What is the height in an isosceles triangle. Features that make up the elements and properties of an isosceles triangle

Simply put, these are vegetables cooked in water according to a special recipe. I will consider two initial components (vegetable salad and water) and the finished result - borscht. Geometrically, this can be thought of as a rectangle with one side representing lettuce and the other side representing water. The sum of these two sides will represent borscht. The diagonal and area of \u200b\u200bsuch a "borscht" rectangle are purely mathematical concepts and are never used in borscht recipes.

How do lettuce and water turn into borscht mathematically? How can the sum of two segments turn into trigonometry? To understand this, we need linear angle functions.

You won't find anything about linear angle functions in mathematics textbooks. But without them there can be no mathematics. The laws of mathematics, like the laws of nature, work regardless of whether we know about their existence or not.

Linear angle functions are addition laws. See how algebra turns into geometry and geometry turns into trigonometry.

Can linear angle functions be dispensed with? It is possible, because mathematicians still do without them. The trick of mathematicians lies in the fact that they always tell us only about those problems that they themselves know how to solve, and never talk about those problems that they cannot solve. Look. If we know the result of addition and one term, we use subtraction to find the other term. All. We do not know other tasks and cannot solve them. What to do if we only know the result of addition and do not know both terms? In this case, the result of addition must be decomposed into two terms using linear angle functions. Then we ourselves choose what one term can be, and the linear angular functions show what the second term should be so that the result of the addition is exactly what we need. There can be an infinite number of such pairs of terms. IN everyday life we can do just fine without decomposing the sum; subtraction is enough for us. But with scientific research the laws of nature, the decomposition of the sum into terms is very useful.

Another law of addition that mathematicians don't like to talk about (another trick of theirs) requires that the terms have the same units of measurement. For salad, water, and borscht, these can be units of weight, volume, value, or units of measure.

The figure shows two levels of difference for math. The first level is the differences in the field of numbers, which are indicated a, b, c... This is what mathematicians do. The second level is the differences in the area of \u200b\u200bunits of measurement, which are shown in square brackets and indicated by the letter U... This is what physicists do. We can understand the third level - differences in the area of \u200b\u200bthe described objects. Different objects can have the same number of identical units. How important this is, we can see on the example of borscht trigonometry. If we add subscripts to the same designation of units of measurement of different objects, we can say exactly which mathematical value describes a specific object and how it changes over time or in relation to our actions. By letter W I will designate water by letter S I will designate the salad and the letter B - borsch. This is what the linear angular functions for borsch would look like.

If we take some part of the water and some part of the salad, together they will turn into one portion of borscht. Here I suggest you take a break from the borscht and recall your distant childhood. Remember how we were taught to put bunnies and ducks together? It was necessary to find how many animals there would be. What then were we taught to do? We were taught to separate units from numbers and add numbers. Yes, any number can be added to any other number. This is a direct path to the autism of modern mathematics - we do not understand what, it is not clear why, and we very poorly understand how this relates to reality, because of the three levels of difference, mathematics operates only one. It would be more correct to learn how to switch from one measurement unit to another.

Bunnies, ducks and animals can be counted in pieces. One common unit of measurement for different objects allows us to add them together. This is a childish version of the problem. Let's take a look at a similar problem for adults. What happens if you add bunnies and money? There are two possible solutions here.

First option... We determine the market value of the bunnies and add it to the available amount of money. We got the total value of our wealth in monetary terms.

Second option... You can add the number of bunnies to the number of banknotes we have. We will receive the number of movable property in pieces.

As you can see, the same law of addition produces different results. It all depends on what exactly we want to know.

But back to our borscht. Now we can see what will happen for different values \u200b\u200bof the angle of the linear angle functions.

The angle is zero. We have salad, but no water. We cannot cook borscht. The amount of borscht is also zero. This does not mean at all that zero borscht is equal to zero water. Zero borscht can be at zero salad (right angle).

