Basic properties of trigonometric functions: even, odd, periodicity. Signs of the values ​​of trigonometric functions by quarters. Positive and negative angles in trigonometry

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, to reach 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 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 at a 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? Stay in constant units measurements of time and do not go to reciprocal quantities. 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 it is not complete solution Problems. 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 a car, you need two photographs taken from different points in space at one point in time, but from them you cannot determine the fact of movement (of course, you still need additional data for calculations, trigonometry will help you). What I want to draw special attention to 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, 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.” Such absurd logic sentient beings never understand. 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: on different coins there is different quantities dirt, crystal structure and atomic arrangement of each coin is unique...

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. We 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” taught by shamans that mathematicians use. But that is 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, it means it has nothing to do with mathematics.

What is real mathematics? This is when the result mathematical operation does not depend on the size of the number, the unit of measurement used and who performs the 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: a minus sign, the number four, a designation of degrees). 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.

Lesson type: systematization of knowledge and intermediate control.

Equipment: trigonometric circle, tests, task cards.

Lesson objectives: systematize the studied theoretical material according to the definitions of sine, cosine, tangent of an angle; check the degree of knowledge acquisition on this topic and application in practice.

Tasks:

• Generalize and consolidate the concepts of sine, cosine and tangent of an angle.
• Form a comprehensive understanding of trigonometric functions.
• To promote students’ desire and need to study trigonometric material; cultivate a culture of communication, the ability to work in groups and the need for self-education.

“Whoever does and thinks for himself from a young age,
Then it becomes 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 announces the topic, goals and objectives of the lesson.

II. Updating knowledge (frontal work with the class)

1) Work in groups on tasks:

1. Formulate the definition of sin angle.

– What signs does sin α have in each coordinate quadrant?
– At what values ​​does the expression sin α make sense, and what values ​​can it take?

2. The second group is the same questions for cos α.

3. The third group prepares answers to the same questions tg α and ctg α.

At this time, three students work independently at the board using cards (representatives of different groups).

Card No. 1.

Practical work.
Using the unit circle, calculate the values ​​of sin α, cos α and tan α for angles of 50, 210 and – 210.

Card No. 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 series of numbers is proposed: 1; 1.2; 3; , 0, , – 1. Among them there are redundant ones. What property of sin α or cos α can these numbers express (Can sin α or cos α take these values).
b) Does the expression make sense: cos (–); sin 2; tg 3: ctg (– 5); ; ctg0;
cotg(–π). Why?
c) Is there a smallest and highest value sin or cos, tg, ctg.
d) Is it true?
1) α = 1000 is the angle of the second quarter;
2) α = – 330 is the angle of the IV quarter.
e) The numbers correspond to the same point on the unit circle.

3) Work at the board

No. 567 (2; 4) – Find the value of the expression
No. 583 (1-3) Determine the sign of the expression

Homework: table in notebook. No. 567(1, 3) No. 578

III. Acquiring additional knowledge. Trigonometry in the palm of your hand

Teacher: It turns out that the values ​​of the sines and cosines of angles are “located” in the palm of your hand. Reach out your hand (either hand) and spread your fingers as far apart as possible (as in the poster). One student is invited. We measure the angles between our fingers.
Take a triangle where there is an angle of 30, 45 and 60 90 and apply the vertex of the angle to the hillock of the Moon in the palm of your 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 there is an angle of 90 between the little finger and the thumb, 30 between the little and ring fingers, 45 between the little and middle fingers, and 60 between the little and index fingers. And this is true for all people without exception.

little finger No. 0 – corresponds to 0,
unnamed No. 1 – corresponds to 30,
average No. 2 – corresponds to 45,
index number 3 – corresponds to 60,
large No. 4 – corresponds to 90.

Thus, we have 4 fingers on our hand and remember the formula:

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 do not change, but are counted from the thumb). A physical pause associated with the signs sin α or cos α.

IV. Checking your knowledge of knowledge and skills

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 turning through an angle of 50?
2) Find the value of the expression: 4cos 60 – 3sin 90.
3) Which number 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 turning by an angle of 10.
2) Find the value of the expression: 4cos 90 – 6sin 30.
3) Which number 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 number is less than zero: sin 40, cos (– 10), tan 210, sin 140.
3) Which quarter angle is angle α, if sin α > 0, cos α< 0.

