1. Trigonometric functions are elementary functions whose argument is corner. By using trigonometric functions describes the relationship between the parties and sharp corners in a right triangle. The areas of application of trigonometric functions are extremely diverse. For example, any periodic processes can be represented as a sum of trigonometric functions (Fourier series). These functions often appear when solving differential and functional equations.
2. Trigonometric functions include the following 6 functions: sinus, cosine, tangent,cotangent, secant And cosecant. For each of these functions there is an inverse trigonometric function.
3. Geometric definition trigonometric functions can be conveniently entered using unit circle. The figure below shows a circle with radius r=1. The point M(x,y) is marked on the circle. The angle between the radius vector OM and the positive direction of the Ox axis is equal to α.
4. Sinus angle α is the ratio of the ordinate y of the point M(x,y) to the radius r:
sinα=y/r.
Since r=1, then the sine is equal to the ordinate of the point M(x,y).
5. Cosine angle α is the ratio of the abscissa x of the point M(x,y) to the radius r:
cosα=x/r
6. Tangent angle α is the ratio of the ordinate y of a point M(x,y) to its abscissa x:
tanα=y/x,x≠0
7. Cotangent angle α is the ratio of the abscissa x of a point M(x,y) to its ordinate y:
cotα=x/y,y≠0
8. Secant angle α is the ratio of the radius r to the abscissa x of the point M(x,y):
secα=r/x=1/x,x≠0
9. Cosecant angle α is the ratio of the radius r to the ordinate y of the point M(x,y):
cscα=r/y=1/y,y≠0
10. In the unit circle, the projections x, y, the points M(x,y) and the radius r form a right triangle, in which x,y are the legs, and r is the hypotenuse. Therefore, the above definitions of trigonometric functions in the appendix to right triangle are formulated as follows:
Sinus angle α is the ratio of the opposite side to the hypotenuse.
Cosine angle α is the ratio of the adjacent leg to the hypotenuse.
Tangent angle α is called the opposite leg to the adjacent one.
Cotangent angle α is called the adjacent side to the opposite side.
Secant angle α is the ratio of the hypotenuse to the adjacent leg.
Cosecant angle α is the ratio of the hypotenuse to the opposite leg.
11. Graph of the sine function
y=sinx, domain of definition: x∈R, range of values: −1≤sinx≤1
12. Graph of the cosine function
y=cosx, domain: x∈R, range: −1≤cosx≤1
13. Graph of the tangent function 14. Graph of the cotangent function 15. Graph of the secant function During the intermission, there was a smell of cold in Helen's box, the door opened and, bending down and trying not to catch anyone, Anatole entered. Only after arriving home, Natasha could clearly think through everything that had happened to her, and suddenly remembering Prince Andrei, she was horrified, and in front of everyone at tea, which everyone sat down to after the theater, she gasped loudly and ran out of the room, flushed. - "My God! I'm dead! she said to herself. How could I let this happen?” she thought. She sat for a long time, covering her flushed face with her hands, trying to give herself a clear account of what had happened to her, and could neither understand what had happened to her, nor what she felt. Everything seemed dark, unclear and scary to her. There, in this huge, illuminated hall, where Duport jumped on the wet boards to the music with bare legs in a jacket with sequins, and girls, and old men, and Helen, naked with a calm and proud smile, shouted “bravo” in delight - there, under the shadow of this Helen , there it was all clear and simple; but now alone, with herself, it was incomprehensible. - “What is this? What was this fear that I felt for him? What is this remorse that I feel now? she thought. Anatol Kuragin lived in Moscow because his father sent him away from St. Petersburg, where he lived more than twenty thousand a year in money and the same amount in debts that creditors demanded from his father. The next day after the theater, the Rostovs did not go anywhere and no one came to them. Marya Dmitrievna, hiding something from Natasha, was talking with her father. Natasha guessed that they were talking about the old prince and making up something, and this bothered and offended her. She waited for Prince Andrei every minute, and twice that day she sent the janitor to Vzdvizhenka to find out if he had arrived. He didn't come. It was now harder for her than the first days of her arrival. Her impatience and sadness about him were joined by an unpleasant memory of her meeting with Princess Marya and the old prince, and fear and anxiety, for which she did not know the reason. It seemed to her that either he would never come, or that something would happen to her before he arrived. She could not, as before, calmly and continuously, alone with herself, think about him. As soon as she began to think about him, the memory of him was joined by the memory of the old prince, of Princess Marya and of the last performance, and of Kuragin. She again wondered if she was guilty, if her loyalty to Prince Andrei had already been violated, and again she found herself remembering in the smallest detail every word, every gesture, every shade of play of expression on the face of this man, who knew how to arouse in her something incomprehensible to her. and a terrible feeling. To the eyes of her family, Natasha seemed more lively than usual, but she was far from being as calm and happy as she had been before. Count Ilya Andreich took his girls to Countess Bezukhova. There were quite a lot of people at the evening. But the whole society was almost unfamiliar to Natasha. Count Ilya Andreich noted with displeasure that this entire society consisted mainly of men and women, known for their freedom of treatment. M lle Georges, surrounded by young people, stood in the corner of the living room. There were several Frenchmen and among them Metivier, who, from the time of Helene's arrival, had been her housemate. Count Ilya Andreich decided not to play cards, not to leave his daughters, and to leave as soon as the Georges performance was over. The morning came with its worries and bustle. Everyone stood up, moved around, started talking, the milliners came again, Marya Dmitrievna came out again and called for tea. Natasha, with her eyes wide open, as if she wanted to intercept everyone looking at her, looked around restlessly at everyone and tried to seem the same as she had always been. The question, as they say, is interesting... It is possible, it is possible to pass with a 4! And at the same time not to burst... The main condition is to exercise regularly. Here is the basic preparation for the Unified State Exam in mathematics. With all the secrets and mysteries of the Unified State Exam, which you will not read about in textbooks... Study this section, solve more tasks from various sources - and everything will work out! It is assumed that the basic section "A C is enough for you!" it doesn't cause you any problems. But