Let's say Achilles runs ten times faster than the tortoise and is a thousand steps behind it. During the time it takes Achilles to run this distance, the tortoise will crawl a hundred steps in the same direction. When Achilles runs a hundred steps, the tortoise crawls another ten steps, and so on. The process will continue ad infinitum, Achilles will never catch up with the tortoise.
This reasoning became a logical shock for all subsequent generations. Aristotle, Diogenes, Kant, Hegel, Hilbert... They all considered Zeno's aporia in one way or another. The shock was so strong that " ... discussions continue to this day; the scientific community has not yet been able to come to a common opinion on the essence of paradoxes ... mathematical analysis, set theory, new physical and philosophical approaches were involved in the study of the issue; none of them became a generally accepted solution to the problem..."[Wikipedia, "Zeno's Aporia". Everyone understands that they are being fooled, but no one understands what the deception consists of.
From a mathematical point of view, Zeno in his aporia clearly demonstrated the transition from quantity to . This transition implies application instead of permanent ones. As far as I understand, the mathematical apparatus for using variable units of measurement has either not yet been developed, or it has not been applied to Zeno’s aporia. Applying our usual logic leads us into a trap. We, due to the inertia of thinking, apply constant units of time to the reciprocal value. From a physical point of view, this looks like time slowing down until it stops completely at the moment when Achilles catches up with the turtle. If time stops, Achilles can no longer outrun the tortoise.
If we turn our usual logic around, everything falls into place. Achilles runs with constant speed. Each subsequent segment of his path is ten times shorter than the previous one. Accordingly, the time spent on overcoming it is ten times less than the previous one. If we apply the concept of “infinity” in this situation, then it would be correct to say “Achilles will catch up with the turtle infinitely quickly.”
How to avoid this logical trap? Remain in constant units of time and do not jump to reciprocals. In Zeno's language it looks like this:
In the time it takes Achilles to run a thousand steps, the tortoise will crawl a hundred steps in the same direction. During the next time interval equal to the first, Achilles will run another thousand steps, and the tortoise will crawl a hundred steps. Now Achilles is eight hundred steps ahead of the tortoise.
This approach adequately describes reality without any logical paradoxes. But this is not a complete solution to the problem. Einstein’s statement about the irresistibility of the speed of light is very similar to Zeno’s aporia “Achilles and the Tortoise”. We still have to study, rethink and solve this problem. And the solution must be sought not in infinitely large numbers, but in units of measurement.
Another interesting aporia of Zeno tells about a flying arrow:
A flying arrow is motionless, since at every moment of time it is at rest, and since it is at rest at every moment of time, it is always at rest.
In this aporia logical paradox it can be overcome very simply - it is enough to clarify that at each moment of time a flying arrow is at rest at different points in space, which, in fact, is motion. Another point needs to be noted here. From one photograph of a car on the road it is impossible to determine either the fact of its movement or the distance to it. To determine whether a car is moving, you need two photographs taken from the same point at different points in time, but you cannot determine the distance from them. To determine the distance to the car, you need two photographs taken from different points space at one point in time, but it is impossible to determine the fact of movement from them (naturally, additional data is still needed for calculations, trigonometry will help you). What I want to point out 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 possibilities for research.
Wednesday, July 4, 2018
The differences between set and multiset are described very well on Wikipedia. Let's see.
As you can see, “there cannot be two identical elements in a set,” but if there are identical elements in a set, such a set is called a “multiset.” Reasonable beings will never understand such absurd logic. This is the level of talking parrots and trained monkeys, who have no intelligence from the word “completely”. Mathematicians act as ordinary trainers, preaching to us their absurd ideas.
Once upon a time, the engineers who built the bridge were in a boat under the bridge while testing 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 register, giving out salaries. So a mathematician comes to us for his money. We count out the entire amount to him and lay it out on our table in different piles, into which we put bills of the same denomination. Then we take one bill from each pile and give the mathematician his “mathematical set of salary.” Let us explain to the mathematician that he will receive the remaining bills only when he proves that a set without identical elements is not equal to a set with identical elements. This is where the fun begins.
