Number circle trigonometry. Trigonometry. Unit circle. you can get acquainted with functions and derivatives

Trigonometry, as a science, originated in the Ancient East. First trigonometric ratios were developed by astronomers to create an accurate calendar and navigate by the stars. These calculations related to spherical trigonometry, while in school course study the ratios of sides and angles of a plane triangle.

Trigonometry is a branch of mathematics that deals with the properties of trigonometric functions and the relationship between the sides and angles of triangles.

During the heyday of culture and science in the 1st millennium AD, knowledge spread from the Ancient East to Greece. But the main discoveries of trigonometry are the merit of the men of the Arab Caliphate. In particular, the Turkmen scientist al-Marazwi introduced functions such as tangent and cotangent, and compiled the first tables of values for sines, tangents and cotangents. The concepts of sine and cosine were introduced by Indian scientists. Trigonometry received a lot of attention in the works of such great figures of antiquity as Euclid, Archimedes and Eratosthenes.

Basic quantities of trigonometry

Basic trigonometric functions numeric argument– these are sine, cosine, tangent and cotangent. Each of them has its own graph: sine, cosine, tangent and cotangent.

The formulas for calculating the values of these quantities are based on the Pythagorean theorem. It is better known to schoolchildren in the formulation: “Pythagorean pants, equal in all directions,” since the proof is given using the example of an isosceles right triangle.

Sine, cosine and other dependencies establish the relationship between sharp corners and sides of any right triangle. Let us present formulas for calculating these quantities for angle A and trace the relationships between trigonometric functions:

As you can see, tg and ctg are inverse functions. If we imagine leg a as the product of sin A and hypotenuse c, and leg b as cos A * c, we get following formulas for tangent and cotangent:

Trigonometric circle

Graphically, the relationship between the mentioned quantities can be represented as follows:

The circle, in this case, represents all possible values of the angle α - from 0° to 360°. As can be seen from the figure, each function takes a negative or positive value depending on the angle. For example, sin α will have a “+” sign if α belongs to the 1st and 2nd quarters of the circle, that is, it is in the range from 0° to 180°. For α from 180° to 360° (III and IV quarters), sin α can only be a negative value.

Let's try to build trigonometric tables for specific angles and find out the value of the quantities.

Values of α equal to 30°, 45°, 60°, 90°, 180° and so on are called special cases. The values of trigonometric functions for them are calculated and presented in the form of special tables.

These angles were not chosen at random. The designation π in the tables is for radians. Rad is the angle at which the length of a circle's arc corresponds to its radius. This value was introduced in order to establish a universal dependence; when calculating in radians, the actual length of the radius in cm does not matter.

Angles in tables for trigonometric functions correspond to radian values:

So, it is not difficult to guess that 2π is a complete circle or 360°.

Properties of trigonometric functions: sine and cosine

In order to consider and compare the basic properties of sine and cosine, tangent and cotangent, it is necessary to draw their functions. This can be done in the form of a curve located in a two-dimensional coordinate system.

Consider the comparative table of properties for sine and cosine:

Sine wave	Cosine
y = sin x	y = cos x
ODZ [-1; 1]	ODZ [-1; 1]
sin x = 0, for x = πk, where k ϵ Z	cos x = 0, for x = π/2 + πk, where k ϵ Z
sin x = 1, for x = π/2 + 2πk, where k ϵ Z	cos x = 1, at x = 2πk, where k ϵ Z
sin x = - 1, at x = 3π/2 + 2πk, where k ϵ Z	cos x = - 1, for x = π + 2πk, where k ϵ Z
sin (-x) = - sin x, i.e. the function is odd	cos (-x) = cos x, i.e. the function is even
	the function is periodic, shortest period- 2π
sin x › 0, with x belonging to the 1st and 2nd quarters or from 0° to 180° (2πk, π + 2πk)	cos x › 0, with x belonging to the I and IV quarters or from 270° to 90° (- π/2 + 2πk, π/2 + 2πk)
sin x ‹ 0, with x belonging to the third and fourth quarters or from 180° to 360° (π + 2πk, 2π + 2πk)	cos x ‹ 0, with x belonging to the 2nd and 3rd quarters or from 90° to 270° (π/2 + 2πk, 3π/2 + 2πk)
increases in the interval [- π/2 + 2πk, π/2 + 2πk]	increases on the interval [-π + 2πk, 2πk]
decreases on intervals [π/2 + 2πk, 3π/2 + 2πk]	decreases on intervals
derivative (sin x)’ = cos x	derivative (cos x)’ = - sin x

Determining whether a function is even or not is very simple. It is enough to imagine a trigonometric circle with the signs of trigonometric quantities and mentally “fold” the graph relative to the OX axis. If the signs coincide, the function is even, otherwise it is odd.