if suddenly... Follow the links, don’t be lazy! And we will start with a great and terrible topic. Attention! This topic causes a lot of problems for students. It is considered one of the most severe. What are sine and cosine? What are tangent and cotangent? What is a number circle? As soon as you ask these harmless questions, the person turns pale and tries to divert the conversation... But in vain. These are simple concepts. And this topic is no more difficult than others. You just need to clearly understand the answers to these very questions from the very beginning. This is very important. If you understand, you will like trigonometry. So, Let's start with ancient times. Don’t worry, we’ll go through all 20 centuries of trigonometry in about 15 minutes. And, without noticing it, we’ll repeat a piece of geometry from 8th grade. Let's draw a right triangle with sides a, b, c and angle X. Here it is. Let me remind you that the sides that form a right angle are called legs. a and c– legs. There are two of them. The remaining side is called the hypotenuse. With– hypotenuse. Triangle and triangle, just think! What to do with it? But the ancient people knew what to do! Let's repeat their actions. Let's measure the side V. In the picture, the cells are specially drawn, as happens in Unified State Examination tasks. Side V equal to four cells. OK. Let's measure the side A. Three cells. Now let's divide the length of the side A per side length V. Or, as they also say, let’s take the attitude A To V. a/v= 3/4. On the contrary, you can divide V on A. We get 4/3. Can V divide by With. Hypotenuse With It’s impossible to count by cells, but it is equal to 5. We get high quality= 4/5. In short, you can divide the lengths of the sides by each other and get some numbers. So what? What is the point of this interesting activity? None yet. A pointless exercise, to put it bluntly.) Now let's do this. Let's enlarge the triangle. Let's extend the sides in and with, but so that the triangle remains rectangular. Corner X, of course, does not change. To see this, hover your mouse over the picture, or touch it (if you have a tablet). Parties a, b and c will turn into m, n, k, and, of course, the lengths of the sides will change. But their relationship is not! Attitude a/v was: a/v= 3/4, became m/n= 6/8 = 3/4. The relationships of other relevant parties are also won't change
. You can change the lengths of the sides in a right triangle as you like, increase, decrease, without changing the angle x – the relationship between the relevant parties will not change
. You can check it, or you can take the ancient people’s word for it. But this is already very important! The ratios of the sides in a right triangle do not depend in any way on the lengths of the sides (at the same angle). This is so important that the relationship between the parties has earned its own special name. Your names, so to speak.) Meet me. What is the sine of angle x
? This is the ratio of the opposite side to the hypotenuse: sinx = a/c What is the cosine of the angle x
? This is the ratio of the adjacent leg to the hypotenuse: Withosx=
high quality What is tangent x
? This is the ratio of the opposite side to the adjacent side: tgx =a/v What is the cotangent of angle x
? This is the ratio of the adjacent side to the opposite: ctgx = v/a It's very simple. Sine, cosine, tangent and cotangent are some numbers. Dimensionless. Just numbers. Each angle has its own. Why am I repeating everything so boringly? Then what is this need to remember. It's important to remember. Memorization can be made easier. Is the phrase “Let’s start from afar…” familiar? So start from afar. Sinus angle is a ratio distant from the leg angle to the hypotenuse. Cosine– the ratio of the neighbor to the hypotenuse. Tangent angle is a ratio distant from the leg angle to the near one. Cotangent- vice versa. It's easier, right? Well, if you remember that in tangent and cotangent there are only legs, and in sine and cosine the hypotenuse appears, then everything will become quite simple. This whole glorious family - sine, cosine, tangent and cotangent is also called trigonometric functions. Now a question for consideration. Why do we say sine, cosine, tangent and cotangent corner? We are talking about the relationship between the parties, like... What does it have to do with it? corner? Let's look at the second picture. Exactly the same as the first one. Hover your mouse over the picture. I changed the angle X. Increased it from x to x. All relationships have changed! Attitude a/v was 3/4, and the corresponding ratio t/v became 6/4. And all other relationships became different! Therefore, the ratios of the sides do not depend in any way on their lengths (at one angle x), but depend sharply on this very angle! And only from him. Therefore, the terms sine, cosine, tangent and cotangent refer to corner. The angle here is the main one. It must be clearly understood that the angle is inextricably linked with its trigonometric functions. Each angle has its own sine and cosine. And almost everyone has their own tangent and cotangent. This is important. It is believed that if we are given an angle, then its sine, cosine, tangent and cotangent we know
! And vice versa. Given a sine, or any other trigonometric function, it means we know the angle. There are special tables where for each angle its trigonometric functions are described. They are called Bradis tables. They were compiled a very long time ago. When there were no calculators or computers yet... Of course, it is impossible to remember the trigonometric functions of all angles. You are required to know them only for a few angles, more on this later. But the spell I know an angle, which means I know its trigonometric functions” - always works! So we repeated a piece of geometry from 8th grade. Do we need it for the Unified State Exam? Necessary. Here is a typical problem from the Unified State Exam. To solve this problem, 8th grade is enough. Given picture: All. There is no more data. We need to find the length of the side of the aircraft. The cells do not help much, the triangle is somehow incorrectly positioned.... On purpose, I guess... From the information there is the length of the hypotenuse. 