In what quarters is the cotangent positive? Basic properties of trigonometric functions: evenness, oddness, periodicity. Signs of the values of trigonometric functions by quarters
Counting angles on a trigonometric circle.
Attention!
There are additional
material in Special Section 555.
For those who strongly "not very..."
And for those who "very much...")
It is almost the same as in the previous lesson. There are axes, a circle, an angle, everything is chin-china. Added numbers of quarters (in the corners of a large square) - from the first to the fourth. And then suddenly who does not know? As you can see, quarters (they are also called beautiful word"quadrants") are numbered against the move clockwise. Added angle values on axes. Everything is clear, no frills.
And added a green arrow. With a plus. What does she mean? Let me remind you that the fixed side of the corner always nailed to the positive axis OH. So, if we twist the moving side of the corner plus arrow, i.e. in ascending quarter numbers, the angle will be considered positive. For example, the picture shows a positive angle of +60°.
If we postpone the corners in reverse side, clockwise, angle will be considered negative. Hover over the picture (or touch the picture on the tablet), you will see a blue arrow with a minus. This is the direction of the negative reading of the angles. A negative angle (-60°) is shown as an example. And you will also see how the numbers on the axes have changed ... I also translated them into negative angles. The numbering of the quadrants does not change.
Here, usually, the first misunderstandings begin. How so!? And if the negative angle on the circle coincides with the positive!? And in general, it turns out that the same position of the movable side (or a point on the numerical circle) can be called both a negative angle and a positive one!?
Yes. Exactly. Let's say a positive angle of 90 degrees takes on a circle exactly the same position as a negative angle of minus 270 degrees. A positive angle, for example +110° degrees, takes exactly the same position as the negative angle is -250°.
No problem. Everything is correct.) The choice of a positive or negative calculation of the angle depends on the condition of the assignment. If the condition says nothing plain text about the sign of the angle, (like "determine the smallest positive angle", etc.), then we work with values that are convenient for us.
An exception (and how without them ?!) are trigonometric inequalities, but there we will master this trick.
And now a question for you. How do I know that the position of the 110° angle is the same as the position of the -250° angle?
I will hint that this is due to the full turnover. In 360°... Not clear? Then we draw a circle. We draw on paper. Marking the corner about 110°. And believe how much remains until a full turn. Just 250° remains...
Got it? And now - attention! If the angles 110° and -250° occupy the circle same
position, then what? Yes, the fact that the angles are 110 ° and -250 ° exactly the same
sine, cosine, tangent and cotangent!
Those. sin110° = sin(-250°), ctg110° = ctg(-250°) and so on. Now this is really important! And in itself - there are a lot of tasks where it is necessary to simplify expressions, and as a basis for the subsequent development of reduction formulas and other intricacies of trigonometry.
Of course, I took 110 ° and -250 ° at random, purely for example. All these equalities work for any angles occupying the same position on the circle. 60° and -300°, -75° and 285°, and so on. I note right away that the corners in these couples - various. But they have trigonometric functions - the same.
I think you understand what negative angles are. It's quite simple. Counter-clockwise is a positive count. Along the way, it's negative. Consider angle positive or negative depends on us. From our desire. Well, and more from the task, of course ... I hope you understand how to move in trigonometric functions from negative to positive angles and vice versa. Draw a circle, an approximate angle, and see how much is missing before a full turn, i.e. up to 360°.
Angles greater than 360°.
Let's deal with angles that are greater than 360 °. And such things happen? There are, of course. How to draw them on a circle? Not a problem! Suppose we need to understand in which quarter an angle of 1000 ° will fall? Easily! We make one full turn counterclockwise (the angle was given to us positive!). Rewind 360°. Well, let's move on! Another turn - it has already turned out 720 °. How much is left? 280°. It is not enough for a full turn ... But the angle is more than 270 ° - and this is the border between the third and fourth quarter. So our angle of 1000° falls into the fourth quarter. Everything.
As you can see, it's quite simple. Let me remind you once again that the angle of 1000° and the angle of 280°, which we obtained by discarding the "extra" full turns, are, strictly speaking, various corners. But the trigonometric functions of these angles exactly the same! Those. sin1000° = sin280°, cos1000° = cos280° etc. If I were a sine, I wouldn't notice the difference between these two angles...