For me personally, this is the main mathematical proof of the fact that. Zero does not change the number when added. This is because the addition itself is impossible if there is only one term and the second term is missing. You can treat this as you like, but remember - all mathematical operations with zero were invented by mathematicians themselves, so discard your logic and stupidly cram definitions invented by mathematicians: "division by zero is impossible", "any number multiplied by zero equals zero" , "for the knockout point zero" and other delirium. It is enough to remember once that zero is not a number, and you will never have a question whether zero is a natural number or not, because such a question generally loses any meaning: how can we consider a number that is not a number. It's like asking what color an invisible color should be. Adding zero to a number is like painting with paint that doesn't exist. We waved with a dry brush and told everyone that "we have painted". But I digress a little.

The angle is greater than zero, but less than forty-five degrees. We have a lot of salad, but not enough water. As a result, we get a thick borscht.

The angle is forty-five degrees. We have equal amounts of water and salad. This is the perfect borscht (forgive me the chefs, it's just math).

The angle is greater than forty-five degrees, but less than ninety degrees. We have a lot of water and little salad. You get liquid borscht.

Right angle. We have water. From the salad, only memories remain, as we continue to measure the angle from the line that once stood for the salad. We cannot cook borscht. The amount of borscht is zero. In that case, hold on and drink water while you have it)))

Here. Something like this. I can tell other stories here that will be more than appropriate here.

Two friends had their shares in the common business. After killing one of them, everything went to the other.

The emergence of mathematics on our planet.

All of these stories are told in the language of mathematics using linear angle functions. Some other time I'll show you the real place of these functions in the structure of mathematics. For now, let's return to the trigonometry of borscht and consider the projections.

saturday, 26 October 2019

wednesday, 7 August 2019

Concluding the conversation about, there is an infinite number to consider. It gave in that the concept of "infinity" acts on mathematicians like a boa constrictor on a rabbit. The trembling horror of infinity robs mathematicians of common sense. Here's an example:

The original source is located. Alpha stands for real number. The equal sign in the above expressions indicates that if you add a number or infinity to infinity, nothing will change, the result will be the same infinity. If we take as an example an infinite set of natural numbers, then the considered examples can be presented in the following form:

For a visual proof of their correctness, mathematicians have come up with many different methods. Personally, I look at all these methods as dancing shamans with tambourines. Essentially, they all boil down to the fact that either some of the rooms are not occupied and new guests are moving in, or that some of the visitors are thrown out into the corridor to make room for guests (very humanly). I presented my view on such decisions in the form of a fantastic story about the Blonde. What is my reasoning based on? Relocating an infinite number of visitors takes an infinite amount of time. After we have vacated the first room for a guest, one of the visitors will always walk along the corridor from his room to the next one until the end of the century. Of course, the time factor can be stupidly ignored, but it will already be from the category "the law is not written for fools." It all depends on what we are doing: adjusting reality to match mathematical theories or vice versa.

What is an "endless hotel"? An endless hotel is a hotel that always has any number of vacant places, no matter how many rooms are occupied. If all the rooms in the endless visitor corridor are occupied, there is another endless corridor with the guest rooms. There will be an infinite number of such corridors. Moreover, the "infinite hotel" has an infinite number of floors in an infinite number of buildings on an infinite number of planets in an infinite number of universes created by an infinite number of Gods. Mathematicians, however, are not able to distance themselves from commonplace everyday problems: God-Allah-Buddha is always only one, the hotel is one, the corridor is only one. Here mathematicians are trying to manipulate the serial numbers of hotel rooms, convincing us that you can "shove in the stuff."

I will demonstrate the logic of my reasoning to you on the example of an infinite set of natural numbers. First you need to answer a very simple question: how many sets of natural numbers are there - one or many? There is no correct answer to this question, since we invented numbers ourselves; in Nature, numbers do not exist. Yes, Nature is excellent at counting, but for this she uses other mathematical tools that are not familiar to us. As Nature thinks, I will tell you another time. Since we invented the numbers, we ourselves will decide how many sets of natural numbers there are. Consider both options, as befits a real scientist.

