Let's say Achilles runs ten times faster than a turtle and is a thousand steps behind it. During the time it takes Achilles to run this distance, the turtle will crawl a hundred steps in the same direction. When Achilles has run a hundred steps, the turtle will crawl ten more steps, and so on. The process will continue indefinitely, Achilles will never catch up with the turtle.
This reasoning came as a logical shock to all subsequent generations. Aristotle, Diogenes, Kant, Hegel, Hilbert ... 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 the scientific community until it succeeded ... mathematical analysis, set theory, new physical and philosophical approaches were involved in the study of the issue; none of them has become a generally accepted solution to the question ..."[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 magnitude to. This transition implies application instead of constants. As far as I understand, the mathematical apparatus of application variable units measurements either have not yet been developed, or they have not been applied to Zeno's aporia. Applying our usual logic leads us into a trap. We, by inertia of thinking, apply constant units of measurement of time to the reciprocal. From a physical point of view, it looks like time dilation until it stops completely at the moment when Achilles is level with the turtle. If time stops, Achilles can no longer overtake the turtle.
If we turn over the logic we are used to, everything falls into place. Achilles flees 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 infinitely quickly catch up with the turtle."
How can you avoid this logical trap? Stay in constant time units and do not go backwards. In Zeno's language, it looks like this:
During the time during which Achilles will run a thousand steps, the turtle will crawl a hundred steps in the same direction. Over the next interval of time, equal to the first, Achilles will run another thousand steps, and the turtle will crawl a hundred steps. Now Achilles is eight hundred steps ahead of the turtle.
This approach adequately describes reality without any logical paradoxes. But it is not complete solution Problems. Einstein's statement about the insuperability of the speed of light is very similar to the Zeno aporia "Achilles and the Turtle". 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 Zeno tells about a flying arrow:
The flying arrow is motionless, since at every moment of time it is at rest, and since it is at rest at every moment of time, it is always at rest.
In this aporia, the logical paradox is overcome very simply - it is enough to clarify that at each moment of time the flying arrow rests at different points in space, which, in fact, is motion. Another point should be noted here. From a single 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 are needed, taken from the same point at different points in time, but it is impossible to determine the distance from them. 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 (of course, you still need additional data for calculations, trigonometry will help you). What do I want to turn Special attention, so it is that two points in time and two points in space are different things that should not be confused, because they provide different opportunities for research.
Wednesday, 4 July 2018
The distinction between set and multiset is very well documented in Wikipedia. We look.
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". Such logic of absurdity will never be understood by rational beings. This is the level of talking parrots and trained monkeys, who lack intelligence from the word "completely". Mathematicians act as ordinary trainers, preaching their absurd ideas to us.
Once the engineers who built the bridge were in a boat under the bridge during the tests of the bridge. If the bridge collapsed, the incompetent engineer died under the rubble of his creation. If the bridge could withstand the load, a talented engineer would build other bridges.
No matter how mathematicians hide behind the phrase "chur, I'm in the house", or rather "mathematics is studying abstract concepts", there is one umbilical cord that inextricably connects them with reality. This umbilical cord is money. Let's apply mathematical set theory to the mathematicians themselves.
We studied mathematics very well and now we are sitting at the checkout, giving out salaries. Here comes a mathematician to us for his money. We count the entire amount to him and lay 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 hand the mathematician his “mathematical set of salary”. Let us explain the mathematics that he will receive the rest of the 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: "You can apply it to others, you can not apply it to me!" Further, we will begin to assure us that there are different banknote numbers on bills of the same denomination, which means that they cannot be considered the same elements. Okay, let's count the salary in coins - there are no numbers on the coins. Here the mathematician will start to frantically recall physics: on different coins there is different amount dirt, crystal structure and arrangement of atoms for each coin is unique ...
And now I have the most interest Ask: where is the line beyond which the elements of the multiset turn into elements of the set and vice versa? Such a line does not exist - everything is decided by shamans, science did not lie anywhere near here.