Why is all this necessary? Why do we need to translate angles from one to another? Yes, all for the same.) In order to simplify expressions. Simplification of expressions, in fact, is the main task of school mathematics. Well, along the way, the head is training.)
Well, shall we practice?)
We answer questions. Simple at first.
1. In which quarter does the angle -325° fall?
2. In which quarter does the angle 3000° fall?
3. In which quarter does the angle -3000° fall?
There is a problem? Or insecurity? We go to Section 555, Practical work with a trigonometric circle. There, in the first lesson of this very " practical work..." everything is detailed ... In such questions of uncertainty shouldn't!
4. What is the sign of sin555°?
5. What is the sign of tg555°?
Determined? Fine! Doubt? It is necessary to Section 555 ... By the way, there you will learn how to draw tangent and cotangent on a trigonometric circle. A very useful thing.
And now the smarter questions.
6. Bring the expression sin777° to the sine of the smallest positive angle.
7. Bring the expression cos777° to the cosine of the largest negative angle.
8. Convert the expression cos(-777°) to the cosine of the smallest positive angle.
9. Bring the expression sin777° to the sine of the largest negative angle.
What, questions 6-9 puzzled? Get used to it, there are not such formulations on the exam ... So be it, I will translate it. Only for you!
The words "reduce the expression to ..." mean to transform the expression so that its value hasn't changed a appearance changed in accordance with the task. So, in tasks 6 and 9, we must get a sine, inside which is the smallest positive angle. Everything else doesn't matter.
I will give the answers in order (in violation of our rules). But what to do, there are only two signs, and only four quarters ... You will not scatter in options.
6. sin57°.
7.cos(-57°).
8.cos57°.
9.-sin(-57°)
I suppose that the answers to questions 6-9 confused some people. Especially -sin(-57°), right?) Indeed, in the elementary rules for counting angles there is room for errors ... That is why I had to make a lesson: "How to determine the signs of functions and give angles on a trigonometric circle?" In Section 555. There tasks 4 - 9 are sorted out. Well sorted, with all the pitfalls. And they are here.)
In the next lesson, we will deal with the mysterious radians and the number "Pi". Learn how to easily and correctly convert degrees to radians and vice versa. And we will be surprised to find that this elementary information on the site enough already to solve some non-standard trigonometry puzzles!
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. Learning - with interest!)
you can get acquainted with functions and derivatives.
Lesson type: systematization of knowledge and intermediate control.
Equipment: trigonometric circle, tests, cards with tasks.
Lesson Objectives: systematize the studied theoretical material according to the definitions of sine, cosine, tangent of an angle; check the degree of assimilation of knowledge on this topic and application in practice.
Tasks:
- Generalize and consolidate the concepts of sine, cosine and tangent of an angle.
- To form a complex idea of trigonometric functions.
- Contribute to the development in students of the desire and need to study trigonometric material; to cultivate a culture of communication, the ability to work in groups and the need for self-education.
“Whoever does and thinks from his youth, he
becomes then, more reliable, stronger, smarter.
(V. Shukshin)
DURING THE CLASSES
I. Organizational moment
The class is represented by three groups. Each group has a consultant.
The teacher reports the topic, goals and objectives of the lesson.
II. Actualization of knowledge (frontal work with the class)
1) Work in groups on assignments:
1. Formulate the definition of the sin angle.
– What signs does sin α have in each coordinate quarter?
– At what values does the expression sin α make sense, and what values can it take?
2. The second group are the same questions for cos α.
3. The third group prepares answers on the same questions tg α and ctg α.
At this time, three students work independently at the board on cards (representatives of different groups).
Card number 1.
Practical work.
Using the unit circle, calculate the values of sin α, cos α and tg α for the angle 50, 210 and -210.
Card number 2.
Determine the sign of the expression: tg 275; cos 370; sin 790; tg 4.1 and sin 2.
Card number 3.
1) Calculate:
2) Compare: cos 60 and cos 2 30 - sin 2 30
2) Orally:
a) A number of numbers are proposed: 1; 1.2; 3; , 0, , – 1. Some of them are redundant. What property sin α or cos α can express these numbers (Can sin α or cos α take these values).
b) Does the expression make sense: cos (-); sin2; tg3:ctg(-5); ; ctg0;
ctg(-π). Why?
c) Is there a least and highest value sin or cos, tg, ctg.
d) Is it true?