Option 4

1) Find the value of the expression: tg 60 – 6ctg 90.
2) Which number is less than zero: sin(– 10), cos 140, tg 250, cos 250.
3) Which quadrant angle is angle α, if ctg α< 0, cos α> 0.

(the key word is trigonometry)

V. Information from the history of trigonometry

Teacher: Trigonometry is a fairly important branch of mathematics for human life. Modern look trigonometry was introduced by the greatest mathematician of the 18th century, Leonhard Euler - Swiss by birth long years worked in Russia and was a member of the St. Petersburg Academy of Sciences. He introduced the 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 through Yakovlev’s book “Leonard Euler”.

(Message from the guys on this topic)

VI. Summing up the lesson

Game "Tic Tac Toe"

The two most active students are participating. They are supported by groups. The solutions to the tasks are written down 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 the angle 300 in radians
4) What is the largest and smallest value the expression can have: 1+ sin α;
5) Determine the sign of the expression: sin 260, cos 300.
6) In which quarter of the number circle is the point located?
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 9th grade, and even now, do you use the concepts of sin α, cos α; tg α; ctg α and for what purpose?

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

Sine Range- segment from -1 before 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- the set of all numbers, since for any number you can find the ordinate of the point representing it.

Cosine Range- segment from -1 before 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- 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- 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.

This article will look at three basic properties of trigonometric functions: sine, cosine, tangent and cotangent.

The first property is the sign of the function depending on which quarter of the unit circle the angle α belongs to. The second property is periodicity. According to this property, the tigonometric function does not change its value when the angle changes by an integer number of revolutions. The third property determines how the values ​​of the functions sin, cos, tg, ctg change at opposite angles α and - α.

Yandex.RTB R-A-339285-1

Often in a mathematical text or in the context of a problem you can find the phrase: “the angle of the first, second, third or fourth coordinate quarter.” What it is?

Let's turn to the unit circle. It is divided into four quarters. Let's mark the starting point A 0 (1, 0) on the circle and, rotating it around the point O by an angle α, we will get to the point A 1 (x, y). Depending on which quarter the point A 1 (x, y) lies in, the angle α will be called the angle of the first, second, third and fourth quarter, respectively.

For clarity, here is an illustration.

The angle α = 30° lies in the first quarter. Angle - 210° is the second quarter angle. The 585° angle is the third quarter angle. The angle - 45° is the fourth quarter angle.

In this case, the angles ± 90 °, ± 180 °, ± 270 °, ± 360 ° do not belong to any quarter, since they lie on the coordinate axes.

Now consider the signs that sine, cosine, tangent and cotangent take, depending on which quadrant the angle lies in.

To determine the signs of the sine by quarters, recall the definition. Sine is the ordinate of point A 1 (x, y). The figure shows that in the first and second quarters it is positive, and in the third and quadruple it is negative.

Cosine is the abscissa of point A 1 (x, y). In accordance with this, we determine the signs of the cosine on the circle. The cosine is positive in the first and fourth quarters, and negative in the second and third quarters.

To determine the signs of tangent and cotangent by quarters, we also recall the definitions of these trigonometric functions. Tangent is the ratio of the ordinate of a point to the abscissa. This means, according to the rule for dividing numbers with different signs, when the ordinate and abscissa have the same signs, the sign of the tangent on the circle will be positive, and when the ordinate and abscissa have different signs, it will be negative. The cotangent signs for quarters are determined in a similar way.

Important to remember!

1. The sine of angle α has a plus sign in the 1st and 2nd quarters, a minus sign in the 3rd and 4th quarters.
2. The cosine of angle α has a plus sign in the 1st and 4th quarters, a minus sign in the 2nd and 3rd quarters.
3. The tangent of the angle α has a plus sign in the 1st and 3rd quarters, a minus sign in the 2nd and 4th quarters.
4. The cotangent of angle α has a plus sign in the 1st and 3rd quarters, a minus sign in the 2nd and 4th quarters.

Periodicity property

The property of periodicity is one of the most obvious properties of trigonometric functions.