First of all, the logic of the deputies will work: “This can be applied to others, but not to me!” Then they will begin to reassure us that bills of the same denomination have different bill numbers, which means they cannot be considered the same elements. Okay, let's count salaries in coins - there are no numbers on the coins. Here the mathematician will begin to frantically remember physics: different coins have different amounts of dirt, the crystal structure and arrangement of atoms is unique for each coin...
And now I have the most interesting question: where is the line beyond which the 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 is not even close to lying here.
Look here. We select football stadiums with the same field area. The areas of the fields are the same - which means we have a multiset. But if we look at the names of these same stadiums, we get many, because the names are different. As you can see, the same set of elements is both a set and a multiset. Which is correct? And here the mathematician-shaman-sharpist pulls out an ace of trumps from his sleeve and begins to tell us either about a set or a multiset. In any case, he will convince us that he is right.
To understand how modern shamans operate with set theory, tying it to reality, it is enough to answer one question: how do the elements of one set differ from the elements of another set? I'll 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 that’s why they are shamans, 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 page "Sum of digits of a number." She doesn't exist. There is no formula in mathematics that can be used to 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 easily.
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 us 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 into a graphical number symbol. This is not a mathematical operation.
2. Cut one resulting picture into several pictures containing individual numbers. Cutting a picture is not a mathematical operation.
3. Convert individual graphic symbols into numbers. This is not a mathematical operation.
4. Add the resulting numbers. Now this is mathematics.
The sum of the digits of the number 12345 is 15. These are the “cutting and sewing courses” from shamans that mathematicians use. But that's not all.
From a mathematical point of view, it does not matter in which number system we write a number. So, in different systems In calculus, 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, let’s look at the number 26 from the article about . Let's write this number in binary, octal, decimal and hexadecimal number systems. We won't look at every step under a microscope; we've 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 determined the area of a rectangle in meters and centimeters, you would get completely different results.
Zero looks the same in all number systems and has no sum of digits. This is another argument in favor of the fact that. Question for mathematicians: how is something that is not a number designated in mathematics? What, for mathematicians nothing exists except numbers? I can allow this for shamans, but not for scientists. Reality is not just about numbers.
The result obtained should be considered as proof that number systems are units of measurement for 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 this has nothing to do with mathematics.
What is real mathematics? This is when the result of a mathematical operation does not depend on the size of the number, the unit of measurement used and on who performs this action.
Oh! Isn't this the women's restroom?
- Young woman! This is a laboratory for the study of the indephilic holiness of souls during their ascension to heaven! Halo on top and arrow up. What other toilet?
Female... The halo on top and the arrow down are male.
If such a work of design art flashes before your eyes several times a day,
Then it’s not surprising that you suddenly find a strange icon in your car:
Personally, I make an effort to see minus four degrees in a pooping person (one picture) (a composition of several pictures: minus sign, number four, degree designation). And I don’t think this girl is a fool who doesn’t know physics. She just has a strong stereotype of perceiving graphic images. And mathematicians teach us this all the time. Here's an example.
1A is not “minus four degrees” or “one a”. This is "pooping man" or the number "twenty-six" in hexadecimal notation. Those people who constantly work in this number system automatically perceive a number and a letter as one graphic symbol.
Counting angles on a trigonometric circle.
Attention!
There are additional
materials in Special Section 555.
For those who are very "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 in order. Added quarter numbers (in the corners of the large square) - from the first to the fourth. What if someone doesn’t know? As you can see, quarters (they are also called a beautiful word"quadrants") are numbered counterclockwise. Added angle values on axes. Everything is clear, no problems.
And a green arrow is added. With a plus. What does it mean? Let me remind you that the fixed side of the angle Always nailed to the positive semi-axis OX. So, if we rotate the movable side of the angle along the arrow with a plus, i.e. in ascending order of quarter numbers, the angle will be considered positive. As an example, the picture shows a positive angle of +60°.
