st or 4th coordinate quadrant. Because only there the x coordinate (aka abscissa) will be greater than zero;
3. tan α > 0 if angle α lies in the I or III coordinate quadrant. This follows from the definition: after all, tan α = y : x, therefore it is positive only where the signs of x and y coincide. This happens in the first coordinate quarter (here x > 0, y > 0) and the third coordinate quarter (x< 0, y < 0).

For clarity, let us note the signs of each trigonometric function - sine, cosine and tangent - on separate “radars”. We get the following picture:

Please note: in my discussions I never spoke about the fourth trigonometric function - cotangent. The fact is that the cotangent signs coincide with the tangent signs - there are no special rules there.

Now I propose to consider examples similar to problems B11 from trial Unified State Exam in mathematics, which took place on September 27, 2011. After all, The best way understanding theory is practice. It is advisable to have a lot of practice. Of course, the conditions of the tasks were slightly changed.

Task. Determine the signs of trigonometric functions and expressions (the values ​​of the functions themselves do not need to be calculated):

1. sin(3π/4);
2. cos(7π/6);
3. tg(5π/3);
4. sin (3π/4) cos (5π/6);
5. cos (2π/3) tg (π/4);
6. sin (5π/6) cos (7π/4);
7. tan (3π/4) cos (5π/3);
8. ctg (4π/3) tg (π/6).

The action plan is this: first we convert all angles from radian measures to degrees (π → 180°), and then look at which coordinate quarter the resulting number lies in. Knowing the quarters, we can easily find the signs - according to the rules just described. We have:

1. sin (3π/4) = sin (3 · 180°/4) = sin 135°. Since 135° ∈ , this is an angle from the II coordinate quadrant. But the sine in the second quarter is positive, so sin (3π/4) > 0;
2. cos (7π/6) = cos (7 · 180°/6) = cos 210°. Because 210° ∈ , this is the angle from the third coordinate quadrant, in which all cosines are negative. Therefore cos(7π/6)< 0;
3. tg (5π/3) = tg (5 · 180°/3) = tg 300°. Since 300° ∈ , we are in the IV quarter, where the tangent takes negative values. Therefore tan (5π/3)< 0;
4. sin (3π/4) cos (5π/6) = sin (3 180°/4) cos (5 180°/6) = sin 135° cos 150°. Let's deal with the sine: because 135° ∈ , this is the second quarter in which the sines are positive, i.e. sin (3π/4) > 0. Now we work with cosine: 150° ∈ - again the second quarter, the cosines there are negative. Therefore cos(5π/6)< 0. Наконец, следуя правилу «плюс на минус дает знак минус», получаем: sin (3π/4) · cos (5π/6) < 0;
5. cos (2π/3) tg (π/4) = cos (2 180°/3) tg (180°/4) = cos 120° tg 45°. We look at the cosine: 120° ∈ is the II coordinate quarter, so cos (2π/3)< 0. Смотрим на тангенс: 45° ∈ — это I четверть (самый обычный угол в тригонометрии). Тангенс там положителен, поэтому tg (π/4) >0. Again we got a product in which the factors have different signs. Since “minus by plus gives minus”, we have: cos (2π/3) tg (π/4)< 0;
6. sin (5π/6) cos (7π/4) = sin (5 180°/6) cos (7 180°/4) = sin 150° cos 315°. We work with the sine: since 150° ∈ , we are talking about the II coordinate quarter, where the sines are positive. Therefore, sin (5π/6) > 0. Similarly, 315° ∈ is the IV coordinate quarter, the cosines there are positive. Therefore cos (7π/4) > 0. We have obtained the product of two positive numbers - such an expression is always positive. We conclude: sin (5π/6) cos (7π/4) > 0;
7. tg (3π/4) cos (5π/3) = tg (3 180°/4) cos (5 180°/3) = tg 135° cos 300°. But the angle 135° ∈ is the second quarter, i.e. tg(3π/4)< 0. Аналогично, угол 300° ∈ — это IV четверть, т.е. cos (5π/3) >0. Since “minus by plus gives a minus sign,” we have: tg (3π/4) cos (5π/3)< 0;
8. ctg (4π/3) tg (π/6) = ctg (4 180°/3) tg (180°/6) = ctg 240° tg 30°. We look at the cotangent argument: 240° ∈ is the III coordinate quarter, therefore ctg (4π/3) > 0. Similarly, for the tangent we have: 30° ∈ is the I coordinate quarter, i.e. the simplest angle. Therefore tan (π/6) > 0. Again we have two positive expressions - their product will also be positive. Therefore cot (4π/3) tg (π/6) > 0.