1) α = 1000 is the angle of the II quarter;
2) α \u003d - 330 is the angle of the IV quarter.
e) Numbers correspond to the same point on the unit circle.
3) Whiteboard work
#567 (2; 4) - Find the value of an expression
#583 (1-3) Determine the sign of the expression
Homework: table in a notebook. No. 567(1, 3) No. 578
III. Acquisition of additional knowledge. Trigonometry in the palm of your hand
Teacher: It turns out that the values of the sines and cosines of the angles "are" in your palm. Reach out your hand (any) and spread your fingers as far as possible (as on the poster). One student is invited. We measure the angles between our fingers.
A triangle is taken, where there is an angle of 30, 45 and 60 90 and we apply the top of the angle to the hillock of the Moon in the palm of our hand. The Mount of the Moon is located at the intersection of the extensions of the little finger and thumb. We combine one side with the little finger, and the other side with one of the other fingers.
It turns out that the angle between the little finger and the thumb is 90, between the little finger and the ring finger - 30, between the little finger and the middle finger - 45, between the little finger and the index finger - 60. And this is for all people without exception
little finger number 0 - corresponds to 0,
nameless number 1 - corresponds to 30,
medium number 2 - corresponds to 45,
index number 3 - corresponds to 60,
large number 4 - corresponds to 90.
Thus, we have 4 fingers on our hand and remember the formula:
finger number |
Injection |
Meaning |
|
||
|
||
|
This is just a mnemonic rule. In general, the value of sin α or cos α must be known by heart, but sometimes this rule will help in difficult times.
Come up with a rule for cos (angles without change, but counting from the thumb). A physical pause associated with the signs sin α or cos α.
IV. Checking the assimilation of ZUN
Independent work with feedback
Each student receives a test (4 options) and the answer sheet is the same for everyone.
Test
Option 1
1) At what angle of rotation will the radius take the same position as when rotated through an angle of 50.
2) Find the value of the expression: 4cos 60 - 3sin 90.
3) Which of the numbers is less than zero: sin 140, cos 140, sin 50, tg 50.
Option 2
1) At what angle of rotation will the radius take the same position as when rotated through an angle of 10.
2) Find the value of the expression: 4cos 90 - 6sin 30.
3) Which of the numbers is greater than zero: sin 340, cos 340, sin 240, tg (- 240).
Option 3
1) Find the value of the expression: 2ctg 45 - 3cos 90.
2) Which of the numbers is less than zero: sin 40, cos (- 10), tg 210, sin 140.
3) The angle of which quarter is the angle α, if sin α > 0, cos α< 0.
Option 4
1) Find the value of the expression: tg 60 - 6ctg 90.
2) Which of the numbers is less than zero: sin (- 10), cos 140, tg 250, cos 250.
3) The angle of which quarter is the angle α, if ctg α< 0, cos α> 0.
BUT |
B |
AT |
G |
D |
E |
F |
W |
And |
L |
M |
|
H |
O |
P |
R |
With |
T |
At |
F |
X |
W |
YU |
I |
(the word is trigonometry is the key)
V. Information from the history of trigonometry
Teacher: Trigonometry is a fairly important branch of mathematics for human life. Modern look trigonometry was given by the greatest mathematician of the 18th century, Leonhard Euler, a Swiss by birth long years who worked in Russia and was a member of the St. Petersburg Academy of Sciences. He introduced famous definitions trigonometric functions formulated and proved well-known formulas, we will learn them later. Euler's life is very interesting and I advise you to get acquainted with it from Yakovlev's book "Leonard Euler".
(Message guys on this topic)
VI. Summing up the lesson
Tic-tac-toe game
The two most active students participate. They are supported by groups. The solution of tasks is recorded in a notebook.
Tasks
1) Find the error
a) sin 225 = - 1.1 c) sin 115< О
b) cos 1000 = 2 d) cos (– 115) > 0
2) Express the angle in degrees
3) Express in radians the angle 300
4) What is the largest and smallest value may have the expression: 1+ sin α;
5) Determine the sign of the expression: sin 260, cos 300.
6) In what quarter of the number circle is the point
7) Determine the signs of the expression: cos 0.3π, sin 195, ctg 1, tg 390
8) Calculate:
9) Compare: sin 2 and sin 350
VII. Lesson reflection
Teacher: Where can we meet trigonometry?