If we put aside the corners in the opposite direction, clockwise, the angle will be considered negative. Hover your cursor over the picture (or touch the picture on your tablet), you will see a blue arrow with a minus sign. This is the direction of negative angle reading. For example, a negative angle (- 60°) is shown. And you will also see how the numbers on the axes have changed... I also converted them to negative angles. The numbering of the quadrants does not change.
This is where the first misunderstandings usually begin. How so!? What if a negative angle on a circle coincides with a positive one!? And in general, it turns out that the same position of the moving side (or point on the number circle) can be called both a negative angle and a positive one!?
Yes. That's right. 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 negative angle -250°.
No question. Anything is correct.) The choice of positive or negative angle calculation depends on the conditions of the task. If the condition says nothing in clear text about the sign of the angle, (like "determine the smallest positive angle", etc.), then we work with values that are convenient for us.
The exception (how could we live without them?!) are trigonometric inequalities, but there we will master this trick.
And now a question for you. How did I know that the position of the 110° angle is the same as the position of the -250° angle?
Let me hint that this is connected with a complete revolution. In 360°... Not clear? Then we draw a circle. We draw it ourselves, on paper. Marking the corner approximately 110°. AND we think, how much time remains until a full revolution. Just 250° will remain...
Got it? And now - attention! If angles 110° and -250° occupy a circle same thing
situation, then what? Yes, 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 you need to simplify expressions, and as a basis for the subsequent mastery of reduction formulas and other intricacies of trigonometry.
Of course, I took 110° and -250° at random, purely as an example. All these equalities work for any angles occupying the same position on the circle. 60° and -300°, -75° and 285°, and so on. Let me note right away that the angles in these pairs are different. But they have trigonometric functions - identical.
I think you understand what negative angles are. It's quite simple. Counterclockwise - positive counting. Along the way - negative. Consider the angle positive or negative depends on us. From our desire. Well, and also from the task, of course... I hope you understand how to move in trigonometric functions from negative angles to positive ones and back. Draw a circle, an approximate angle, and see how much is missing to complete a full revolution, i.e. up to 360°.
Angles greater than 360°.
Let's deal with angles that are greater than 360°. Are there such things? There are, of course. How to draw them on a circle? No problem! Let's say we need to understand which quarter an angle of 1000° will fall into? Easily! We make one full turn counterclockwise (the angle we were given is positive!). We rewinded 360°. Well, let's move on! One more turn - it’s already 720°. How many are left? 280°. It’s not enough for a full turn... But the angle is more than 270° - and this is the border between the third and fourth quarter. Therefore, our angle of 1000° falls into the fourth quarter. All.
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 revolutions, are, strictly speaking, different 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 needed? Why do we need to convert angles from one to another? Yes, all for the same thing.) In order to simplify expressions. Simplification of expressions, in fact, main task school mathematics. Well, and, along the way, the head is trained.)
Well, let's practice?)
We answer questions. Simple ones first.
1. Which quarter does the -325° angle fall into?
2. Which quarter does the 3000° angle fall into?
3. Which quarter does the angle -3000° fall into?
Any problems? Or uncertainty? Go to Section 555, Trigonometric Circle Practice. There, in the first lesson of this very " Practical work..." all in detail... In such questions of uncertainty to be shouldn't!
4. What sign does sin555° have?
5. What sign does tg555° have?
Have you determined? Great! Do you have any doubts? You need to go to Section 555... By the way, there you will learn to draw tangent and cotangent on a trigonometric circle. A very useful thing.
And now the questions are more sophisticated.
6. Reduce the expression sin777° to the sine of the smallest positive angle.
7. Reduce the expression cos777° to the cosine of the largest negative angle.
8. Reduce the expression cos(-777°) to the cosine of the smallest positive angle.
9. Reduce the expression sin777° to the sine of the largest negative angle.
What, questions 6-9 puzzled you? Get used to it, on the Unified State Exam you don’t find such formulations... So be it, I’ll translate it. Just for you!