8 cells. For some reason, the angle was given. This is where you need to immediately remember about trigonometry. There is an angle, which means we know all its trigonometric functions. Which of the four functions should we use? Let's see, what do we know? We know the hypotenuse and the angle, but we need to find adjacent catheter to this corner! It’s clear, the cosine needs to be put into action! Here we go. We simply write, by the definition of cosine (the ratio adjacent leg to hypotenuse): cosC = BC/8 Angle C is 60 degrees, its cosine is 1/2. You need to know this, without any tables! So: 1/2 = BC/8 Elementary linear equation. Unknown – Sun. Those who have forgotten how to solve equations, take a look at the link, the rest solve: BC = 4 When ancient people realized that each angle has its own set of trigonometric functions, they had a reasonable question. Are sine, cosine, tangent and cotangent somehow related to each other? So that knowing one angle function, you can find the others? Without calculating the angle itself? They were so restless...) Of course, the sine, cosine, tangent and cotangent of the same angle are related to each other. Any connection between expressions is given in mathematics by formulas. In trigonometry there are a colossal number of formulas. But here we will look at the most basic ones. These formulas are called: basic trigonometric identities. Here they are: You need to know these formulas thoroughly. Without them, there is generally nothing to do in trigonometry. Three more auxiliary identities follow from these basic identities: I warn you right away that the last three formulas quickly fall out of your memory. For some reason.) You can, of course, derive these formulas from the first three. But, in difficult times... You understand.) In standard problems, like the ones below, there is a way to avoid these forgettable formulas. AND dramatically reduce errors due to forgetfulness, and in calculations too. This practice is in Section 555, lesson "Relationships between trigonometric functions of the same angle." In what tasks and how are the basic trigonometric identities used? The most popular task is to find some angle function if another is given. In the Unified State Examination such a task is present from year to year.) For example: Find the value of sinx if x is an acute angle and cosx=0.8. The task is almost elementary. We are looking for a formula that contains sine and cosine. Here is the formula: sin 2 x + cos 2 x = 1 We substitute here a known value, namely, 0.8 instead of the cosine: sin 2 x + 0.8 2 = 1 Well, we count as usual: sin 2 x + 0.64 = 1 sin 2 x = 1 - 0.64 That's practically all. We have calculated the square of the sine, all that remains is to extract the square root and the answer is ready! The root of 0.36 is 0.6. The task is almost elementary. But the word “almost” is there for a reason... The fact is that the answer sinx= - 0.6 is also suitable... (-0.6) 2 will also be 0.36. There are two different answers. And you need one. The second one is wrong. How to be!? Yes, as usual.) Read the assignment carefully. For some reason it says:... if x is an acute angle... And in tasks, every word has a meaning, yes... This phrase is additional information for the solution. An acute angle is an angle less than 90°. And at such corners All trigonometric functions - sine, cosine, and tangent with cotangent - positive. Those. We simply discard the negative answer here. We have the right. Actually, eighth graders don’t need such subtleties. They only work with right triangles, where the corners can only be acute. And they don’t know, happy ones, that there are both negative angles and angles of 1000°... And all these terrible angles have their own trigonometric functions, both plus and minus... But for high school students, without taking into account the sign - no way. Much knowledge multiplies sorrows, yes...) And for the correct solution, additional information is necessarily present in the task (if it is necessary). For example, it can be given by the following entry: Or some other way. You will see in the examples below.) To solve such examples you need to know Which quarter does the given angle x fall into and what sign does the desired trigonometric function have in this quarter? These basics of trigonometry are discussed in the lessons on what a trigonometric circle is, the measurement of angles on this circle, the radian measure of an angle. Sometimes you need to know the table of sines, cosines of tangents and cotangents. So, let's note the most important thing: Practical tips: 1. Remember the definitions of sine, cosine, tangent and cotangent. It will be very useful. 2. We clearly understand: sine, cosine, tangent and cotangent are tightly connected with angles. We know one thing, which means we know another. 3. We clearly understand: sine, cosine, tangent and cotangent of one angle are related to each other by basic trigonometric identities. We know one function, which means we can (if we have the necessary additional information) calculate all the others. Now let's decide, as usual. First, tasks in the scope of 8th grade. But high school students can do it too...) 1. Calculate the value of tgA if ctgA = 0.4. 2. β is an angle in a right triangle. Find the value of tanβ if sinβ = 12/13. 3. Determine the sine of the acute angle x if tgх = 4/3. 4. Find the meaning of the expression: 6sin 2 5° - 3 + 6cos 2 5° 5. Find the meaning of the expression: (1-cosx)(1+cosx), if sinx = 0.3 Answers (separated by semicolons, in disarray): 0,09; 3; 0,8; 2,4; 2,5 Did it work? Great! Eighth graders can already go get their A's.) Didn't everything work out? Tasks 2 and 3 are somehow not very good...? No problem! There is one beautiful technique for such tasks. Everything can be solved practically without formulas at all! And, therefore, without errors. This technique is described in the lesson: “Relationships between trigonometric functions of one angle” in Section 555. All other tasks are also dealt with there. These were problems like the Unified State Exam, but in a stripped-down version. Unified State Exam - light). And now almost the same tasks, but in a full-fledged format. For knowledge-burdened high school students.) 6. Find the value of tanβ if sinβ = 12/13, and 7. Determine sinх if tgх = 4/3, and x belongs to the interval (- 540°; - 450°). 8. Find the value of the expression sinβ cosβ if ctgβ = 1. Answers (in disarray): 0,8; 0,5; -2,4. Here in problem 6 the angle is not specified very clearly... But in problem 8 it is not specified at all! This is on purpose). Additional information is taken not only from the task, but also from the head.) But if you decide, one correct task is guaranteed! What if you haven't decided? Hmm... Well, Section 555 will help here. There the solutions to all these tasks are described in detail, it is difficult not to understand. This lesson provides a very limited understanding of trigonometric functions. Within 8th grade. And the elders still have questions... For example, if the angle X(look at the second picture on this page) - make it stupid!? The triangle will completely fall apart! So what should we do? There will be no leg, no hypotenuse... The sine has disappeared... If ancient people had not found a way out of this situation, we would not have cell phones, TV, or electricity now. Yes, yes! The theoretical basis for all these things without trigonometric functions is zero without a stick. But the ancient people did not disappoint. How they got out is in the next lesson. 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.
y=tanx, range of definition: x∈R,x≠(2k+1)π/2, range of values: −∞ 
y=cotx, domain: x∈R,x≠kπ, range: −∞ 
y=secx, domain of definition: x∈R,x≠(2k+1)π/2, range of values: secx∈(−∞,−1]∪∪ – said Helen, turning to her.