In what lessons in grade 9, and even now, do you use the concepts of sin α, cos α; tgα; ctg α and for what purpose?
The sign of the trigonometric function depends solely on the coordinate quarter in which the numeric argument is located. Last time we learned how to translate arguments from a radian measure into a degree measure (see the lesson “ Radian and degree measure of an angle”), and then determine this same coordinate quarter. Now let's deal, in fact, with the definition of the sign of the sine, cosine and tangent.
The sine of the angle α is the ordinate (coordinate y) of a point on a trigonometric circle, which occurs when the radius is rotated through the angle α.
The cosine of the angle α is the abscissa (x coordinate) of a point on a trigonometric circle, which occurs when the radius rotates through the angle α.
The tangent of the angle α is the ratio of the sine to the cosine. Or, equivalently, the ratio of the y-coordinate to the x-coordinate.
Notation: sin α = y ; cosα = x; tgα = y : x .
All these definitions are familiar to you from the high school algebra course. However, we are not interested in the definitions themselves, but in the consequences that arise on the trigonometric circle. Take a look:
The blue color indicates the positive direction of the OY axis (ordinate axis), the red color indicates the positive direction of the OX axis (abscissa axis). On this "radar" the signs of trigonometric functions become obvious. In particular:
- sin α > 0 if the angle α lies in the I or II coordinate quarter. This is because, by definition, a sine is an ordinate (y coordinate). And the y coordinate will be positive precisely in the I and II coordinate quarters;
- cos α > 0 if the angle α lies in the I or IV coordinate quarter. Because only there the x coordinate (it is also the abscissa) will be greater than zero;
- tg α > 0 if the angle α lies in the I or III coordinate quadrant. This follows from the definition: after all, tg α = y : x , so it is positive only where the signs of x and y coincide. This happens in the 1st coordinate quarter (here x > 0, y > 0) and the 3rd coordinate quarter (x< 0, y < 0).
For clarity, we note the signs of each trigonometric function - sine, cosine and tangent - on separate "radar". We get the following picture:
Note: in my reasoning, I never spoke about the fourth trigonometric function - the cotangent. The fact is that the signs of the cotangent coincide with the signs of the tangent - there are no special rules there.
Now I propose to consider examples similar to problems B11 from trial exam in mathematics, which took place on September 27, 2011. After all The best way understanding theory is practice. Preferably 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 considered):
- sin(3π/4);
- cos(7π/6);
- tan (5π/3);
- sin(3π/4) cos(5π/6);
- cos (2π/3) tg (π/4);
- sin(5π/6) cos(7π/4);
- tan (3π/4) cos (5π/3);
- ctg (4π/3) tg (π/6).
The action plan is as follows: first, we convert all angles from radian measure to degree measure (π → 180°), and then look in which coordinate quarter the resulting number lies. Knowing the quarters, we can easily find the signs - according to the rules just described. We have:
- 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;
- cos (7π/6) = cos (7 180°/6) = cos 210°. Because 210° ∈ , this is an angle from the III coordinate quadrant in which all cosines are negative. Therefore, cos (7π/6)< 0;
- tg (5π/3) = tg (5 180°/3) = tg 300°. Since 300° ∈ , we are in the fourth quadrant, where the tangent takes negative values. Therefore tg (5π/3)< 0;
- 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 the 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;
- 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 factors of different signs. Since “a minus times a plus gives a minus”, we have: cos (2π/3) tg (π/4)< 0;
- 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 got the product of two positive numbers - such an expression is always positive. We conclude: sin (5π/6) cos (7π/4) > 0;
- 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. tan (3π/4)< 0. Аналогично, угол 300° ∈ — это IV четверть, т.е. cos (5π/3) >0. Since “a minus plus gives a minus sign”, we have: tg (3π/4) cos (5π/3)< 0;
- 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. easiest corner. Therefore, tg (π/6) > 0. Again, we got two positive expressions - their product will also be positive. Therefore ctg (4π/3) tg (π/6) > 0.