Look here. We select football stadiums with the same pitch. The area of the fields is the same, which means we have got 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 is it correct? And here the mathematician-shaman-shuller takes a trump ace out of his sleeve and begins to tell us either about the set or about the 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 "thinkable as not a single whole" or "not thinkable as a whole."
Sunday, 18 March 2018
The sum of the digits of the 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 is why they are shamans in order to teach their descendants their skills and wisdom, otherwise shamans will simply die out.
Need proof? Open Wikipedia and try to find the Sum of Digits of a Number page. It 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 - it is elementary.
Let's see 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 should be done in order to find the sum of the digits of this number? Let's go through all the steps in order.
1. We write down the number on a piece of paper. What have we done? We have converted the number to the graphic symbol of the number. This is not a mathematical operation.
2. We cut one resulting picture into several pictures containing separate numbers. Cutting a picture is not a mathematical operation.
3. Convert individual graphic symbols to numbers. This is not a mathematical operation.
4. Add up the resulting numbers. Now that's mathematics.
The sum of the digits of 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 systems reckoning, 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 do not 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 look at every step under a microscope, we have already done that. Let's see 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 for the fact that. A question for mathematicians: how is something that is not a number designated in mathematics? What, for mathematicians, nothing but numbers exists? For shamans, I can allow this, but for scientists - no. Reality is not all 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 action does not depend on the magnitude of the number, the unit of measurement used and on who performs this action.
Ouch! Isn't this a women's toilet?
- Young woman! This is a laboratory for the study of the indiscriminate holiness of souls during the ascension to heaven! Halo on top and arrow pointing up. What other toilet?
Female ... The nimbus above and the down arrow is male.
If a piece of design art like this flashes 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 so that in a pooping person (one picture), I can see minus four degrees (a composition of several pictures: minus sign, number four, degrees designation). And I don’t think this girl is a fool who doesn’t know physics. She just has a stereotype of perception of graphic images. And mathematicians constantly teach us this. 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 the number and the letter as one graphic symbol.
At the very beginning of this article, we examined the concept of trigonometric functions. Their main purpose is to study the basics of trigonometry and the study of periodic processes. And we drew a trigonometric circle for a reason, because in most cases trigonometric functions are defined as the ratio of the sides of a triangle or its specific segments in the unit circle. I also mentioned the undeniably great importance of trigonometry in modern life. But science does not stand still, as a result, we can significantly expand the scope of trigonometry and transfer its provisions to real, and sometimes complex numbers.
Trigonometry formulas are of several types. Let's consider them in order.
Ratios of trigonometric functions of the same angle
Expressions of trigonometric functions through each other
(The choice of the sign in front of the root is determined by which of the quarters of the circle is the corner?)
The following are the formulas for adding and subtracting angles:
Double, triple and half angle formulas.
Note that they all follow from the previous formulas.
Trigonometric conversion formulas:
Here we come to the consideration of such a concept as the main trigonometric identities .
Trigonometric identity is an equality that consists of trigonometric ratios and which is satisfied for all values of the angles that are included in it.
Consider the most important trigonometric identities and their proofs:
The first identity follows from the very definition of tangent.
Let's take right triangle in which there is sharp corner x at vertex A.
To prove the identities, it is necessary to use the Pythagorean theorem:
(BC) 2 + (AC) 2 = (AB) 2
Now we divide by (AB) 2 both sides of the equality and remembering the definitions of sin and cos of the angle, we get the second identity:
(ВС) 2 / (AB) 2 + (AC) 2 / (AB) 2 = 1
sin x = (BC) / (AB)
cos x = (AC) / (AB)
sin 2 x + cos 2 x = 1
To prove the third and fourth identities, we use the previous proof.
To do this, we divide both sides of the second identity by cos 2 x:
sin 2 x / cos 2 x + cos 2 x / cos 2 x = 1 / cos 2 x
sin 2 x / cos 2 x + 1 = 1 / cos 2 x
Based on the first identity tg x = sin x / cos x we get the third:
1 + tg 2 x = 1 / cos 2 x
Now we divide the second identity by sin 2 x:
sin 2 x / sin 2 x + cos 2 x / sin 2 x = 1 / sin 2 x
1+ cos 2 x / sin 2 x = 1 / sin 2 x
cos 2 x / sin 2 x is nothing but 1 / tan 2 x, so we get the fourth identity:
1 + 1 / tg 2 x = 1 / sin 2 x
It's time to remember the sum theorem inner corners triangle, which says that the sum of the angles of the triangle = 180 0. It turns out that at the vertex B of the triangle there is an angle, the value of which is 180 0 - 90 0 - x = 90 0 - x.
