Formulas for reducing trigonometric functions
Given training manual corresponds to Chapter V “Trigonometric expressions and their transformations” from the textbook “Algebra, 9” by the same authors previous years publications
DEFINITION OF SINE, COSINE, TANGENT AND COTANGENT.
Let's mark point A on the x-axis to the right of the origin and draw a circle through it with the center at point O (Fig. 1). We will call the radius OA the initial radius.
Let's rotate the initial radius near point O by 70° counterclockwise. At the same time, it will move into the OB radius. The rotation angle is said to be 70°. If you rotate the initial radius near point O by 70° clockwise, it will become the radius OS. In this case, the rotation angle is said to be -70°. The rotation angles of 70° and -70° are shown by arrows in Figure 64.
In general, when turning counterclockwise, the angle of rotation is considered positive, and when turning clockwise, it is negative.
From the geometry course it is known that the measure of an angle in degrees is expressed as a number from 0 to 180. As for the angle of rotation, it can be expressed in degrees in any way real number from -∞ to +∞. So, if the initial radius is rotated counterclockwise by 180°, and then another 30°, then the angle of rotation will be equal to 210°. If the initial radius makes a full rotation counterclockwise, the rotation angle will be 360°; if he makes one and a half turns in the same direction, then the angle of rotation will be 540°, etc. In Figure 2, the arrows indicate the angles of rotation of 405° and -200°.
§1. Trigonometric functions of any angle
1. Definition of sine, cosine, tangent and cotangent
2. Properties of sine, cosine, tangent and cotangent
3. Radian measure of angle. Calculating Values trigonometric functions using a microcalculator
§2. Basic trigonometric formulas
4. Relationships between trigonometric functions of the same angle
5. Application of basic trigonometric formulas to transform expressions
6. Reduction formulas
§3. Addition formulas and their consequences
7. Addition formulas
8. Double angle formulas
9. Formulas for the sum and difference of trigonometric functions
Additional exercises
Answers.
Name: Trigonometry. 2003.
This book, written by a group of authors under the leadership of one of the greatest mathematicians of the 20th century, Academician I. M. Gelfand, is designed to refute the popular opinion about trigonometry as a boring and incomprehensible section school course mathematics. The reader is invited to look at a familiar subject in a new way. Presentation accompanied by a large number tasks, starts from scratch and reaches material that goes quite far beyond the scope school curriculum; trigonometric formulas are illustrated with examples from physics and geometry.
A separate chapter is devoted to typical methods for solving trigonometric problems offered in entrance exams to higher educational institutions.
The book will an indispensable assistant for high school students, teachers, parents and everyone interested in mathematics.
What is trigonometry? Almost all high school students will say boring and useless formulas. However, we want to dissuade you from this.
To take a fresh look at trigonometry, we talk about it from scratch. Therefore, it is better to read the manual from the very beginning and consecutively, although you will need some information. Most likely, you already know.
Our definitions are equivalent to those from school textbooks, but do not always coincide verbatim with them.
There is no need to strive to solve all the problems from the book (we deliberately placed them with a reserve), but it is worth solving a number of problems after each paragraph. If the tasks for a paragraph don’t work out at all, it means you haven’t learned something, and it makes sense to re-read this paragraph.
More difficult problems are marked with an asterisk, more difficult text is printed in small font. You can skip all this on your first reading.
Table of contents
1. First introduction to trigonometry 7
§ 1. How to measure steepness 7
1.1. Sine 7
1.2. Measuring angles 9
§ 2. Tangent 11
§ 3. Cosine 13
§ 4. Small angles 16
2. Initial properties of trigonometric functions 21
§ 5. Clock, or modern look for trigonometry 21
5.1. Clocks and processes 21
5.2. Speed 24
§ 6. Definition of trigonometric functions 26
6.1. Tangent axis 31
6.2. Signs of trigonometric functions 32
§ 7. The simplest formulas 34
§ 8. Periods of trigonometric functions 36
§ 9. Reduction formulas 40
§ 10. Protozoa trigonometric equations 45
§ 11. Graphs of sine and cosine 55
§ 12. Graphs of tangent and cotangent 62
§ 13. What is sin x + cos x 65?
3. Solving triangles 67
§ 14. Cosine theorem 67
§ 15. Around the area of a triangle 71
§ 16. Theorem of sines 76
4. Addition formulas and their consequences 81
§ 17. Vectors 81
17.1. Directed segments and vectors 81
17.2. Vector addition 87
17.3. Subtraction and multiplication by the number 90
17.4. About vectors in physics 94
§ 18. Dot product 95
§ 19. Trigonometric formulas addition 99
§ 20. Auxiliary angle formula, or addition of oscillations of equal frequency 105
§ 21. Double, triple and half angles 111
§ 22. Converting a product into a sum and a sum into a product 118
§ 23. Derivatives of trigonometric functions 126
5. Trigonometry for applicants 137
§ 24. How to solve trigonometric equations 137
§ 25. Selection of numbers on the trigonometric circle 151
§ 26. How to solve trigonometric inequalities 159
§ 27. Repetition problems 165
6. Complex numbers 168
§ 28. What are complex numbers 168
§ 29. Module and argument complex number 173
§ 30. Exponential function and Euler's formula 182
Answers and directions to some problems 189
Subject index 196
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This book, written by a group of authors under the leadership of one of the greatest mathematicians of the 20th century, Academician I.M. Gelfand, is designed to refute the popular opinion about trigonometry as a boring and incomprehensible section of the school mathematics course. The reader is invited to look at a familiar subject in a new way. The presentation, accompanied by a large number of tasks, starts from scratch? and reaches material that goes quite far beyond the scope of the school curriculum; trigonometric formulas are illustrated with examples from physics and geometry.