Option one. "Let us be given" a single set of natural numbers, which lies serenely on the shelf. We take this set from the shelf. That's it, there are no other natural numbers on the shelf and there is nowhere to take them. We cannot add one to this set, since we already have one. And if you really want to? No problem. We can take one from the set we have already taken and return it to the shelf. After that, we can take a unit from the shelf and add it to what we have left. As a result, we again get an infinite set of natural numbers. You can write all our manipulations like this:

I wrote down the actions in algebraic notation and in the notation used in set theory, with a detailed listing of the elements of the set. The subscript indicates that we have one and only set of natural numbers. It turns out that the set of natural numbers will remain unchanged only if one subtracts from it and adds the same unit.

Option two. We have many different infinite sets of natural numbers on our shelf. I emphasize - DIFFERENT, despite the fact that they are practically indistinguishable. We take one of these sets. Then we take one from another set of natural numbers and add it to the set we have already taken. We can even add two sets of natural numbers. Here's what we get:

Subscripts "one" and "two" indicate that these items belonged to different sets. Yes, if you add one to the infinite set, the result will also be an infinite set, but it will not be the same as the original set. If you add another infinite set to one infinite set, the result is a new infinite set consisting of the elements of the first two sets.

Many natural numbers are used for counting in the same way as a ruler for measurements. Now imagine that you have added one centimeter to the ruler. This will already be a different line, not equal to the original.

You can accept or not accept my reasoning - it's your own business. But if you ever run into mathematical problems, think about whether you are not following the path of false reasoning trodden by generations of mathematicians. After all, doing mathematics, first of all, form a stable stereotype of thinking in us, and only then add to us mental abilities (or vice versa, they deprive us of free thought).

pozg.ru

sunday, 4 August 2019

I was writing a postscript to an article about and saw this wonderful text on Wikipedia:

We read: "... the rich theoretical basis of the mathematics of Babylon did not have a holistic character and was reduced to a set of disparate techniques, devoid of a common system and evidence base."

Wow! How smart we are and how well we can see the shortcomings of others. Is it hard for us to look at modern mathematics in the same context? Slightly paraphrasing the above text, I personally got the following:

The rich theoretical basis of modern mathematics does not have an integral character and is reduced to a set of disparate sections devoid of a common system and evidence base.

I will not go far to confirm my words - it has a language and conventions that are different from the language and conventions of many other branches of mathematics. The same names in different branches of mathematics can have different meanings. I want to devote a whole series of publications to the most obvious blunders of modern mathematics. See you soon.

saturday, 3 August 2019

How to subdivide a set into subsets? To do this, it is necessary to enter a new unit of measurement that is present for some of the elements of the selected set. Let's look at an example.

Let us have many ANDconsisting of four people. This set is formed on the basis of "people" Let us denote the elements of this set by the letter and, a subscript with a digit will indicate the ordinal number of each person in this set. Let's introduce a new unit of measurement "gender" and denote it by the letter b... Since sexual characteristics are inherent in all people, we multiply each element of the set AND by gender b... Note that now our multitude of "people" has become a multitude of "people with sex characteristics." After that we can divide sex characteristics into masculine bm and women bw sexual characteristics. Now we can apply a mathematical filter: we select one of these sex characteristics, no matter which one is male or female. If a person has it, then we multiply it by one, if there is no such sign, we multiply it by zero. And then we apply the usual school mathematics. See what happened.

After multiplication, contraction and rearrangement, we got two subsets: the subset of men Bm and a subset of women Bw... Mathematicians think about the same when they apply set theory in practice. But they do not devote us to the details, but give a finished result - "a lot of people consist of a subset of men and a subset of women." Naturally, you may wonder how correctly the mathematics is applied in the above transformations? I dare to assure you, in fact, everything was done correctly, it is enough to know the mathematical basis of arithmetic, Boolean algebra and other branches of mathematics. What it is? I'll tell you about it some other time.