Periodicity property

When the angle changes by an integer number of full revolutions, the values ​​of the sine, cosine, tangent and cotangent of the given angle remain unchanged.

Indeed, when the angle changes by an integer number of revolutions, we will always get from the initial point A on the unit circle to point A 1 with the same coordinates. Accordingly, the values ​​of sine, cosine, tangent and cotangent will not change.

Mathematically, this property is written as follows:

sin α + 2 π z = sin α cos α + 2 π z = cos α t g α + 2 π z = t g α c t g α + 2 π z = c t g α

How is this property used in practice? The periodicity property, like reduction formulas, is often used to calculate the values ​​of sines, cosines, tangents and cotangents of large angles.

Let's give examples.

sin 13 π 5 = sin 3 π 5 + 2 π = sin 3 π 5

t g (- 689 °) = t g (31 ° + 360 ° (- 2)) = t g 31 ° t g (- 689 °) = t g (- 329 ° + 360 ° (- 1)) = t g (- 329 °)

Let's look again at the unit circle.

Point A 1 (x, y) is the result of rotating the initial point A 0 (1, 0) around the center of the circle by angle α. Point A 2 (x, - y) is the result of rotating the starting point by an angle - α.

Points A 1 and A 2 are symmetrical about the abscissa axis. In the case where α = 0 °, ± 180 °, ± 360 ° points A 1 and A 2 coincide. Let one point have coordinates (x, y) and the second - (x, - y). Let us recall the definitions of sine, cosine, tangent, cotangent and write:

sin α = y , cos α = x , t g α = y x , c t g α = x y sin - α = - y , cos - α = x , t g - α = - y x , c t g - α = x - y

This implies the property of sines, cosines, tangents and cotangents of opposite angles.

Property of sines, cosines, tangents and cotangents of opposite angles

sin - α = - sin α cos - α = cos α t g - α = - t g α c t g - α = - c t g α

According to this property, the equalities are true

sin - 48 ° = - sin 48 ° , c t g π 9 = - c t g - π 9 , cos 18 ° = cos - 18 °

The considered property is often used when solving practical problems in cases where you need to get rid of negative angle signs in the arguments of trigonometric functions.

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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| - length of the arc of a circle with center at point A.
α is the angle expressed in radians.

Tangent ( tan α) 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 side |BC| to the length of the adjacent leg |AB| .

Cotangent ( ctg α) 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.

In Western literature, tangent is denoted as follows:
.
;
;
.

Graph of the tangent function, y = tan x

Cotangent

Where n- whole.

In Western literature, cotangent is denoted as follows:
.
The following notations are also accepted:
;
;
.

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 tangent and cotangent functions are odd.

Areas of definition and values, increasing, decreasing

The tangent and cotangent functions are continuous in their domain of definition (see proof of continuity). The main properties of tangent and cotangent are presented in the table ( n- whole).

Formulas

Expressions using sine and cosine

; ;
; ;
;

Formulas for tangent and cotangent from sum and difference

The remaining formulas are easy to obtain, for example

Product of tangents

Formula for the sum and difference of tangents

This table presents the values ​​of tangents and cotangents for certain values ​​of the argument.

Expressions using complex numbers

Expressions through hyperbolic functions

;
;

Derivatives

; .

.
Derivative of the nth order with respect to the variable x of the function:
.
Deriving formulas for tangent > > > ; for cotangent > > >

Integrals

Series expansions

To obtain the expansion of the tangent in powers of x, you need to take several terms of the expansion in a power series for the functions sin x And cos x and divide these polynomials by each other, . In this case it turns out following formulas.

At .

at .
Where Bn- Bernoulli numbers. They are determined either from the recurrence relation:
;
;
Where .
Or according to Laplace's formula:

Inverse functions

The inverse functions of tangent and cotangent are arctangent and arccotangent, respectively.

Arctangent, arctg

, Where n- whole.

Arccotangent, arcctg

, Where n- whole.

References:
I.N. Bronstein, K.A. Semendyaev, Handbook of mathematics for engineers and college students, “Lan”, 2009.
G. Korn, Handbook of Mathematics for Scientists and Engineers, 2012.