“Oh, oui, [Oh, yes,”] Natasha answered.
“Let me introduce you to my brother,” Helen said, nervously flicking her eyes from Natasha to Anatole. Natasha turned her pretty head over her bare shoulder to the handsome man and smiled. Anatole, who was as good-looking up close as he was from afar, sat down next to her and said that he had long wanted to have this pleasure, ever since the Naryshkin Ball, at which he had the pleasure, which he had not forgotten, of seeing her. Kuragin was much smarter and simpler with women than in male society. He spoke boldly and simply, and Natasha was strangely and pleasantly struck by the fact that not only was there nothing so terrible about this man about whom they talked so much, but that on the contrary, he had the most naive, cheerful and good-natured smile.
Kuragin asked about the impression of the performance and told her about how Semenova fell while playing in the last performance.
“You know, Countess,” he said, suddenly addressing her as if he were an old acquaintance, “we are having a carousel in costumes; you should take part in it: it will be a lot of fun. Everyone gathers at the Karagins'. Please come, right? - he said.
As he said this, he did not take his smiling eyes off Natasha’s face, neck, and bare arms. Natasha undoubtedly knew that he admired her. She was pleased with this, but for some reason his presence made her feel cramped and heavy. When she was not looking at him, she felt that he was looking at her shoulders, and she involuntarily intercepted his gaze so that he would look better at her eyes. But, looking into his eyes, she felt with fear that between him and her there was absolutely no barrier of modesty that she had always felt between herself and other men. She, without knowing how, after five minutes felt terribly close to this man. When she turned away, she was afraid that he would take her bare hand from behind and kiss her neck. They talked about the simplest things and she felt that they were close, like she had never been with a man. Natasha looked back at Helen and her father, as if asking them what this meant; but Helen was busy talking with some general and did not respond to her glance, and her father’s gaze did not tell her anything other than what he always said: “It’s fun, well, I’m glad.”
In one of the moments of awkward silence, during which Anatole calmly and persistently looked at her with his bulging eyes, Natasha, in order to break this silence, asked him how he liked Moscow. Natasha asked and blushed. It constantly seemed to her that she was doing something indecent when talking to him. Anatole smiled, as if encouraging her.
– At first I didn’t like it much, because what makes a city pleasant, ce sont les jolies femmes, [pretty women,] isn’t it? Well, now I really like it,” he said, looking at her significantly. – Will you go to the carousel, Countess? “Go,” he said, and, stretching out his hand to her bouquet and lowering his voice, he said: “Vous serez la plus jolie.” Venez, chere comtesse, et comme gage donnez moi cette fleur. [You will be the prettiest. Go, dear Countess, and give me this flower as a pledge.]
Natasha did not understand what he said, just like he himself, but she felt that there was indecent intent in his incomprehensible words. She didn't know what to say and turned away as if she hadn't heard what he said. But as soon as she turned away, she thought that he was there behind her, so close to her.
“What is he now? Is he confused? Angry? Should I fix this? she asked herself. She couldn't help but look back. She looked straight into his eyes, and his closeness and confidence, and the good-natured tenderness of his smile defeated her. She smiled just like him, looking straight into his eyes. And again she felt with horror that there was no barrier between him and her.
The curtain rose again. Anatole left the box, calm and cheerful. Natasha returned to her father’s box, completely subjugated to the world in which she found herself. Everything that happened in front of her already seemed completely natural to her; but for that all her previous thoughts about the groom, about Princess Marya, about village life never once entered her head, as if all that was a long, long time ago.
In the fourth act there was some kind of devil who sang, waving his hand until the boards were pulled out under him and he sat down there. Natasha saw only this from the fourth act: something worried and tormented her, and the cause of this excitement was Kuragin, whom she involuntarily followed with her eyes. When they left the theater, Anatole approached them, called their carriage and picked them up. As he sat Natasha down, he shook her hand above the elbow. Natasha, excited and red, looked back at him. He looked at her, his eyes sparkling and smiling tenderly.
Natasha would be able to tell the old countess alone in bed at night everything that she thought. Sonya, she knew, with her stern and integral gaze, either would not have understood anything, or would have been horrified by her confession. Natasha, alone with herself, tried to resolve what was tormenting her.
“Did I die for the love of Prince Andrei or not? she asked herself and with a reassuring smile answered herself: What kind of fool am I that I ask this? What happened to me? Nothing. I didn't do anything, I didn't do anything to cause this. No one will know, and I will never see him again, she told herself. It became clear that nothing had happened, that there was nothing to repent of, that Prince Andrei could love me just like that. But what kind? Oh God, my God! Why isn’t he here?” Natasha calmed down for a moment, but then again some instinct told her that although all this was true and although nothing had happened, instinct told her that all the former purity of her love for Prince Andrei had perished. And again in her imagination she repeated her entire conversation with Kuragin and imagined the face, gestures and gentle smile of this handsome and brave man, while he shook her hand.
The father announced to his son that he was paying half of his debts for the last time; but only so that he would go to Moscow to the post of adjutant to the commander-in-chief, which he procured for him, and would finally try to make a good match there. He pointed him to Princess Marya and Julie Karagina.
Anatole agreed and went to Moscow, where he stayed with Pierre. Pierre accepted Anatole reluctantly at first, but then got used to him, sometimes went with him on his carousings and, under the pretext of a loan, gave him money.
Anatole, as Shinshin rightly said about him, since he arrived in Moscow, drove all the Moscow ladies crazy, especially because he neglected them and obviously preferred gypsies and French actresses to them, with the head of which, Mademoiselle Georges, as they said, he was in intimate relations. He did not miss a single revelry with Danilov and other merry fellows of Moscow, drank all night long, outdrinking everyone, and attended all the evenings and balls of high society. They talked about several of his intrigues with Moscow ladies, and at balls he courted some. But he did not get close to girls, especially rich brides, who for the most part were all bad, especially since Anatole, which no one knew except his closest friends, had been married two years ago. Two years ago, while his regiment was stationed in Poland, a poor Polish landowner forced Anatole to marry his daughter.