As for supersets, you can combine two sets into one superset by choosing the unit of measurement that is present for the elements of these two sets.

As you can see, units and common mathematics make set theory a thing of the past. An indication that set theory is not all right is that mathematicians have come up with their own language and notation for set theory. Mathematicians did what shamans once did. Only shamans know how to "correctly" apply their "knowledge". They teach us this "knowledge".

Finally, I want to show you how mathematicians manipulate with.

monday, 7 January 2019

In the fifth century BC, the ancient Greek philosopher Zeno of Elea formulated his famous aporias, the most famous of which is the aporia "Achilles and the tortoise". This is how it sounds:

Let's say Achilles runs ten times faster than a turtle and is a thousand paces 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 ... They all in one way or another considered Zeno's aporias. The shock was so strong that " ... the discussions continue at the present time, the scientific community has not yet managed to come to a common opinion about the essence of paradoxes ... 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 Aporia"]. 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 involves applying instead of constants. As far as I understand, the mathematical apparatus for applying variable units of measurement has either not yet been developed, or it has not been applied to Zeno's aporia. Applying our usual logic leads us into a trap. We, by inertia of thinking, apply constant time units 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 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 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 this is not a complete solution to the problem. 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:

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 the flying arrow rests at different points in space, which, in fact, is motion. Another point should be noted here. It is impossible to determine either the fact of its movement or the distance to it from a single photograph of a car on the road. To determine the fact of car movement, two photographs are needed, taken from the same point at different moments in time, but the distance cannot be determined from them. To determine the distance to the car, you need two photographs taken from different points in space at the same time, but it is impossible to determine the fact of movement from them (of course, additional data are still needed for calculations, trigonometry will help you). What I want to draw particular 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.
Let me show you the process with an example. We select "red solid in a pimple" - this is our "whole". At the same time, we see that these things are with a bow, but there are no bows. After that, we select part of the "whole" and form a set "with a bow". This is how shamans feed themselves by tying their set theory to reality.

Now let's do a little dirty trick. Take "solid in a pimple with a bow" and combine these "wholes" by color, selecting the red elements. We got a lot of "red". Now a question to fill in: the resulting sets "with a bow" and "red" are the same set or are two different sets? Only shamans know the answer. More precisely, they themselves do not know anything, but as they say, so be it.

This simple example shows that set theory is completely useless when it comes to reality. What's the secret? We have formed a set of "red solid into a bump with a bow". The formation took place according to four different units of measurement: color (red), strength (solid), roughness (in a pimple), ornaments (with a bow). Only a set of units of measurement makes it possible to adequately describe real objects in the language of mathematics... This is what it looks like.

The letter "a" with different indices indicates different units of measurement. Units of measurement are highlighted in brackets, by which the "whole" is allocated at the preliminary stage. The unit of measurement by which the set is formed is taken out of the brackets. The last line shows the final result - an element of the set. As you can see, if we use units of measurement to form a set, then the result does not depend on the order of our actions. And this is mathematics, and not dancing shamans with tambourines. Shamans can “intuitively” come to the same result, arguing it “by evidence,” because units of measurement are not included in their “scientific” arsenal.

It is very easy to use units to split one or combine several sets into one superset. Let's take a closer look at the algebra of this process.

USE for 4? Won't you burst with happiness?

The question, as they say, is interesting ... You can, you can pass at 4! And while not bursting ... The main condition is to practice regularly. Here is the basic preparation for the exam in mathematics. With all the secrets and secrets of the exam, which you will not read about in textbooks ... Study this section, solve more tasks from various sources - and everything will work out! It is assumed that the base section "You've had enough of a three!" does not cause any difficulties for you. But if suddenly ... Follow the links, do not be lazy!

And we'll start with a great and terrible topic.

Trigonometry

Attention!
There are additional
materials in Special section 555.
For those who are "not very ..."
And for those who are "very even ...")