The words "bring an expression to..." mean to transform the expression so that its meaning hasn't changed A appearance changed according to the assignment. So, in tasks 6 and 9 we must get a sine, inside of which there is smallest positive angle. Everything else doesn't matter.
I will give out the answers in order (in violation of our rules). But what to do, there are only two signs, and there are only four quarters... You won’t be spoiled for choice.
6. sin57°.
7. cos(-57°).
8. cos57°.
9. -sin(-57°)
I assume that the answers to questions 6-9 confused some people. Especially -sin(-57°), really?) Indeed, in the elementary rules for calculating angles there is room for errors... That is why I had to do a lesson: “How to determine the signs of functions and give angles on a trigonometric circle?” In Section 555. Tasks 4 - 9 are covered there. 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". Let's learn how to easily and correctly convert degrees to radians and vice versa. And we will be surprised to discover that this basic information on the site enough already to solve some custom trigonometry problems!
If you like this site...
By the way, I have a couple more interesting sites for you.)
You can practice solving examples and find out your level. Testing with instant verification. Let's learn - with interest!)
You can get acquainted with functions and derivatives.
Allows you to establish a number of characteristic results - properties of sine, cosine, tangent and cotangent. In this article we will look at three main properties. The first of them indicates the signs of the sine, cosine, tangent and cotangent of the angle α depending on the angle of which coordinate quarter is α. Next we will consider the property of periodicity, which establishes the invariance of the values of sine, cosine, tangent and cotangent of the angle α when this angle changes by an integer number of revolutions. The third property expresses the relationship between the values of sine, cosine, tangent and cotangent of opposite angles α and −α.
If you are interested in the properties of the functions sine, cosine, tangent and cotangent, then you can study them in the corresponding section of the article.
Page navigation.
Signs of sine, cosine, tangent and cotangent by quarters
Below in this paragraph the phrase “angle of I, II, III and IV coordinate quarter” will appear. Let's explain what these angles are.
Let's take a unit circle, mark the starting point A(1, 0) on it, and rotate it around the point O by an angle α, and we will assume that we will get to the point A 1 (x, y).
They say that angle α is the angle of the I, II, III, IV coordinate quadrant, if point A 1 lies in the I, II, III, IV quarters, respectively; if the angle α is such that point A 1 lies on any of the coordinate lines Ox or Oy, then this angle does not belong to any of the four quarters.
For clarity, here is a graphic illustration. The drawings below show rotation angles of 30, −210, 585, and −45 degrees, which are the angles of the I, II, III, and IV coordinate quarters, respectively.
Angles 0, ±90, ±180, ±270, ±360, … degrees do not belong to any of the coordinate quarters.
Now let's figure out what signs have the values of sine, cosine, tangent and cotangent of the angle of rotation α, depending on which quadrant angle α is.
For sine and cosine this is easy to do.
By definition, the sine of angle α is the ordinate of point A 1. Obviously, in the I and II coordinate quarters it is positive, and in the III and IV quarters it is negative. Thus, the sine of angle α has a plus sign in the 1st and 2nd quarters, and a minus sign in the 3rd and 6th quarters.
In turn, the cosine of the angle α is the abscissa of point A 1. In the I and IV quarters it is positive, and in the II and III quarters it is negative. Consequently, the values of the cosine of the angle α in the I and IV quarters are positive, and in the II and III quarters they are negative.
To determine the signs by quarters of tangent and cotangent, you need to remember their definitions: tangent is the ratio of the ordinate of point A 1 to the abscissa, and cotangent is the ratio of the abscissa of point A 1 to the ordinate. Then from rules for dividing numbers with the same and different signs it follows that tangent and cotangent have a plus sign when the abscissa and ordinate signs of point A 1 are the same, and have a minus sign when the abscissa and ordinate signs of point A 1 are different. Consequently, the tangent and cotangent of the angle have a + sign in the I and III coordinate quarters, and a minus sign in the II and IV quarters.