Again, recall the definitions for sin and cos and obtain the fifth and sixth identities:
sin x = (BC) / (AB)
cos (90 0 - x) = (BC) / (AB)
cos (90 0 - x) = sin x
Now let's do the following:
cos x = (AC) / (AB)
sin (90 0 - x) = (AC) / (AB)
sin (90 0 - x) = cos x
As you can see, everything is elementary here.
There are other identities that are used to solve mathematical identities, I will give them simply in the form reference information, because they all stem from the above.
sin 2x = 2sin x * cos x
cos 2x = cos 2x -sin 2x = 1-2sin 2x = 2cos 2x -1
tg 2x = 2tgx / (1 - tg 2 x)
сtg 2x = (сtg 2 x - 1) / 2сtg x
sin3x = 3sin x - 4sin 3 x
cos3x = 4cos 3x - 3cosx
tg 3x = (3tgx - tg 3 x) / (1 - 3tg 2 x)
сtg 3x = (сtg 3 x - 3сtg x) / (3сtg 2 x - 1)
Trigonometric identities- these are equalities that establish a relationship between the sine, cosine, tangent and cotangent of one angle, which allows you to find any of these functions, provided that any other is known.
tg \ alpha = \ frac (\ sin \ alpha) (\ cos \ alpha), \ enspace ctg \ alpha = \ frac (\ cos \ alpha) (\ sin \ alpha)
tg \ alpha \ cdot ctg \ alpha = 1
This identity says that the sum of the square of the sine of one angle and the square of the cosine of one angle is equal to one, which in practice makes it possible to calculate the sine of one angle when its cosine is known and vice versa.
When converting trigonometric expressions, this identity is very often used, which allows you to replace the sum of the squares of the cosine and sine of one angle with a unit and also perform the replacement operation in the reverse order.
Finding tangent and cotangent in terms of sine and cosine
tg \ alpha = \ frac (\ sin \ alpha) (\ cos \ alpha), \ enspace
These identities are formed from the definitions of sine, cosine, tangent and cotangent. After all, if you look at it, then by definition the ordinate of y is the sine, and the abscissa of x is the cosine. Then the tangent will be equal to the ratio \ frac (y) (x) = \ frac (\ sin \ alpha) (\ cos \ alpha) and the ratio \ frac (x) (y) = \ frac (\ cos \ alpha) (\ sin \ alpha)- will be a cotangent.
We add that only for such angles \ alpha for which the trigonometric functions included in them make sense will the identities hold, ctg \ alpha = \ frac (\ cos \ alpha) (\ sin \ alpha).
For example: tg \ alpha = \ frac (\ sin \ alpha) (\ cos \ alpha) is valid for angles \ alpha that are different from \ frac (\ pi) (2) + \ pi z, a ctg \ alpha = \ frac (\ cos \ alpha) (\ sin \ alpha)- for an angle \ alpha other than \ pi z, z - is an integer.
Relationship between tangent and cotangent
tg \ alpha \ cdot ctg \ alpha = 1
This identity is valid only for angles \ alpha that are different from \ frac (\ pi) (2) z... Otherwise, either cotangent or tangent will not be specified.
Based on the above points, we find that tg \ alpha = \ frac (y) (x), a ctg \ alpha = \ frac (x) (y)... Hence it follows that tg \ alpha \ cdot ctg \ alpha = \ frac (y) (x) \ cdot \ frac (x) (y) = 1... Thus, the tangent and cotangent of the same angle at which they make sense are reciprocal numbers.