Finally, let's take a look at a few more challenging tasks. In addition to finding out the sign of the trigonometric function, here you have to do a little calculation - just like it is done in real problems B11. In principle, these are almost real tasks that are really found in the exam 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. It remains to decide: plus or minus? By assumption, the 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 assumption, the angle α ∈ [π; 3π/2], i.e. we are talking about the III coordinate quarter. There, all cosines 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. Again we look at the angle: α ∈ is the IV coordinate quarter, in which, as you know, the sine will be negative. Thus, we conclude: sin α = −0.5.
Task. Find tg α if tg 2 α = 9 and α ∈ .
Everything is the same, only for the tangent. We take the square root: tg 2 α = 9 ⇒ tg α = ±3. But by the condition, the angle α ∈ is the I coordinate quadrant. All trigonometric functions, incl. tangent, there are positive, so tg α = 3. That's it!
In the fifth century BC, the ancient Greek philosopher Zeno of Elea formulated his famous aporias, the most famous of which is the aporia "Achilles and the tortoise". Here's how it sounds:Let's say Achilles runs ten times faster than the tortoise and is a thousand paces behind it. During the time during which Achilles runs this distance, the tortoise crawls a hundred steps in the same direction. When Achilles has run a hundred steps, the tortoise will crawl another ten steps, and so on. The process will continue indefinitely, Achilles will never catch up with the tortoise.
This reasoning became a logical shock for all subsequent generations. Aristotle, Diogenes, Kant, Hegel, Gilbert... All of them, in one way or another, considered Zeno's aporias. The shock was so strong that " ... discussions continue at the present time, to come to 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 universally accepted solution to the problem ..."[Wikipedia," Zeno's Aporias "]. Everyone understands that they are being fooled, but no one understands what the deception is.
From the point of view of mathematics, Zeno in his aporia clearly demonstrated the transition from the value to. This transition implies applying instead of constants. As far as I understand, the mathematical apparatus for applying variable units of measurement has either not yet been developed, or it has not been applied to Zeno's aporia. The application of our usual logic leads us into a trap. We, by the inertia of thinking, apply constant units of time to the reciprocal. From a physical point of view, it looks like time slowing down to a complete stop at the moment when Achilles catches up with the tortoise. If time stops, Achilles can no longer overtake the tortoise.
If we turn the logic we are used to, everything falls into place. Achilles runs at a constant speed. Each subsequent segment of its path is ten times shorter than the previous one. Accordingly, the time spent on overcoming it is ten times less than the previous one. If we apply the concept of "infinity" in this situation, then it would be correct to say "Achilles will infinitely quickly overtake the tortoise."
How to avoid this logical trap? Remain in constant units of time and do not switch to reciprocal values. In Zeno's language, it looks like this:
In the time it takes Achilles to run a thousand steps, the tortoise crawls a hundred steps in the same direction. During the next time interval, equal to the first, Achilles will run another thousand steps, and the tortoise will crawl one hundred steps. Now Achilles is eight hundred paces ahead of the tortoise.
This approach adequately describes reality without any logical paradoxes. But it is not complete solution Problems. Einstein's statement about the insurmountability of the speed of light is very similar to Zeno's aporia "Achilles and the tortoise". We have yet to study, rethink and solve this problem. And the solution must be sought not in infinitely large numbers, but in units of measurement.
Another interesting aporia of Zeno tells of a flying arrow:
A flying arrow is motionless, since at each moment of time it is at rest, and since it is at rest at every moment of time, it is always at rest.
In this aporia logical paradox it is overcome very simply - it is enough to clarify that at each moment of time the flying arrow rests at different points in space, which, in fact, is movement. There is another point to be noted here. From one photograph of a car on the road, it is impossible to determine either the fact of its movement or the distance to it. To determine the fact of the movement of the car, two photographs taken from the same point at different points in time are needed, but they cannot be used to determine the distance. To determine the distance to the car, you need two photographs taken from different points in space at the same time, but you cannot determine the fact of movement from them (naturally, you still need additional data for calculations, trigonometry will help you). What do I want to focus on Special attention, 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 exploration.
Wednesday, July 4, 2018
Very well the differences between set and multiset are described in Wikipedia. We look.
As you can see, "the set cannot have two identical elements", but if there are identical elements in the set, such a set is called a "multiset". Similar logic of absurdity sentient beings never understand. This is the level of talking parrots and trained monkeys, in which the mind is absent from the word "completely." Mathematicians act as ordinary trainers, preaching their absurd ideas to us.