The introduction of radians and the listing of the basic properties of sine and cosine waves allow us to present the following pattern:

It is very easy to verify that the formula is correct. For example, for x = π/2, the sine is 1, as is the cosine of x = 0. The check can be done by consulting tables or by tracing function curves for given values.

Properties of tangentsoids and cotangentsoids

The graphs of the tangent and cotangent functions differ significantly from the sine and cosine functions. The values tg and ctg are reciprocals of each other.

Y = tan x.
The tangent tends to the values of y at x = π/2 + πk, but never reaches them.
The smallest positive period of the tangentoid is π.
Tg (- x) = - tg x, i.e. the function is odd.
Tg x = 0, for x = πk.
The function is increasing.
Tg x › 0, for x ϵ (πk, π/2 + πk).
Tg x ‹ 0, for x ϵ (— π/2 + πk, πk).
Derivative (tg x)’ = 1/cos 2 ⁡x.

Let's consider graphic image cotangentoids below in the text.

Main properties of cotangentoids:

Y = cot x.
Unlike the sine and cosine functions, in the tangentoid Y can take on the values of the set of all real numbers.
The cotangentoid tends to the values of y at x = πk, but never reaches them.
The smallest positive period of a cotangentoid is π.
Ctg (- x) = - ctg x, i.e. the function is odd.
Ctg x = 0, for x = π/2 + πk.
The function is decreasing.
Ctg x › 0, for x ϵ (πk, π/2 + πk).
Ctg x ‹ 0, for x ϵ (π/2 + πk, πk).
Derivative (ctg x)’ = - 1/sin 2 ⁡x Correct

Back forward

Attention! Slide previews are for informational purposes only and may not represent all the features of the presentation. If you are interested this work, please download the full version.

Target: teach how to use the unit circle when solving various trigonometric problems.

In a school mathematics course, various options for introducing trigonometric functions are possible. The most convenient and frequently used is the “numerical unit circle”. Its application in the topic “Trigonometry” is very extensive.

Unit circle is used for:

– definitions of sine, cosine, tangent and cotangent of an angle;
– finding the values of trigonometric functions for some values of numerical and angle argument;
– derivation of basic trigonometry formulas;
– derivation of reduction formulas;
– finding the domain of definition and range of values of trigonometric functions;
– determining the periodicity of trigonometric functions;
– determination of parity and oddness of trigonometric functions;
– determination of intervals of increasing and decreasing trigonometric functions;
– determination of intervals of constant sign of trigonometric functions;
– radian measurement of angles;
– finding the values of inverse trigonometric functions;
– solution to the simplest trigonometric equations;
– solving simple inequalities, etc.

Thus, students’ active, conscious mastery of this type of visualization provides undeniable advantages for mastering the “Trigonometry” section of mathematics.

The use of ICT in mathematics teaching lessons makes it easier to master the numerical unit circle. Certainly, interactive board has a wide range of applications, but not all classes have it. If we talk about the use of presentations, there is a wide choice on the Internet, and every teacher can find the most suitable option for their lessons.

What is special about the presentation I am presenting?

This presentation suggests various use cases and is not intended to be a demonstration of a specific lesson in the topic “Trigonometry”. Each slide of this presentation can be used separately, both at the stage of explaining the material, developing skills, and for reflection. When creating this presentation, special attention was paid to its “readability” from a long distance, since the number of students with low vision is constantly growing. The color scheme has been thought out, logically related objects are united by a single color. The presentation is animated in such a way that the teacher can comment on a fragment of the slide and the student can ask a question. Thus, this presentation is a kind of “moving” tables. The last slides are not animated and are used to test mastery of the material while solving trigonometric tasks. The circle on the slides is simplified as much as possible in appearance and is as close as possible to the one depicted on the notebook paper by the students. I consider this condition to be fundamental. It is important for students to form an opinion about the unit circle as an accessible and mobile (although not the only) form of clarity when solving trigonometric tasks.