Dependencies between tangent and cosine, cotangent and sine
tg ^ (2) \ alpha + 1 = \ frac (1) (\ cos ^ (2) \ alpha)- the sum of the square of the tangent of the angle \ alpha and 1, is equal to the inverse square of the cosine of this angle. This identity is valid for all \ alpha different from \ frac (\ pi) (2) + \ pi z.
1 + ctg ^ (2) \ alpha = \ frac (1) (\ sin ^ (2) \ alpha)- the sum of 1 and the square of the cotangent of the angle \ alpha, is equal to the inverse square of the sine of the given angle. This identity is valid for any \ alpha other than \ pi z.
Examples with solutions to problems on the use of trigonometric identities
Example 1
Find \ sin \ alpha and tg \ alpha if \ cos \ alpha = - \ frac12 and \ frac (\ pi) (2)< \alpha < \pi ;
Show solution
Solution
The \ sin \ alpha and \ cos \ alpha functions are bound by a formula \ sin ^ (2) \ alpha + \ cos ^ (2) \ alpha = 1... Substituting into this formula \ cos \ alpha = - \ frac12, we get:
\ sin ^ (2) \ alpha + \ left (- \ frac12 \ right) ^ 2 = 1
This equation has 2 solutions:
\ sin \ alpha = \ pm \ sqrt (1- \ frac14) = \ pm \ frac (\ sqrt 3) (2)
By condition \ frac (\ pi) (2)< \alpha < \pi ... In the second quarter, the sine is positive, therefore \ sin \ alpha = \ frac (\ sqrt 3) (2).
In order to find tg \ alpha, we use the formula tg \ alpha = \ frac (\ sin \ alpha) (\ cos \ alpha)
tg \ alpha = \ frac (\ sqrt 3) (2): \ frac12 = \ sqrt 3
Example 2
Find \ cos \ alpha and ctg \ alpha if and \ frac (\ pi) (2)< \alpha < \pi .
Show solution
Solution
Substituting into the formula \ sin ^ (2) \ alpha + \ cos ^ (2) \ alpha = 1 conditionally given number \ sin \ alpha = \ frac (\ sqrt3) (2), we get \ left (\ frac (\ sqrt3) (2) \ right) ^ (2) + \ cos ^ (2) \ alpha = 1... This equation has two solutions \ cos \ alpha = \ pm \ sqrt (1- \ frac34) = \ pm \ sqrt \ frac14.
By condition \ frac (\ pi) (2)< \alpha < \pi ... In the second quarter, the cosine is negative, so \ cos \ alpha = - \ sqrt \ frac14 = - \ frac12.
In order to find ctg \ alpha, use the formula ctg \ alpha = \ frac (\ cos \ alpha) (\ sin \ alpha)... We know the corresponding values.
ctg \ alpha = - \ frac12: \ frac (\ sqrt3) (2) = - \ frac (1) (\ sqrt 3).
Trigonometric functions- Request "sin" is redirected here; see also other meanings. The "sec" request is redirected here; see also other meanings. The Sinus request is redirected here; see also other meanings ... Wikipedia
Tan
Rice. 1 Graphs of trigonometric functions: sine, cosine, tangent, secant, cosecant, cotangent Trigonometric functions are the form of elementary functions. Usually they include sine (sin x), cosine (cos x), tangent (tg x), cotangent (ctg x), ... ... Wikipedia
Cosine- Rice. 1 Graphs of trigonometric functions: sine, cosine, tangent, secant, cosecant, cotangent Trigonometric functions are the form of elementary functions. Usually they include sine (sin x), cosine (cos x), tangent (tg x), cotangent (ctg x), ... ... Wikipedia
Cotangent- Rice. 1 Graphs of trigonometric functions: sine, cosine, tangent, secant, cosecant, cotangent Trigonometric functions are the form of elementary functions. Usually they include sine (sin x), cosine (cos x), tangent (tg x), cotangent (ctg x), ... ... Wikipedia
Secant- Rice. 1 Graphs of trigonometric functions: sine, cosine, tangent, secant, cosecant, cotangent Trigonometric functions are the form of elementary functions. Usually they include sine (sin x), cosine (cos x), tangent (tg x), cotangent (ctg x), ... ... Wikipedia
History of trigonometry- Geodetic measurements (XVII century) ... Wikipedia
Half Angle Tangent Formula- In trigonometry, the half-angle tangent formula relates the half-angle tangent to the trigonometric functions of the full angle: Various variations this formula look like this ... Wikipedia
Trigonometry- (from the Greek τρίγονο (triangle) and the Greek μετρειν (to measure), that is, the measurement of triangles) a branch of mathematics in which trigonometric functions and their applications to geometry are studied. This term first appeared in 1595 as ... ... Wikipedia
Solving triangles- (Latin solutio triangulorum) historical term, meaning the solution to the main trigonometric problem: from the known data about the triangle (sides, angles, etc.), find the rest of its characteristics. The triangle can be located on ... ... Wikipedia