Once upon a time, the engineers who built the bridge were in a boat under the bridge during the tests of the bridge. If the bridge collapsed, the mediocre engineer died under the rubble of his creation. If the bridge could withstand the load, the talented engineer built other bridges.
No matter how mathematicians hide behind the phrase "mind me, I'm in the house", or rather "mathematics studies abstract concepts", there is one umbilical cord that inextricably connects them with reality. This umbilical cord is money. Let us apply mathematical set theory to mathematicians themselves.
We studied mathematics very well and now we are sitting at the cash desk, paying salaries. Here a mathematician comes to us for his money. We count the whole amount to him and lay it out on our table into different piles, in which we put bills of the same denomination. Then we take one bill from each pile and give the mathematician his "mathematical salary set". We explain the mathematics that he will receive the rest of the bills only when he proves that the set without identical elements is not equal to the set with identical elements. This is where the fun begins.
First of all, the deputies' logic will work: "you can apply it to others, but not to me!" Further, assurances will begin that there are different banknote numbers on banknotes of the same denomination, which means that they cannot be considered identical elements. Well, we count the salary in coins - there are no numbers on the coins. Here the mathematician will begin to convulsively recall physics: different coins available different amount dirt, crystal structure and atomic arrangement of each coin is unique...
And now I have the most interest Ask: where is the boundary beyond which elements of a multiset turn into elements of a set and vice versa? Such a line does not exist - everything is decided by shamans, science here is not even close.
Look here. We select football stadiums with the same field area. The area of the fields is the same, which means we have a multiset. But if we consider the names of the same stadiums, we get a lot, because the names are different. As you can see, the same set of elements is both a set and a multiset at the same time. How right? And here the mathematician-shaman-shuller takes out a trump ace from his sleeve and begins to tell us about either a set or a multiset. In any case, he will convince us that he is right.
To understand how modern shamans operate with set theory, tying it to reality, it is enough to answer one question: how do the elements of one set differ from the elements of another set? I will show you, without any "conceivable as not a single whole" or "not conceivable as a single whole."
Sunday, March 18, 2018
The sum of the digits of a number is a dance of shamans with a tambourine, which has nothing to do with mathematics. Yes, in mathematics lessons we are taught to find the sum of the digits of a number and use it, but they are shamans for that, to teach their descendants their skills and wisdom, otherwise shamans will simply die out.
Do you need proof? Open Wikipedia and try to find the "Sum of Digits of a Number" page. She doesn't exist. There is no formula in mathematics by which you can find the sum of the digits of any number. After all, numbers are graphic symbols with which we write numbers, and in the language of mathematics, the task sounds like this: "Find the sum of graphic symbols representing any number." Mathematicians cannot solve this problem, but shamans can do it elementarily.
Let's figure out what and how we do in order to find the sum of the digits of a given number. And so, let's say we have the number 12345. What needs to be done in order to find the sum of the digits of this number? Let's consider all the steps in order.
1. Write down the number on a piece of paper. What have we done? We have converted the number to a number graphic symbol. This is not a mathematical operation.
2. We cut one received picture into several pictures containing separate numbers. Cutting a picture is not a mathematical operation.
3. Convert individual graphic characters to numbers. This is not a mathematical operation.
4. Add up the resulting numbers. Now that's mathematics.
The sum of the digits of the number 12345 is 15. These are the "cutting and sewing courses" from shamans used by mathematicians. But that is not all.
From the point of view of mathematics, it does not matter in which number system we write the number. So, in different number systems, the sum of the digits of the same number will be different. In mathematics, the number system is indicated as a subscript to the right of the number. With a large number 12345 I don’t want to fool my head, consider the number 26 from the article about. Let's write this number in binary, octal, decimal and hexadecimal number systems. We will not consider each step under a microscope, we have already done that. Let's look at the result.
As you can see, in different number systems, the sum of the digits of the same number is different. This result has nothing to do with mathematics. It's the same as if you would get completely different results when determining the area of a rectangle in meters and centimeters.
Zero in all number systems looks the same and has no sum of digits. This is another argument in favor of the fact that . A question for mathematicians: how is it denoted in mathematics that which is not a number? What, for mathematicians, nothing but numbers exists? For shamans, I can allow this, but for scientists, no. Reality is not just about numbers.