Finally, let's look at some more complex problems. In addition to figuring out the sign of the trigonometric function, you will have to do a little math here - exactly as it is done in real problems B11. In principle, these are almost real problems that actually appear in the Unified State Examination in mathematics.

Task. Find sin α if sin 2 α = 0.64 and α ∈ [π/2; π].

Since sin 2 α = 0.64, we have: sin α = ±0.8. All that remains is to decide: plus or minus? By condition, angle α ∈ [π/2; π] is the II coordinate quarter, where all sines are positive. Therefore, sin α = 0.8 - the uncertainty with signs is eliminated.

Task. Find cos α if cos 2 α = 0.04 and α ∈ [π; 3π/2].

We act similarly, i.e. extract Square root: cos 2 α = 0.04 ⇒ cos α = ±0.2. By condition, angle α ∈ [π; 3π/2], i.e. We are talking about the third coordinate quarter. All cosines there are negative, so cos α = −0.2.

Task. Find sin α if sin 2 α = 0.25 and α ∈ .

We have: sin 2 α = 0.25 ⇒ sin α = ±0.5. We look at the angle again: α ∈ is the IV coordinate quarter, in which, as we know, the sine will be negative. Thus, we conclude: sin α = −0.5.

Task. Find tan α if tan 2 α = 9 and α ∈ .

Everything is the same, only for the tangent. Extract the square root: tan 2 α = 9 ⇒ tan α = ±3. But according to the condition, the angle α ∈ is the I coordinate quarter. All trigonometric functions, incl. tangent, there are positive, so tan α = 3. That's it!

Lesson No. 1

Trigonometric functions of any argument.

Definition and properties of sine, cosine, tangent and cotangent.

Radian measure of angle.

Let us mark point A on the Ox axis from the origin of coordinates and draw a circle through it with the center at point O. We will call the radius OA initial radius.

Angle P (OM; OE) can be described as resulting from rotation around the origin of the beam with the origin at point O from the initial position OM to the final position OE. This rotation can occur either counterclockwise or clockwise, and

a) either by partial rotation,

b) either by an integer full revolutions;

c) either by an integer number of full revolutions and a partial revolution.

Measures of angles oriented counterclockwise are considered positive, and measures oriented clockwise are considered negative.

We will consider as equal angles those angles for which, when their initial rays are combined in some way, the final rays are also combined, and the movement from initial beam to the final one is carried out in the same direction for the same number of full and incomplete revolutions around point O.

Zero angles are considered equal.

Properties of angle measures:

There is an angle whose measure is 1 - the unit of measurement for angles. Equal angles have equal measures. The measure of the sum of two angles is equal to the sum of the measures of the angles. The measure of the zero angle is zero.

The most common measures of angles are degrees and radians.

The unit of measurement for angles in degree measure is an angle of one degree - 1/180 of the unfolded angle. From the geometry course we know that the measure of an angle in degrees is expressed as a number from 01/01/01. As for the angle of rotation, it can be expressed in degrees in any way real number from -∞ to + ∞.

As a circle with a center at the origin, we will take a circle of unit radius, designating the points of its intersection with the coordinate axes A (1;0), B (0;1), C (-1;0), D (0;-1). The ray OA will be taken as the initial angle for the angles under consideration.

The coordinate axes of abscissa and ordinate are mutually perpendicular and divide the plane into four coordinate quarters: I, II, III, IV (see picture).

Depending on which coordinate quarter the OM radius is in, the angleα will also be the angle of this quarter.

So, if 00< α <900 , то угол α – first quarter angle;

If 900< α <1800 , то угол α – angle of the second quarter;

If 1800< α <2700 , то угол α – third quarter angle;

If 2700< α <3600 , то угол α - fourth quarter angle.

Obviously, when adding a whole number of revolutions to an angle, the angle of the same quarter is obtained.

For example, an angle of 4300 is an angle I – oh quarter, since 4300 = 3600 + 700 = 700;

Angle 9200 is the angle III -th quarter, since 9200 = 3600 2 + 2000 = 2000

(i.e. the number of whole revolutions can be ignored!)

Angles 00, ± 900, ± 1800, ± 2700, ± 3600 – do not belong to any quarter .