A separate chapter is devoted to typical methods for solving trigonometric problems offered in entrance exams to higher educational institutions.
The book will be an indispensable assistant for high school students, teachers, parents and anyone interested in mathematics.
1. First introduction to trigonometry
? 1. How to measure slope
1.1. Sinus
1.2. Measuring angles
? 2. Tangent
? 3. Cosine
? 4. Small angles
2. Initial properties of trigonometric functions
? 5. Clock, or a modern view of trigonometry
5.1. Clocks and processes
5.2. Speed
? 6. Definition of trigonometric functions
6.1. Tangent axis
6.2. Signs of trigonometric functions
? 7. The simplest formulas
? 8. Periods of trigonometric functions
? 9. Reduction formulas
? 10. The simplest trigonometric equations
? 11. Sine and cosine graphs
? 12. Graphs of tangent and cotangent
? 13. What is sin x + cos x equal?
3. Solving triangles
? 14. Cosine theorem
? 15. Around the area of a triangle
? 16. Theorem of sines
4. Addition formulas and their consequences
? 17. Vectors
17.1. Directed Segments and Vectors
17.2. Vector addition
17.3. Subtracting and multiplying by a number
17.4. About vectors in physics
? 18. Dot product
? 19. Trigonometric addition formulas
? 20. Auxiliary angle formula, or addition of oscillations of equal frequency
? 21. Double, triple and half angles
? 22. Converting a product into a sum and a sum into a product
? 23. Derivatives of trigonometric functions
5. Trigonometry for applicants
? 24. How to solve trigonometric equations
? 25. Selection of numbers on the trigonometric circle
? 26. How to solve trigonometric inequalities
? 27. Repetition problems
6. Complex numbers
? 28. What are complex numbers
? 29. Modulus and argument of a complex number
? 30. Exponential function and Euler's formula
Answers and directions to some problems
Subject index
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And even if there was no logic in my actions
I can't live any other way...
Hello. My name is Gabriela, for acquaintances and friends you can simply call me Gabi. I work as a designer, although this is not entirely true, I have a small company, in which I hold the honorary position of designer, director and a bunch of other part-time positions. When I get tired of all this, I call my friend a taxi driver, rent a car from him and this way I take a break from my worries. Each person has his own hobby, but I just like to drive around the city and look at the people I am driving, happy, sad, who like to chat or, on the contrary, remain silent.
And then one day, one of these days, a stunning girl with tears in her eyes got into my taxi. To my question “Where are you going?” the answer was “anywhere, just away from this place.” The client’s desire is the law, I started the car and, looking in the rearview mirror, I saw a girl run out of the restaurant where my passenger came out and start looking around, followed by a guy. He tried to tell her something but, apparently, she sent him somewhere further and, having caught a taxi, left. We had been driving for about three minutes, and I decided to try again to clarify the address. They answered me more clearly, but nevertheless gave me a very interesting address. Literally, “You take me to a place where I will always be happy.” At first I wanted to ask again, but after looking at my passenger, I changed my mind. Instead, I invited her, unexpectedly for herself, to come to me. She just nodded and continued to shed tears, looking out the window. We got to my house, I got out of the car and helped her get out. Apparently, all her strength was in tears, and she had just shed them all. Therefore, I had no choice but to pick her up and carry her into the room onto the bed, where she immediately fell asleep. I went down to the kitchen to drink coffee and think. I was very interested in this girl... Not with her story, no. I felt something familiar in her. It feels as if you have found something that you lost a long time ago, although until now you didn’t even know about it. And before that it seemed to me that I had lost the ability to feel... But am I rushing things too much, I don’t know her at all, I thought. I wonder what happened to her? She probably caught her boyfriend with a girl. Although the girl ran out first, it was her friend. Then it’s clear why she was so upset, losing both a boyfriend and a girlfriend in one day is not very fun, I thought. Having finished my coffee, I rattled the dishes, waved the rag and decided that it was time to look at my sudden, but so exciting tenant. Entering the bedroom (renovation was completed only in it and in the kitchen, since I had only recently purchased this house), I saw that she was no longer sleeping, but admiring the ceiling. And I, in turn, secretly admired her.