Anatole very soon abandoned his wife and, for the money that he agreed to send to his father-in-law, he negotiated for himself the right to be considered a single man.
Anatole was always pleased with his position, himself and others. He was instinctively convinced with his whole being that he could not live differently than the way he lived, and that he had never done anything bad in his life. He was unable to think about how his actions might affect others, nor what might come of such or such an action. He was convinced that just as a duck was created in such a way that it should always live in water, so he was created by God in such a way that he should live with an income of thirty thousand and always occupy the highest position in society. He believed in this so firmly that, looking at him, others were convinced of this and did not deny him either the highest position in the world or the money, which he obviously borrowed without return from those he met and those who met him.
He was not a gambler, at least he never wanted to win. He wasn't vain. He didn't care at all what anyone thought about him. Still less could he be guilty of ambition. He teased his father several times, ruining his career, and laughed at all the honors. He was not stingy and did not refuse anyone who asked him. The only thing he loved was fun and women, and since, according to his concepts, there was nothing ignoble in these tastes, and he could not think about what came out of satisfying his tastes for other people, in his soul he believed considered himself an impeccable person, sincerely despised scoundrels and bad people and carried his head high with a calm conscience.
The revelers, these male Magdalenes, have a secret sense of consciousness of innocence, the same as the female Magdalenes, based on the same hope of forgiveness. “Everything will be forgiven to her, because she loved a lot, and everything will be forgiven to him, because he had a lot of fun.”
Dolokhov, who this year appeared again in Moscow after his exile and Persian adventures, and led a luxurious gambling and carousing life, became close to his old St. Petersburg comrade Kuragin and used him for his own purposes.
Anatole sincerely loved Dolokhov for his intelligence and daring. Dolokhov, who needed the name, nobility, connections of Anatoly Kuragin to lure rich young people into his gambling society, without letting him feel this, used and amused himself with Kuragin. In addition to the calculation for which he needed Anatole, the very process of controlling someone else’s will was a pleasure, a habit and a need for Dolokhov.
Natasha made a strong impression on Kuragin. At dinner after the theater, with the techniques of a connoisseur, he examined in front of Dolokhov the dignity of her arms, shoulders, legs and hair, and announced his decision to drag himself after her. What could come out of this courtship - Anatole could not think about it and know, just as he never knew what would come out of each of his actions.
“It’s good, brother, but not about us,” Dolokhov told him.
“I’ll tell my sister to call her for dinner,” said Anatole. - A?
- You better wait until she gets married...
“You know,” said Anatole, “j”adore les petites filles: [I adore girls:] - now he’ll get lost.
“You’ve already fallen for a petite fille [girl],” said Dolokhov, who knew about Anatole’s marriage. - Look!
- Well, you can’t do it twice! A? – Anatole said, laughing good-naturedly.
On Sunday morning, Marya Dmitrievna invited her guests to mass at her parish of the Assumption on Mogiltsy.
“I don’t like these fashionable churches,” she said, apparently proud of her free-thinking. - There is only one God everywhere. Our priest is wonderful, he serves decently, it’s so noble, and so is the deacon. Does this make it so sacred that people sing concerts in the choir? I don’t like it, it’s just self-indulgence!
Marya Dmitrievna loved Sundays and knew how to celebrate them. Her house was all washed and cleaned on Saturday; people and she were not working, everyone was dressed up for the holidays, and everyone was attending mass. Food was added to the master's dinner, and people were given vodka and roast goose or pig. But nowhere in the whole house was the holiday more noticeable than on Marya Dmitrievna’s broad, stern face, which on that day assumed an unchanging expression of solemnity.
When they drank coffee after mass, in the living room with the covers removed, Marya Dmitrievna was informed that the carriage was ready, and she, with a stern look, dressed in the ceremonial shawl in which she made visits, stood up and announced that she was going to Prince Nikolai Andreevich Bolkonsky to explain to him about Natasha.
After Marya Dmitrievna left, a milliner from Madame Chalmet came to the Rostovs, and Natasha, having closed the door in the room next to the living room, very pleased with the entertainment, began trying on new dresses. While she was putting on a sour cream bodice still without sleeves and bending her head, looking in the mirror at how the back was sitting, she heard in the living room the animated sounds of her father’s voice and another, female voice, which made her blush. It was Helen's voice. Before Natasha had time to take off the bodice she was trying on, the door opened and Countess Bezukhaya entered the room, beaming with a good-natured and affectionate smile, in a dark purple, high-necked velvet dress.
- Ah, ma delicieuse! [Oh, my charming one!] - she said to the blushing Natasha. - Charmante! [Charming!] No, this is not like anything, my dear Count,” she said to Ilya Andreich, who came in after her. – How to live in Moscow and not travel anywhere? No, I won't leave you alone! This evening M lle Georges is reciting and some people will gather; and if you don’t bring your beauties, who are better than m lle Georges, then I don’t want to know you. My husband is gone, he left for Tver, otherwise I would have sent him for you. Be sure to come, definitely, at nine o'clock. “She nodded her head to the milliner she knew, who sat down respectfully to her, and sat down on a chair near the mirror, picturesquely spreading out the folds of her velvet dress. She did not stop chatting good-naturedly and cheerfully, constantly admiring Natasha’s beauty. She examined her dresses and praised them, and boasted about her new dress en gaz metallique, [made of metal-colored gas], which she received from Paris and advised Natasha to do the same.
“However, everything suits you, my lovely,” she said.
The smile of pleasure never left Natasha's face. She felt happy and blossoming under the praises of this dear Countess Bezukhova, who had previously seemed to her such an unapproachable and important lady, and who was now so kind to her. Natasha felt cheerful and felt almost in love with this so beautiful and such a good-natured woman. Helen, for her part, sincerely admired Natasha and wanted to amuse her. Anatole asked her to set him up with Natasha, and for this she came to the Rostovs. The thought of setting up her brother with Natasha amused her.