This presentation will help teachers introduce students to the unit circle in 9th grade geometry lessons when studying the topic “Relationships between the sides and angles of a triangle.” And, of course, it will help expand and deepen the skill of working with the unit circle when solving trigonometric problems for senior students in algebra lessons.

Slides 3, 4 explain the construction of a unit circle; the principle of determining the location of a point on the unit circle in the 1st and 2nd coordinate quarters; transfer from geometric definitions functions sine and cosine (in right triangle) to algebraic on the unit circle.

Slides 5-8 explain how to find the values of trigonometric functions for the main angles of the first coordinate quadrant.

Slides 9-11 explains the signs of functions in coordinate quarters; determination of intervals of constant sign of trigonometric functions.

Slide 12 used to form ideas about positive and negative angle values; familiarization with the concept of periodicity of trigonometric functions.

Slides 13, 14 are used when switching to a radian angle measure.

Slides 15-18 are not animated and are used when solving various trigonometric tasks, consolidating and checking the results of mastering the material.

Title page.
Goal setting.
Construction of a unit circle. Basic values of angles in degrees.
Determination of sine and cosine of an angle on a unit circle.
Table values for sine in ascending order.
Table values for cosine in ascending order.
Table values for tangent in ascending order.
Table values for cotangent in ascending order.
Function signs sin α.
Function signs cos α.
Function signs tan α And ctg α.
Positive and negative values angles on the unit circle.
Radian measure of angle.
Positive and negative angle values in radians on the unit circle.
Various options unit circle to consolidate and check the results of mastering the material.

In the fifth century BC, the ancient Greek philosopher Zeno of Elea formulated his famous aporias, the most famous of which is the “Achilles and the Tortoise” aporia. Here's what it sounds like:

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 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? Remain in constant units of time and do not switch to reciprocal units. 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, the logical paradox is 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.” 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 “screw 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. Apply mathematical theory sets to the 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. 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 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, 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.

Sign on the door He opens the door and says:

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.

If you are already familiar with trigonometric circle , and you just want to refresh your memory of certain elements, or you are completely impatient, then here it is:

Here we will analyze everything in detail step by step.

The trigonometric circle is not a luxury, but a necessity

Trigonometry Many people associate it with an impenetrable thicket. Suddenly, so many values of trigonometric functions, so many formulas pile up... But it’s like, it didn’t work out at the beginning, and... off we go... complete misunderstanding...

It is very important not to give up values of trigonometric functions, - they say, you can always look at the spur with a table of values.

If you constantly look at a table with values trigonometric formulas, let's get rid of this habit!

He will help us out! You will work with it several times, and then it will pop up in your head. How is it better than a table? Yes, in the table you will find a limited number of values, but on the circle - EVERYTHING!

For example, say while looking at standard table of values of trigonometric formulas , what is the sine equal to, say, 300 degrees, or -45.

No way?.. you can, of course, connect reduction formulas... And looking at the trigonometric circle, you can easily answer such questions. And you will soon know how!

And when solving trigonometric equations and inequalities without a trigonometric circle, it’s absolutely nowhere.

Introduction to the trigonometric circle

Let's go in order.

First, let's write out this series of numbers:

And now this:

And finally this one:

Of course, it is clear that, in fact, in first place is , in second place is , and in last place is . That is, we will be more interested in the chain.

But how beautiful it turned out! If something happens, we will restore this “miracle ladder.”

And why do we need it?

This chain is the main values of sine and cosine in the first quarter.

Let us draw a circle of unit radius in a rectangular coordinate system (that is, we take any radius in length, and declare its length to be unit).

From the “0-Start” beam we lay the corners in the direction of the arrow (see figure).

We get the corresponding points on the circle. So, if we project the points onto each of the axes, then we will get exactly the values from the above chain.

Why is this, you ask?

Let's not analyze everything. Let's consider principle, which will allow you to cope with other, similar situations.

Triangle AOB is rectangular and contains . And we know that opposite the angle b lies a leg half the size of the hypotenuse (we have the hypotenuse = the radius of the circle, that is, 1).

This means AB= (and therefore OM=). And according to the Pythagorean theorem