Indeed, for example, in the first quarter both the abscissa x and the ordinate y of point A 1 are positive, then both the quotient x/y and the quotient y/x are positive, therefore, tangent and cotangent have + signs. And in the second quarter, the abscissa x is negative, and the ordinate y is positive, therefore both x/y and y/x are negative, hence the tangent and cotangent have a minus sign.
Let's move on to the next property of sine, cosine, tangent and cotangent.
Periodicity property
Now we will look at perhaps the most obvious property of sine, cosine, tangent and cotangent of an angle. It is as follows: when the angle changes by an integer number of full revolutions, the values of the sine, cosine, tangent and cotangent of this angle do not change.
This is understandable: when the angle changes by an integer number of revolutions, from the starting point A we will always get to point A 1 by unit circle therefore, the values of sine, cosine, tangent and cotangent remain unchanged, since the coordinates of point A 1 are unchanged.
Using formulas, the considered property of sine, cosine, tangent and cotangent can be written as follows: sin(α+2·π·z)=sinα, cos(α+2·π·z)=cosα, tan(α+2·π· z)=tgα, ctg(α+2·π·z)=ctgα, where α is the angle of rotation in radians, z is any, the absolute value of which indicates the number of full revolutions by which the angle α changes, and the sign of the number z indicates the direction turn.
If the rotation angle α is specified in degrees, then the indicated formulas will be rewritten as sin(α+360° z)=sinα , cos(α+360° z)=cosα , tg(α+360° z)=tgα , ctg(α+360°·z)=ctgα .
Let's give examples of using this property. For example, 
, because 
, A 
. Here's another example: or .
This property, together with reduction formulas, is very often used when calculating the values of sine, cosine, tangent and cotangent of “large” angles.
The considered property of sine, cosine, tangent and cotangent is sometimes called the property of periodicity.
Properties of sines, cosines, tangents and cotangents of opposite angles
Let A 1 be the point obtained by rotating the initial point A(1, 0) around point O by an angle α, and point A 2 be the result of rotating point A by an angle −α, opposite to angle α.
The property of sines, cosines, tangents and cotangents of opposite angles is based on a fairly obvious fact: the points A 1 and A 2 mentioned above either coincide (at) or are located symmetrically relative to the Ox axis. That is, if point A 1 has coordinates (x, y), then point A 2 will have coordinates (x, −y). From here, using the definitions of sine, cosine, tangent and cotangent, we write the equalities and .
Comparing them, we come to relationships between sines, cosines, tangents and cotangents of opposite angles α and −α of the form.
This is the property under consideration in the form of formulas.
Let's give examples of using this property. For example, the equalities and 
.
It remains only to note that the property of sines, cosines, tangents and cotangents of opposite angles, like the previous property, is often used when calculating the values of sine, cosine, tangent and cotangent, and allows you to completely avoid negative angles.
References.
- Algebra: Textbook for 9th grade. avg. school/Yu. N. Makarychev, N. G. Mindyuk, K. I. Neshkov, S. B. Suvorova; Ed. S. A. Telyakovsky. - M.: Education, 1990. - 272 pp.: ill. - ISBN 5-09-002727-7
- Algebra and the beginning of analysis: Proc. for 10-11 grades. general education institutions / A. N. Kolmogorov, A. M. Abramov, Yu. P. Dudnitsyn and others; Ed. A. N. Kolmogorov. - 14th ed. - M.: Education, 2004. - 384 pp.: ill. - ISBN 5-09-013651-3.
- Bashmakov M. I. Algebra and the beginnings of analysis: Textbook. for 10-11 grades. avg. school - 3rd ed. - M.: Education, 1993. - 351 p.: ill. - ISBN 5-09-004617-4.