The result obtained should be considered as proof that number systems are units of measurement of numbers. After all, we cannot compare numbers with different units of measurement. If the same actions with different units of measurement of the same quantity lead to different results after comparing them, then it has nothing to do with mathematics.
What is real mathematics? This is when the result of a mathematical action does not depend on the value of the number, the unit of measurement used, and on who performs this action.
Ouch! Isn't this the women's restroom?
- Young woman! This is a laboratory for studying the indefinite holiness of souls upon ascension to heaven! Nimbus on top and arrow up. What other toilet?
Female... A halo on top and an arrow down is male.
If you have such a work of design art flashing before your eyes several times a day,
Then it is not surprising that you suddenly find a strange icon in your car:
Personally, I make an effort on myself to see minus four degrees in a pooping person (one picture) (composition of several pictures: minus sign, number four, degrees designation). And I do not consider this girl a fool who does not know physics. She just has an arc stereotype of perception of graphic images. And mathematicians teach us this all the time. Here is an example.
1A is not "minus four degrees" or "one a". This is a "pooping man" or the number "twenty-six" in hexadecimal system reckoning. Those people who constantly work in this number system automatically perceive the number and letter as one graphic symbol.
Reference data for tangent (tg x) and cotangent (ctg x). Geometric definition, properties, graphs, formulas. Table of tangents and cotangents, derivatives, integrals, series expansions. Expressions through complex variables. Connection with hyperbolic functions.
Geometric definition
|BD| - the length of the arc of a circle centered at point A.
α is the angle expressed in radians.
Tangent ( tgα) is a trigonometric function depending on the angle α between the hypotenuse and the leg right triangle, equal to the ratio of the length of the opposite leg |BC| to the length of the adjacent leg |AB| .
Cotangent ( ctgα) is a trigonometric function depending on the angle α between the hypotenuse and the leg of a right triangle, equal to the ratio of the length of the adjacent leg |AB| to the length of the opposite leg |BC| .
Tangent
Where n- whole.
AT Western literature tangent is defined as follows:
.
;
;
.
Graph of the tangent function, y = tg x
Cotangent
Where n- whole.
In Western literature, the cotangent is denoted as follows:
.
The following notation has also been adopted:
;
;
.
Graph of the cotangent function, y = ctg x
Properties of tangent and cotangent
Periodicity
Functions y= tg x and y= ctg x are periodic with period π.
Parity
The functions tangent and cotangent are odd.
Domains of definition and values, ascending, descending
The functions tangent and cotangent are continuous on their domain of definition (see the proof of continuity). The main properties of the tangent and cotangent are presented in the table ( n- integer).
| y= tg x | y= ctg x | |
| Scope and continuity | ||
| Range of values | -∞ < y < +∞ | -∞ < y < +∞ |
| Ascending | - | |
| Descending | - | |
| Extremes | - | - |
| Zeros, y= 0 | ||
| Points of intersection with the y-axis, x = 0 | y= 0 | - |
Formulas
Expressions in terms of sine and cosine
;
;
;
;
;
Formulas for tangent and cotangent of sum and difference
The rest of the formulas are easy to obtain, for example
Product of tangents
The formula for the sum and difference of tangents
This table shows the values of tangents and cotangents for some values of the argument. 
Expressions in terms of complex numbers
Expressions in terms of hyperbolic functions
;
;
Derivatives
; .
.
Derivative of the nth order with respect to the variable x of the function :
.
Derivation of formulas for tangent > > > ; for cotangent > > >
Integrals
Expansions into series
To get the expansion of the tangent in powers of x, you need to take several terms of the expansion in power series for functions sin x and cos x and divide these polynomials into each other , . This results in the following formulas.
At .
at .
where B n- Bernoulli numbers. They are determined either from the recurrence relation:
;
;
where .
Or according to the Laplace formula: 
Inverse functions
The inverse functions to tangent and cotangent are arctangent and arccotangent, respectively.
Arctangent, arctg
, where n- whole.
Arc tangent, arcctg
, where n- whole.
References:
I.N. Bronstein, K.A. Semendyaev, Handbook of Mathematics for Engineers and Students of Higher Educational Institutions, Lan, 2009.
G. Korn, Handbook of Mathematics for Researchers and Engineers, 2012.