Books
- A set of tables. Algebra and the beginning of analysis. Grade 10. 17 tables + methodology,. The tables are printed on thick polygraphic cardboard 680 x 980 mm in size. Includes a brochure with guidelines for the teacher. Educational album of 17 sheets. ... Buy for 3944 rub
- Tables of integrals and other mathematical formulas, Dwight G.B .. The tenth edition of the famous handbook contains very detailed tables of indefinite and definite integrals, as well as a large number of other mathematical formulas: series expansions, ...
The relationships between the main trigonometric functions - sine, cosine, tangent and cotangent - are set trigonometric formulas... And since there are a lot of connections between trigonometric functions, this explains the abundance of trigonometric formulas. Some formulas connect trigonometric functions of the same angle, others - functions of a multiple angle, others - allow you to lower the degree, the fourth - to express all functions through the tangent of a half angle, etc.
In this article, we will list all the main trigonometric formulas, which are sufficient to solve the overwhelming majority of trigonometry problems. For ease of memorization and use, we will group them by purpose and enter them into tables.
Page navigation.
Basic trigonometric identities
Basic trigonometric identities set the relationship between sine, cosine, tangent and cotangent of one angle. They follow from the definitions of sine, cosine, tangent and cotangent, as well as the concept of the unit circle. They allow you to express one trigonometric function in terms of any other.
For a detailed description of these trigonometry formulas, their derivation and examples of application, see the article.
Casting formulas
Casting formulas follow from the properties of sine, cosine, tangent and cotangent, that is, they reflect the property of periodicity of trigonometric functions, the property of symmetry, as well as the property of shift by given angle... These trigonometric formulas allow from working with arbitrary angles go to work with angles ranging from zero to 90 degrees.
The rationale for these formulas, the mnemonic rule for memorizing them and examples of their application can be studied in the article.
Addition formulas
Trigonometric addition formulas show how the trigonometric functions of the sum or difference of two angles are expressed in terms of the trigonometric functions of these angles. These formulas serve as the basis for deriving the following trigonometric formulas.
Formulas for double, triple, etc. corner
Formulas for double, triple, etc. angle (also called multiple angle formulas) show how the trigonometric functions of double, triple, etc. angles () are expressed in terms of trigonometric functions of a single angle. Their derivation is based on addition formulas.
More detailed information is collected in the article formulas for double, triple, etc. corner.
Half angle formulas
Half angle formulas show how trigonometric functions of a half angle are expressed in terms of the cosine of an integer angle. These trigonometric formulas follow from the double angle formulas.
Their conclusion and examples of application can be found in the article.
Degree reduction formulas
Trigonometric Degree Reduction Formulas are designed to facilitate the transition from natural degrees of trigonometric functions to sines and cosines in the first degree, but multiple angles. In other words, they allow you to lower the degrees of trigonometric functions to the first.
Sum and difference formulas for trigonometric functions
main destination formulas for the sum and difference of trigonometric functions is to go to the product of functions, which is very useful when simplifying trigonometric expressions. These formulas are also widely used to solve trigonometric equations, since they allow you to factor the sum and difference of sines and cosines.
Formulas for the product of sines, cosines and sine by cosine
The transition from the product of trigonometric functions to the sum or difference is carried out using the formulas for the product of sines, cosines and sine by cosine.
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