Let's determine which quadrant angle is the angleα if:

α =2830 (IV) α = 1900 (III) α =1000 (II) α = -200 (IV h – negative direction)

And now for yourself:

α = 1790 α = 3250 α =8000 α = -1200

In the geometry course, sine, cosine, tangent and cotangent of angle α were defined at

00 ≤ α ≤ 1800 . Now we will consider these definitions for the case of an arbitrary angle α.

font-size:12.0pt;line-height:115%">Let when turning near point O by an angleα the initial radius OA becomes the radius OM.

Sine of the angleα is called the ratio of the ordinate of point M to the length of the radius, i.e.

Cosine of the angleα is called the ratio of the abscissa of point M to the length of the radius, i.e.

Tangent of the angleα is called the ratio of the ordinate of a point M to its abscissa, i.e.

Cotangent of the angle α is called the ratio of the abscissa of a point M to its ordinate, i.e.

Let's look at examples of calculating trigonometric functions using tables of values ​​of some angles. Dashes are made when the expression does not make sense.

α (hail)	00	300	450	600	900	1800	2700	3600
(glad)	0					π		2π
sin α
cos α
tan α
ctg α

Example №1. Find sin300; cos450; tg600.

Solution: a) find in the table column sinα and in line 300, at the intersection of column and line we find the value sin 300 is a number. They write like this: sin 300 =

b) find in the table column cosα and in line 450, at the intersection of column and line we find the value cos 450 is a number. They write like this: cos 450 =

c) find in the table column tgα and in line 600, at the intersection of column and line we find the value tg 600 is the number EN-US style="font-size:12.0pt;line-height:115%"">tg600 =font-size:12.0pt;line-height:115%">Example No. 2

Calculate a) 2c os 600 + EN-US" style="font-size: 12.0pt;line-height:115%">cos300 = 2 font-size:12.0pt;line-height:115%"> b)3 tg 450 tg 600 = 3 1 https://pandia.ru/text/79/454/images/image017_6.gif" width="24" height="24 src=">

Calculate it yourself : a) 5 sin 300 - ctg 450 b) 2 sin 300 + 6 cos 600 – 4 tg 450

c) 4tg 600 sin 600 c) 2cossin 900 + 5tg 1800

Let's look at some properties of trigonometric functions.

Let's find out what signs sine, cosine, tangent and cotangent have in each of the coordinate quarters.

Let, when turning the radius OA equal to R, by angle α , point A has moved to point M with coordinates x and y. Because(R = 1), then the sign depends on the sign of y.

In I and II in quarters y>0, and in II and IV quarters - at<0.

Sign depends on x, since, then for the angles of the 1st and 4th quarters – x >0, and in

II and III quarters x<0.

Because ; , then in the 1st and 3rd quarters and have a “+” sign, and in II and IV in quarters they have a minus sign.

Trigonometric circle. Unit circle. Number circle. What it is?

Attention!
There are additional
materials in Special Section 555.
For those who are very "not very..."
And for those who “very much…”)

Very often terms trigonometric circle, unit circle, number circle poorly understood by students. And completely in vain. These concepts are a powerful and universal assistant in all areas of trigonometry. In fact, this is a legal cheat sheet! I drew a trigonometric circle and immediately saw the answers! Tempting? So let's learn, it would be a sin not to use such a thing. Moreover, it is not at all difficult.

To successfully work with the trigonometric circle, you need to know only three things.

If you like this site...

By the way, I have a couple more interesting sites for you.)

You can practice solving examples and find out your level. Testing with instant verification. Let's learn - with interest!)

You can get acquainted with functions and derivatives.

Example 1.

Find the radian measure of an angle equal to a) 40°, b)120°, c)105°

a) 40° = 40 π / 180 = 2π/9

b) 120° = 120 π/180 = 2π/3

c) 105° = 105 π/180 = 7π/12

Example 2.

Find the degree measure of the angle expressed in radians a) π/6, b) π/9, c) 2 π/3

a) π/6 = 180°/6 = 30°

b) π/9 = 180°/9 = 20°

c) 2π/3 = 2 180°/6 = 120°

Definition of sine, cosine, tangent and cotangent

The sine of the acute angle t of a right triangle is equal to the ratio of the opposite side to the hypotenuse (Fig. 1):

The cosine of the acute angle t of a right triangle is equal to the ratio of the adjacent leg to the hypotenuse (Fig. 1):

These definitions apply to a right triangle and are special cases of the definitions presented in this section.