Despite the fact that she had previously been annoyed with Natasha for having taken Boris away from her in St. Petersburg, she now did not even think about it, and with all her soul, in her own way, wished Natasha well. Leaving the Rostovs, she withdrew her protegee aside.
- Yesterday my brother dined with me - we were dying of laughter - he didn’t eat anything and sighed for you, my precious. Il est fou, mais fou amoureux de vous, ma chere. [He goes crazy, but he goes crazy with love for you, my dear.]
Natasha blushed crimson hearing these words.
- How she blushes, how she blushes, ma delicieuse! [my precious!] - said Helen. - Definitely come. Si vous aimez quelqu"un, ma delicieuse, ce n"est pas une raison pour se cloitrer. Si meme vous etes promise, je suis sure que votre promis aurait desire que vous alliez dans le monde en son absence plutot que de deperir d"ennui. [Just because you love someone, my lovely, you should not live like a nun. Even if you are a bride, I am sure that your groom would prefer that you go out into society in his absence than die of boredom.]
“So she knows that I’m a bride, so she and her husband, with Pierre, with this fair Pierre,” thought Natasha, talked and laughed about it. So it’s nothing.” And again, under the influence of Helen, what had previously seemed terrible seemed simple and natural. “And she is such a grande dame, [important lady,] so sweet and obviously loves me with all her heart,” Natasha thought. And why not have fun? thought Natasha, looking at Helen with surprised, wide-open eyes.
Marya Dmitrievna returned to dinner, silent and serious, obviously defeated by the old prince. She was still too excited from the collision to be able to calmly tell the story. To the count's question, she answered that everything was fine and that she would tell him tomorrow. Having learned about Countess Bezukhova’s visit and invitation to the evening, Marya Dmitrievna said:
“I don’t like hanging out with Bezukhova and wouldn’t recommend it; Well, if you promised, go, you’ll be distracted,” she added, turning to Natasha.
Anatole was obviously at the door waiting for the Rostovs to enter. He immediately greeted the count, approached Natasha and followed her. As soon as Natasha saw him, just like in the theater, a feeling of vain pleasure that he liked her and fear from the absence of moral barriers between her and him overwhelmed her. Helen joyfully received Natasha and loudly admired her beauty and dress. Soon after their arrival, M lle Georges left the room to get dressed. In the living room they began to arrange chairs and sit down. Anatole pulled out a chair for Natasha and wanted to sit next to her, but the count, who had not taken his eyes off Natasha, sat down next to her. Anatole sat down behind.
M lle Georges, with bare, dimpled, thick arms, wearing a red shawl worn over one shoulder, walked out into the empty space left for her between the chairs and stopped in an unnatural pose. An enthusiastic whisper was heard. M lle Georges looked sternly and gloomily at the audience and began to speak some poems in French, which dealt with her criminal love for her son. In some places she raised her voice, in others she whispered, raising her head solemnly, in others she stopped and wheezed, rolling her eyes.
- Adorable, divin, delicieux! [Delightful, divine, wonderful!] - was heard from all sides. Natasha looked at fat Georges, but did not hear anything, did not see and did not understand anything of what was happening in front of her; she only felt again completely irrevocably in that strange, crazy world, so far from the previous one, in that world in which it was impossible to know what was good, what was bad, what was reasonable and what was crazy. Anatole was sitting behind her, and she, feeling his closeness, fearfully waited for something.
After the first monologue, the whole company stood up and surrounded m lle Georges, expressing their delight to her.
- How good she is! - Natasha said to her father, who, along with others, stood up and moved through the crowd towards the actress.
“I don’t find it, looking at you,” said Anatole, following Natasha. He said this at a time when she alone could hear him. “You are lovely... from the moment I saw you, I haven’t stopped....”
“Come on, let’s go, Natasha,” said the count, returning for his daughter. - How good!
Natasha, without saying anything, walked up to her father and looked at him with questioning, surprised eyes.
After several receptions of recitation, M lle Georges left and Countess Bezukhaya asked for company in the hall.
The Count wanted to leave, but Helen begged him not to spoil her impromptu ball. The Rostovs remained. Anatole invited Natasha to a waltz and during the waltz he, shaking her waist and hand, told her that she was ravissante [charming] and that he loved her. During the eco-session, which she again danced with Kuragin, when they were left alone, Anatole did not say anything to her and only looked at her. Natasha was in doubt whether she had seen what he said to her during the waltz in a dream. At the end of the first figure he shook her hand again. Natasha raised her frightened eyes to him, but there was such a self-confidently tender expression in his affectionate gaze and smile that she could not look at him and say what she had to say to him. She lowered her eyes.
“Don’t tell me such things, I’m engaged and love someone else,” she said quickly... “She looked at him. Anatole was not embarrassed or upset by what she said.
- Don't tell me about this. What do I care? - he said. “I’m saying that I’m madly, madly in love with you.” Is it my fault that you are amazing? Let's start.
Natasha, animated and anxious, looked around her with wide, frightened eyes and seemed more cheerful than usual. She remembered almost nothing of what happened that evening. They danced the Ecossaise and Gros Vater, her father invited her to leave, she asked to stay. Wherever she was, no matter who she spoke to, she felt his gaze on her. Then she remembered that she asked her father for permission to go to the dressing room to straighten her dress, that Helen followed her, told her laughing about her brother’s love, and that in the small sofa room she again met Anatole, that Helen disappeared somewhere, they were left alone and Anatole, Taking her hand, he said in a gentle voice:
- I can’t go to you, but will I really never see you? I love you madly. Really never?...” and he, blocking her path, brought his face closer to hers.
His brilliant, large, masculine eyes were so close to hers that she saw nothing but these eyes.
- Natalie?! – his voice whispered questioningly, and someone painfully squeezed her hands.
- Natalie?!
“I don’t understand anything, I have nothing to say,” said her look.