This topic presents a lot of problems for students. It is considered one of the most severe. What is sine and cosine? What are tangent and cotangent? What is a number circle? It is worth asking these harmless questions, as a person turns pale and tries to divert the conversation to the side ... But in vain. These are simple concepts. And this topic is no more complicated than others. You just need to clearly understand the answers to these very questions from the very beginning. It is very important. If you understand, you will like trigonometry. So,

What are sine and cosine? What are tangent and cotangent?

Let's start with deep antiquity. Do not worry, we will go through all 20 centuries of trigonometry in about 15 minutes. And, unnoticed by ourselves, we will repeat a piece of geometry from grade 8.

Draw a right triangle with sides a, b, c and angle x... Here it is.

Let me remind you that the sides that form a right angle are called legs. and in - legs. There are two of them. The remaining side is called the hypotenuse. from - hypotenuse.

Triangle and triangle, think about it! What to do with him? But ancient people knew what to do! Let's repeat their actions. Measure the side in... In the figure, the cells are specially drawn, as in tasks of the exam it happens. Side in equal to four cells. Okay. Measure the side and. Three cells.

Now divide the length of the side and by side length in... Or, as they say, take the attitude and to in. a / b= 3/4.

On the contrary, you can divide in on the and. We get 4/3. Can in divide into from. Hypotenuse from cannot be counted by cells, but it is equal to 5. We get a / c \u003d 4/5. In short, you can divide the lengths of the sides by each other and get some numbers.

So what? What is the point of this interesting lesson? Not yet. Stupid occupation, frankly.)

Now let's do this. Let's increase the triangle. Extend sides in and with, but so that the triangle remains rectangular. Angle xnaturally does not change. To see this, move the mouse cursor over the picture, or tap it (if you have a tablet). Parties a, b and c will turn into m, n, k, and, of course, the lengths of the sides will change.

But their relationship is not!

Attitude a / b It was: a / b \u003d 3/4, now m / n \u003d 6/8 \u003d 3/4. The relationship of other relevant parties is also will not change ... You can change the lengths of the sides in a right-angled triangle, increase, decrease, without changing the angle x – the relationship of the parties concerned will not change ... You can check, but you can take the ancient people at their word.

But this is already very important! The aspect ratios in a right-angled triangle do not depend on the lengths of the sides (at the same angle). This is so important that the relationship between the parties has earned its special names. Your names, so to speak.) Meet.

What is the sine of the angle x ? This is the ratio of the opposite leg to the hypotenuse:

sinx \u003d a / s

What is the cosine of angle x ? This is the ratio of the adjacent leg to the hypotenuse:

fromosx= a / c

What is the tangent of the angle x ? This is the ratio of the opposite leg to the adjacent one:

tgx \u003da / b

What is angle cotangent x ? This is the ratio of the adjacent leg to the opposite one:

ctgx \u003d in / a

Everything is very simple. Sine, cosine, tangent and cotangent are some of the numbers. Dimensionless. Just numbers. Each corner has its own.

Why am I repeating everything so boringly? Then what is it need to remember... It's hard to remember. Memorization can be made easier. Does the phrase "Let's start from afar ..." sound familiar? So start from afar.

Sinus angle is the ratio far from the angle of the leg to the hypotenuse. Cosine - the ratio of the neighbor to hypotenuse.

Tangent angle is the ratio far from the corner of the leg to the near. Cotangent - on the contrary.

It's already easier, right?

Well, if you remember that only legs sit in the tangent and cotangent, and the hypotenuse appears in the sine and cosine, then everything will become quite simple.

This whole glorious family - sine, cosine, tangent and cotangent are also called trigonometric functions.

And now a question for consideration.

Why do we say sine, cosine, tangent and cotangent corner? It's about the relationship between the parties, sort of ... What does it have to do with it angle?

We look at the second picture. Exactly the same as the first one.

Move the mouse over the picture. I changed the angle x... Increased it from x to x. All relationships have changed! Attitude a / b was 3/4, and the corresponding ratio t / in became 6/4.

And all other relationships have become different!