- Gusev V. A., Mordkovich A. G. Mathematics (a manual for those entering technical schools): Proc. allowance.- M.; Higher school, 1984.-351 p., ill.
Sinus numbers A is called the ordinate of the point representing this number on the number circle. Sine of angle in A radian is called the sine of a number A.
Sinus- number function x. Her domain of definition
Sine Range- segment from -1 to 1 , since any number of this segment on the ordinate axis is a projection of any point on the circle, but no point outside this segment is a projection of any of these points.
Sine period
Sine sign:
1. sine is equal to zero at , where n- any integer;
2. sine is positive at , where n- any integer;
3. sine is negative when
Where n- any integer.
Sinus- function odd x And -x, then their ordinates - sines - will also turn out to be opposite. That is 
for anyone x.
1. Sine increases on segments 
, Where n- any integer.
2. Sine decreases on the segment 
, Where n- any integer.
At 
;
at 
.
Cosine
Cosine numbers A The abscissa of the point representing this number on the number circle is called. Cosine of the angle in A radian is called the cosine of a number A.
Cosine- function of number. Her domain of definition- the set of all numbers, since for any number you can find the ordinate of the point representing it.
Cosine Range- segment from -1 to 1 , since any number of this segment on the x-axis is a projection of any point on the circle, but no point outside this segment is a projection of any of these points.
Cosine period equal to . After all, every time the position of the point representing the number is exactly repeated.
Cosine sign:
1. cosine is equal to zero at , where n- any integer;
2. cosine is positive when 
, Where n- any integer;
3. cosine is negative when 
, Where n- any integer.
Cosine- function even. Firstly, the domain of definition of this function is the set of all numbers, and therefore is symmetrical with respect to the origin. And secondly, if we set aside two opposite numbers from the beginning: x And -x, then their abscissas - cosines - will be equal. That is
for anyone x.
1. Cosine increases on segments 
, Where n- any integer.
2. Cosine decreases on segments 
, Where n- any integer.
at ;
at 
.
Tangent
Tangent of a number is called the ratio of the sine of this number to the cosine of this number: .
Tangent angle in A radian is the tangent of a number A.
Tangent- function of number. Her domain of definition- the set of all numbers whose cosine is not equal to zero, since there are no other restrictions in determining the tangent. And since the cosine is equal to zero at , then 
, Where .
Tangent range
Tangent period x(not equal), differing from each other by , and draw a straight line through them, then this straight line will pass through the origin of coordinates and intersect the line of tangents at some point t. So it turns out that , that is, the number is the period of the tangent.
Tangent sign: tangent is the ratio of sine to cosine. So he
1. is equal to zero when the sine is zero, that is, when , where n- any integer.
2. positive when sine and cosine have the same signs. This happens only in the first and third quarters, that is, when 
, Where A- any integer.
3. negative when sine and cosine have different signs. This happens only in the second and fourth quarters, that is, when 
, Where A- any integer.
Tangent- function odd. Firstly, the domain of definition of this function is symmetrical relative to the origin. And secondly, 
. Due to the oddness of the sine and the evenness of the cosine, the numerator of the resulting fraction is equal to , and its denominator is equal to , which means that this fraction itself is equal to .
So it turned out that .
Means, the tangent increases in each section of its domain of definition, that is, on all intervals of the form 
, Where A- any integer.
Cotangent
Cotangent of a number is called the ratio of the cosine of this number to the sine of this number: . Cotangent angle in A radian is called the cotangent of a number A. Cotangent- function of number. Her domain of definition- the set of all numbers whose sine is not equal to zero, since there are no other restrictions in the definition of cotangent. And since the sine is equal to zero at , then where
Cotangent range- the set of all real numbers.
Cotangent period equal to . After all, if we take any two valid values x(not equal), differing from each other by , and draw a straight line through them, then this straight line will pass through the origin of coordinates and intersect the line of cotangents at some point t. So it turns out that , that is, that the number is the period of the cotangent.