Hot lips pressed against hers and at that very moment she felt free again, and the noise of Helen’s steps and dress was heard in the room. Natasha looked back at Helen, then, red and trembling, looked at him with frightened questioning and went to the door.
“Un mot, un seul, au nom de Dieu, [One word, only one, for God’s sake,” said Anatole.
She stopped. She really needed him to say this word, which would explain to her what had happened and to which she would answer him.
“Nathalie, un mot, un seul,” he kept repeating, apparently not knowing what to say, and he repeated it until Helen approached them.
Helen and Natasha went out into the living room again. Without staying for dinner, the Rostovs left.
Returning home, Natasha did not sleep all night: she was tormented by the insoluble question of who she loved, Anatole or Prince Andrei. She loved Prince Andrei - she remembered clearly how much she loved him. But she loved Anatole too, that was certain. “Otherwise, how could all this have happened?” she thought. “If after that, when I said goodbye to him, I could answer his smile with a smile, if I could allow this to happen, then it means that I fell in love with him from the first minute. This means that he is kind, noble and beautiful, and it was impossible not to love him. What should I do when I love him and love another? she told herself, not finding answers to these terrible questions.
After breakfast, Marya Dmitrievna (this was her best time), sitting down in her chair, called Natasha and the old count to her.
“Well, my friends, now I’ve thought about the whole matter and here’s my advice to you,” she began. – Yesterday, as you know, I was with Prince Nikolai; Well, I talked to him... He decided to shout. You can't shout me down! I sang everything to him!
- What is he? - asked the count.
- What is he? madman... doesn’t want to hear; Well, what can I say, and so we tormented the poor girl,” said Marya Dmitrievna. “And my advice to you is to finish things off and go home to Otradnoye... and wait there...
- Oh, no! – Natasha screamed.
“No, let’s go,” said Marya Dmitrievna. - And wait there. “If the groom comes here now, there won’t be a quarrel, but here he will talk everything over alone with the old man and then come to you.”
Ilya Andreich approved this proposal, immediately understanding its reasonableness. If the old man relents, then all the better it will be to come to him in Moscow or Bald Mountains, later; if not, then it will be possible to get married against his will only in Otradnoye.
“And the true truth,” he said. “I regret that I went to see him and took her,” said the old count.
- No, why regret it? Having been here, it was impossible not to pay respects. Well, if he doesn’t want to, that’s his business,” said Marya Dmitrievna, looking for something in her reticule. - Yes, and the dowry is ready, what else do you have to wait for? and what’s not ready, I’ll send it to you. Although I feel sorry for you, it’s better to go with God. “Having found what she was looking for in the reticule, she handed it to Natasha. It was a letter from Princess Marya. - He writes to you. How she suffers, poor thing! She is afraid that you will think that she does not love you.
“Yes, she doesn’t love me,” said Natasha.
“Nonsense, don’t talk,” shouted Marya Dmitrievna.
- I won’t trust anyone; “I know that he doesn’t love me,” Natasha said boldly, taking the letter, and her face expressed dry and angry determination, which made Marya Dmitrievna look at her more closely and frown.
“Don’t answer like that, mother,” she said. – What I say is true. Write your answer.
Natasha did not answer and went to her room to read Princess Marya’s letter.
Princess Marya wrote that she was in despair over the misunderstanding that had occurred between them. Whatever her father’s feelings, Princess Marya wrote, she asked Natasha to believe that she could not help but love her as the one chosen by her brother, for whose happiness she was ready to sacrifice everything.
“However,” she wrote, “don’t think that my father was ill-disposed towards you. He is a sick and old man who needs to be excused; but he is kind, generous and will love the one who will make his son happy.” Princess Marya further asked that Natasha set a time when she could see her again.
After reading the letter, Natasha sat down at the desk to write a response: “Chere princesse,” [Dear princess], she wrote quickly, mechanically and stopped. “What could she write next after everything that happened yesterday? Yes, yes, all this happened, and now everything is different,” she thought, sitting over the letter she had begun. “Should I refuse him? Is it really necessary? This is terrible!”... And in order not to think these terrible thoughts, she went to Sonya and together with her began to sort out the patterns.
After dinner, Natasha went to her room and again took Princess Marya’s letter. - “Is it really all over? she thought. Did all this really happen so quickly and destroy everything that was before”! She recalled with all her former strength her love for Prince Andrei and at the same time felt that she loved Kuragin. She vividly imagined herself as Prince Andrei’s wife, imagined the picture of happiness with him repeated so many times in her imagination, and at the same time, flushed with excitement, imagined all the details of her yesterday’s meeting with Anatole.
“Why couldn’t it be together? sometimes, in complete eclipse, she thought. Then only I would be completely happy, but now I have to choose and without either of both I cannot be happy. One thing, she thought, to say what was meant to Prince Andrei or to hide it is equally impossible. And nothing is spoiled with this. But is it really possible to part forever with this happiness of Prince Andrei’s love, which I lived with for so long?”
“Young lady,” the girl said in a whisper with a mysterious look, entering the room. – One person told me to tell it. The girl handed over the letter. “Only for Christ’s sake,” the girl was still saying when Natasha, without thinking, broke the seal with a mechanical movement and read Anatole’s love letter, of which she, without understanding a word, understood only one thing - that this letter was from him, from that man, whom she loves. “Yes, she loves, otherwise how could what happened happen? Could there be a love letter from him in her hand?Unified State Exam for 4? Won't you burst with happiness?
Trigonometry
There are additional
materials in Special Section 555.
For those who are very "not very..."
And for those who “very much…”)What are sine and cosine? What are tangent and cotangent?
Relationship between trigonometric functions of one angle.
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Definitions
Definitions of trigonometric functions are given using the trigonometric circle, which is understood as a circle of unit radius with a center at the origin.
Let's consider two radii of this circle: stationary (where the point is) and moving (where the point is). Let the moving radius form an angle with the fixed one.
The number equal to the ordinate of the end of a unit radius forming an angle with a fixed radius is called sine of the angle : .