Therefore, the relationship of the sides does not depend in any way on their lengths (at one angle x), but sharply depend on this very angle! And only from him. Therefore, the terms sine, cosine, tangent and cotangent refer to corner. The corner here is the main one.

It must be ironically understood that the angle is inextricably linked with its trigonometric functions. Each angle has its own sine and cosine. And almost everyone has their own tangent and cotangent. It is important. It is believed that if we are given an angle, then its sine, cosine, tangent and cotangent we know ! And vice versa. Given a sine, or any other trigonometric function, it means that we know the angle.

There are special tables where trigonometric functions are described for each angle. Bradis tables are called. They are very long drawn up. Before there were no calculators or computers ...

Of course, the trigonometric functions of all angles cannot be remembered. You must know them only for a few angles, more on that later. But the spell " i know the angle, which means I know its trigonometric functions "-always works!

So we repeated a piece of geometry from the 8th grade. Do we need it for the exam? It is necessary. Here's a typical puzzle from the exam. To solve which, 8th grade is enough. Given a picture:

All. No more data. It is necessary to find the length of the BC leg.

The cells do not help much, the triangle is somehow incorrectly located .... Specially, go ... From the information there is the length of the hypotenuse. 8 cells. For some reason, the angle is given.

Here it is necessary to immediately recall trigonometry. There is an angle, which means that we know all of its trigonometric functions. Which function of the four to put into action? Let's see what we know? We know the hypotenuse, the angle, but we need to find adjacent to this corner legs! It's clear that the cosine needs to be put into operation! So we launch. Just write, by definition, cosine (relation adjacent catheta to hypotenuse):

cosC \u003d BC / 8

Angle C is 60 degrees, its cosine is 1/2. You need to know this, without any tables! That is:

1/2 \u003d Sun / 8

Elementary linear equation. Unknown - Sun... Who has forgotten how to solve equations , follow the link, the rest decide:

Sun \u003d 4

When ancient people realized that each corner has its own set trigonometric functions, they have a reasonable question. Are not sine, cosine, tangent and cotangent related in some way?So that knowing one function of the angle, one can find the rest? Without calculating the angle itself?

They were so restless ...)

Relationship between trigonometric functions of one angle.

Of course, the sine, cosine, tangent and cotangent of the same angle are related. Every connection between expressions is given in mathematics by formulas. In trigonometry formulas - a huge amount. But here we will look at the most basic ones. These formulas are called: basic trigonometric identities. Here they are:

These formulas must be known ironically. Without them, there is nothing to do in trigonometry at all. Three more auxiliary identities follow from these basic identities:

I warn you right away that the last three formulas quickly fall out of memory. For some reason.) You can, of course, derive these formulas from the first three. But, in difficult times ... you understand.)

In standard tasks, such as those listed below, there is a way to do without these unforgettable formulas. AND sharply reduce errors for forgetfulness, and in computing too. This practice is in Section 555 lesson "Relationship between trigonometric functions of one angle."

In what tasks and how are the main trigonometric identities used? The most popular task is to find some function of the angle if another is given. In the exam, such a task is present from year to year.) For example:

Find the value of sinx if x is an acute angle and cosx \u003d 0.8.

The task is almost elementary. We are looking for a formula where there are sine and cosine. This is the formula:

sin 2 x + cos 2 x \u003d 1

We substitute here the known value, namely, 0.8 instead of the cosine:

sin 2 x + 0.8 2 \u003d 1

Well, we consider, as usual:

sin 2 x + 0.64 \u003d 1

sin 2 x \u003d 1 - 0.64

That's practically all. We calculated the square of the sine, it remains to extract square root and the answer is ready! The root of 0.36 is 0.6.

The task is almost elementary. But the word "almost" is not in vain here ... The fact is that the answer sinx \u003d - 0.6 also fits ... (-0.6) 2 will also be 0.36.

Two different answers are obtained. And you need one. The second is wrong. How to be !? Yes, as usual.) Read the assignment carefully. For some reason it says: ... if x is an acute angle ... And in the tasks, each word has a meaning, yes ... This phrase is additional information to the solution.