Let's place the same right triangle on the number circle (Fig. 2).

We see that the leg b equal to a certain value y on the Y axis (ordinate axis), leg A equal to a certain value x on the X-axis (x-axis). And the hypotenuse With equal to the radius of the circle (R).

Thus, our formulas take on a different form.

Since b = y, a = x, c = R, then:

y x
sin t = -- , cos t = --.
R R

By the way, then, naturally, the tangent and cotangent formulas take on a different form.

Since tg t = b/a, ctg t = a/b, then other equations are also true:

tg t = y/x,

ctg = x/y.

But let's return to sine and cosine. We are dealing with a number circle in which the radius is 1. This means:

y
sin t = -- = y,
1

x
cos t = -- = x.
1

So we come to the third, simpler type of trigonometric formulas.

These formulas apply not only to acute, but also to any other angle (obtuse or developed).

Definitions and formulas cos t, sin t, tg t, ctg t.

From the tangent and cotangent formulas another formula follows:

Number circle equations.

Signs of sine, cosine, tangent and cotangent in quarter circles:

	1st quarter	2nd quarter	3rd quarter	4th quarter
cos t	+	–	–	+
sint	+	+	–	–
tg t, ctg t	+	–	+	–

Cosine and sine of the main points of the number circle:

How to remember the values ​​of cosines and sines of the main points of the number circle.

First of all, you need to know that in each pair of numbers the cosine values ​​come first, the sine values ​​come second.

1) Please note: with all the many points on the number circle, we are dealing with only five numbers (per module):

1 √2 √3
0; -; --; --; 1.
2 2 2

Make this “discovery” for yourself - and you will remove the psychological fear of the abundance of numbers: there are actually only five of them.

2) Let's start with the integers 0 and 1. They are only on the coordinate axes.

There is no need to learn by heart where, for example, the cosine in the modulus has one and where it has 0.

At the ends of the axle cosines(axes X), of course, cosines equal modulo 1, and the sines are equal to 0.

At the ends of the axle sinuses(axes at) sines are equal to modulus 1, and the cosines are equal to 0.

Now about the signs. Zero has no sign. As for 1 - here you just need to remember the simplest thing: from the 7th grade course you know that on the axis X to the right of the center of the coordinate plane are positive numbers, to the left are negative numbers; on the axis at positive numbers go up from the center, negative numbers go down. And then you won’t be mistaken with sign 1.

3) Now let's move on to fractional values.

All denominators of fractions contain the same number 2. We will no longer be mistaken about what to write in the denominator.

In the middles of the quarters, cosine and sine have absolutely the same absolute value: √2/2. In which case they are with a plus or minus sign - see the table above. But you hardly need such a table: you know this from the same 7th grade course.

All closest to the axis X the points have absolutely identical cosine and sine values: (√3/2; 1/2).

Values ​​of all closest to the axis at The points are also absolutely identical in modulus - and they have the same numbers, only they have “swapped” places: (1/2; √3/2).

Now about the signs - there is an interesting alternation here (although we believe you should be able to figure out the signs easily anyway).

If in the first quarter the values ​​of both cosine and sine have a plus sign, then in the diametrically opposite (third) they have a minus sign.

If in the second quarter with a minus sign there are only cosines, then in the diametrically opposite (fourth) there are only sines.

It remains only to recall that in each combination of cosine and sine values, the first number is the cosine value, the second number is the sine value.

Pay attention to one more regularity: the sine and cosine of all diametrically opposite points of the circle are absolutely equal in magnitude. Take, for example, the opposite points π/3 and 4π/3:

cos π/3 = 1/2, sin π/3 = √3/2
cos 4π/3 = -1/2, sin 4π/3 = -√3/2

The values ​​of the cosines and sines of two opposite points differ only in sign. But here, too, there is a pattern: the sines and cosines of diametrically opposite points always have opposite signs.

It is important to know:

The values ​​of the cosines and sines of points on the number circle sequentially increase or decrease in a strictly defined order: from the smallest value to the largest and vice versa (see the section “Increasing and decreasing trigonometric functions” - however, this is easy to verify by just looking at the number circle higher).

In descending order, the following alternation of values ​​is obtained:

√3 √2 1 1 √2 √3
1; --; --; -; 0; – -; – --; – --; –1
2 2 2 2 2 2

They increase strictly in the reverse order.

Once you understand this simple pattern, you will learn how to determine the values ​​of sine and cosine quite easily.