How are you? – I asked. She looked up at me with her blue eyes, the color of the summer sky, as if remembering who I was, but at that moment I forgot it myself, drowning in this pool. I was brought out of this stupor by her words, accompanied by the saddest smile I have ever seen.
Lousy...
Do you want anything? – I thought that saying “you” to the girl who is lying in my bed is stupid, a little indecent, and I don’t want to...
No, I’ve already abused your kindness, I’d better go home,” she answered, getting up. Hearing this phrase, I felt so upset that I surprised myself. My feelings were starting to scare me. But, despite this, I decided that I would not let her leave so easily.
Nothing. By the way, my name is Ella,” I introduced myself, but only for some reason as someone else. diminutive name. “Are you sure that you need this so much, is there someone waiting for you there?”
And I'm Dana. No, I'm not sure...
“Hurray,” I thought – maybe I’ll be able to detain her. Although I couldn’t understand what this would give me...
Then what do you have to lose? We don’t know each other at all (we’ll fix that, I thought) and you need to speak out so as not to get bogged down in your tears even more. And who, if not a stranger, is best to open up?
My new friend frowned, but then shook her head decisively, as if she had decided something for herself, and answered, “You’re right.” Only you have something to drink, otherwise it’s disgusting to even live sober.
Is wine okay? – I asked, taking a deep breath of air, because unexpectedly for myself I discovered that I had even held my breath, waiting for her answer.
Yes, quite. And, Ella, I’m sorry for being so annoying, but do you have some kind of robe, otherwise I’m already tired of this dress.
Sorry, but there is no robe, but there is a whole closet of shirts and T-shirts with trousers, choose,” I answered and waved towards my dressing room. I didn’t wait to see what she would find there and watch the process of changing clothes, although I won’t say that I didn’t want to, I went to the kitchen to get some wine, trying to put my raging feelings in order and distract myself from the pictures that my imagination helpfully gave me. I took a bottle of red, then, after thinking, white, then I thought again and took both, glasses, a corkscrew and a basket of fruit.
When I arrived, Dana was waiting for me, having changed into my blue shirt, managing to look even more incomparable in it than in the dress.
“You look great, it suits you,” I said, trying to normalize my blood pressure, although without much success. “What’s wrong with me? – I thought. “The girl feels bad, but I can’t think about anything except her body and eyes.” Like a sexual maniac, honestly. We need to pull ourselves together."
Thanks for the compliment. Is it okay that I put it on?
No, I said, whatever you want. By the way, what wine do you prefer, red or white? - I asked, trying to distract myself from thoughts about... Oh God, where do I get such thoughts in my head?
Red or not, white. “What difference does it make, open any one,” Dana answered, waving her hand and looking into my eyes. It would be better if she didn’t do this... A few minutes later I came to my senses and decided that the first word was more valuable than the second and opened the red one. As I poured the wine into glasses, I thought that I couldn’t drink a lot today, otherwise I wouldn’t be able to restrain myself and I’d do something that I’d regret later. Handing one of the glasses to Dana, I turned away, unable to watch as she, unwillingly, but nevertheless very sexually touched it with her lips. Having sipped the wine, Dana said in amazement: “Ella, isn’t this wine too expensive for a taxi driver?”
“I don’t know, I like it,” I answered, freaking out about my sexual attraction to this girl. “Maybe the whole point is that I haven’t been with anyone for a long time?” – I thought. Naive...
“I see,” Dana answered me and, sighing, continued. - Well, okay, I think you were right when you said that I need to talk out. My story, on the one hand, is terribly banal, and on the other hand, it’s not quite simple, as it seems at first glance... You see, the fact is that I love girls,” Dana began her revelations and looked into my eyes to find out how this will affect me. Ha, I won’t admit to her that I myself am guilty of this. Moreover, now I’m trying to figure out what attracted me so much to her. Therefore, I shrugged my shoulders at this phrase and pretended that my wine was very, very, very good.
Well, so,” Dana continued, seeing such a reaction and her eyes warming slightly, “I had a business meeting scheduled at the restaurant where you picked me up.” We had almost discussed all the issues when a pretty girl took a table next to us. unusual couple- he and she. Unusual in that SHE was my girlfriend... She didn’t see me, but I saw everything perfectly and even heard it. It turned out that they were celebrating something like half a year nearby. Imagine my situation, my girlfriend, with whom everything seems to be fine, is playing tricks with some guy behind my back. She told him about her love. I learned a lot of new things for myself... but when they started kissing, my nerves could not stand it. Fortunately, my clients and I had already agreed on everything, and I didn’t have to say goodbye for long. The only problem was that there was no way to go around the table where my girlfriend was sitting. You should have seen her face when she noticed me. So many feelings flashed through her eyes in a second: indifference, recognition, surprise, confusion and, finally, horror. She wanted to tell me something, but I was somehow not in the mood for it, and you know the rest. – Dana finished her story, and her eyes filled with tears again. I wanted to hug her, but instead I decided to tell her what I saw in the mirror, thinking that she should know it.