The number equal to the abscissa of the end of a unit radius forming an angle with a fixed radius is called cosine of the angle : .
Thus, the point that is the end of the moving radius forming an angle has coordinates.
Tangent of the angle The ratio of the sine of this angle to its cosine is called: , .
Cotangent of the angle The ratio of the cosine of this angle to its sine is called: , .
Geometric meaning of trigonometric functions
The geometric meaning of sine and cosine on a trigonometric circle is clear from the definition: this is the abscissa and ordinate of the point of intersection of the moving radius, which makes an angle with the fixed radius, and the trigonometric circle. That is, .
Let us now consider the geometric meaning of tangent and cotangent. Triangles are similar at three angles (,), then the relation holds. On the other hand, in, therefore.
Also similar at three angles (,), then the relation holds. On the other hand, in, therefore.
Taking into account the geometric meaning of tangent and cotangent, the concept of tangent axis and cotangent axis is introduced.
Tangent axes are axes, one of which touches the trigonometric circle at a point and is directed upward, the second touches the circle at a point and is directed downward.
Cotangent axes are axes, one of which touches the trigonometric circle at a point and is directed to the right, the second touches the circle at a point and is directed to the left.
Properties of trigonometric functions
Let's look at some basic properties of trigonometric functions. Other properties will be discussed in the section on graphs of trigonometric functions.
Domain and range of values
As mentioned earlier, sine and cosine exist for any angles, i.e. the domain of definition of these functions is the set of real numbers. By definition, tangent does not exist for angles, and cotangent does not exist for angles, .
Since sine and cosine are the ordinate and abscissa of a point on a trigonometric circle, their values lie in between. The range of tangent and cotangent values is the set of real numbers (this is easy to see by looking at the axes of tangents and cotangents).
Even/odd
Let's consider the trigonometric functions of two angles (which corresponds to the moving radius) and (which corresponds to the moving radius). Because that means the point has coordinates. Therefore, i.e. sine is an odd function; , i.e. cosine - even function; , i.e. tangent is odd; , i.e. The cotangent is also odd.
Intervals of sign constancy
The signs of trigonometric functions for various coordinate quarters follow from the definition of these functions. It should be noted that since tangent and cotangent are ratios of sine and cosine, they are positive when the sine and cosine of the angle have the same sign and negative when they are different.
Periodicity
The periodicity of sine and cosine is based on the fact that angles that differ by an integer number of full revolutions correspond to the same relative position of the moving and stationary rays. Accordingly, the coordinates of the intersection point of the moving beam and the trigonometric circle will be the same for angles that differ by an integer number of full revolutions. Thus, the period of sine and cosine is and, where.
Obviously, this is also the period for tangent and cotangent. But is there a shorter period for these functions? Let us prove that the smallest period for tangent and cotangent is.
Consider two angles and. On the geometric meaning of tangent and cotangent,. The side and adjacent angles of the triangles are equal and, therefore, their sides are equal, which means and. Similarly, you can prove where. Thus, the period of tangent and cotangent is.
Trigonometric functions of fundamental angles
Trigonometry formulas
To successfully solve trigonometric problems, you must know numerous trigonometric formulas. However, there is no need to memorize all the formulas. You only need to know the most basic ones by heart, and you need to be able to derive the rest of the formulas if necessary.
Basic trigonometric identity and consequences from it
All trigonometric functions of an arbitrary angle are interconnected, i.e. Knowing one function you can always find the rest. This connection is given by the formulas discussed in this section.
Theorem 1 (Basic trigonometric identity). For anyone the identity is true
The proof consists of applying the Pythagorean theorem to a right triangle with legs and a hypotenuse.
A more general theorem is also true.
Theorem 2. In order for two numbers to be taken as the cosine and sine of the same real angle, it is necessary and sufficient that the sum of their squares be equal to one:
Let us consider the consequences of the main trigonometric identity.
Let's express sine through cosine and cosine through sine:
In this formula, the plus or minus sign in front of the root is chosen depending on the quadrant in which the angle lies.
Substituting the formulas obtained above into the formulas defining tangent and cotangent, we obtain:
Dividing the main trigonometric identity term by term by or we obtain, respectively:
These relations can be rewritten as:
The following formulas give the relationship between tangent and cotangent. Since at, and at, then the equality holds:
Reduction formulas
Using reduction formulas, you can express the values of trigonometric functions of arbitrary angles through the values of acute angle functions. All reduction formulas can be generalized using the following rule.
Any trigonometric function of an angle is equal in absolute value to the same function of the angle if the number is even, and to the co-function of the angle if the number is odd. Moreover, if the angle function is positive, when it is an acute positive angle, then the signs of both functions are the same; if it is negative, then they are different.
Sum formulas and angle difference
Theorem 3 . For any real and the following formulas are valid:
The proof of the remaining formulas is based on the reduction formulas and even/odd trigonometric functions.
Q.E.D.
Theorem 4. For any real and such that
1. , the following formulas are valid
2. , the following formulas are valid
Proof. By definition of tangent
The last transformation is obtained by dividing the numerator and denominator of this fraction by.
Similarly for the cotangent (the numerator and denominator in this case are divided by):
Q.E.D.
Attention should be paid to the fact that the right and left sides of the last equalities have different ranges of acceptable values. Therefore, using these formulas without restrictions on possible angle values may lead to incorrect results.
Double and half angle formulas
Double angle formulas allow you to express trigonometric functions of an arbitrary angle in terms of functions of an angle half the original angle. These formulas are consequences of the formulas for the sum of two angles, if we put the angles in them equal to each other.
The last formula can be transformed using the basic trigonometric identity:
Thus, for the cosine of a double angle there are three formulas:
It should be noted that this formula is valid only when
The last formula is valid for, .
Similar to double angle functions, triple angle functions can be obtained. Here these formulas are given without proof:
The half-angle formulas are corollaries of the double-angle formulas and allow us to express the trigonometric functions of a certain angle in terms of functions of an angle twice the original.