An acute angle is an angle less than 90 °. And at such corners all trigonometric functions - and sine, and cosine, and tangent with cotangent - positive. Those. we simply reject the negative answer here. We have the right.

Actually, eighth-graders do not need such subtleties. They only work with right-angled triangles, where the corners can only be sharp. And they don’t know, happy that there are negative angles and angles of 1000 ° ... And all these nightmarish angles have their trigonometric functions with plus and minus ...

But high school students do not take into account the sign. Many knowledge multiplies sorrows, yes ...) And for the right decision in the task, additional information is necessarily present (if it is necessary). For example, it can be written like this:

Or something else. You will see in the examples below.) To solve such examples, you need to know in which quarter does the given angle x fall and what sign has the required trigonometric function in this quarter.

These basics of trigonometry are covered in the lessons what is a trigonometric circle, counting the angles on this circle, radian measure of angle. Sometimes you need to know and table of sines of cosines of tangents and cotangents.

So, let's note the most important thing:

Practical advice:

1. Remember the definitions of sine, cosine, tangent, and cotangent. Very useful.

2. We understand clearly: sine, cosine, tangent and cotangent are firmly connected with the angles. We know one thing - it means we know another.

3. We clearly understand: sine, cosine, tangent and cotangent of one angle are connected with each other by basic trigonometric identities. If we know one function, then we can (if we have the necessary additional information) calculate all the others.

And now we will decide, as usual. First, assignments in the scope of the 8th grade. But high school students can also ...)

1. Calculate the tgА value if ctgА \u003d 0.4.

2. β is the angle in a right triangle. Find tgβ if sinβ \u003d 12/13.

3. Determine the sine of an acute angle x if tgx \u003d 4/3.

4. Find the value of the expression:

6sin 2 5 ° - 3 + 6cos 2 5 °

5. Find the value of the expression:

(1-cosx) (1 + cosx) if sinx \u003d 0.3

Answers (separated by semicolons, in disarray):

0,09; 3; 0,8; 2,4; 2,5

Happened? Fine! Eighth graders may already be able to get their A.)

Not everything worked out? Tasks 2 and 3 are somehow not very good ...? No problem! There is one nice trick for such tasks. Everything is solved, practically, without formulas at all! Well, and, therefore, no mistakes. This trick in the lesson: “The Relationship Between the Trigonometric Functions of a Single Angle” in Section 555 described. All other tasks are also sorted out there.

These were the puzzles. type of exam, but in a stripped-down version. Unified State Exam - light). And now there are almost the same tasks, but in a full-fledged cache format. For high school students burdened with knowledge.)

6. Find the value of tgβ if sinβ \u003d 12/13, and

7. Determine sinx, if tgx \u003d 4/3, and x belongs to the interval (- 540 °; - 450 °).

8. Find the value of the expression sinβ · cosβ if ctgβ \u003d 1.

Answers (in disarray):

0,8; 0,5; -2,4.

Here, in Problem 6, the angle is somehow not very unambiguous ... And in Problem 8 it is not specified at all! This is on purpose). Additional information is taken not only from the task, but also from the head.) But if you decide - one right task is guaranteed!

And if you haven't decided? Um ... well, here Section 555 will help. There, the solutions to all these tasks are detailed, it’s hard not to figure it out.

This lesson introduces a very limited concept of trigonometric functions. Within the 8th grade. And the older ones have questions ...

For example, if the angle x (see the second picture on this page) - make it dumb !? The triangle will fall apart! And how to be? There will be no leg, no hypotenuse ... The sinus is gone ...

If ancient people had not found a way out of this situation, we would not have cell phones, TV, or electricity. Yes Yes! Theoretical basis all these things without trigonometric functions - zero without a wand. But the ancient people did not disappoint. How they got out - in the next lesson.

If you like this site ...

By the way, I have a couple more interesting sites for you.)

You can practice solving examples and find out your level. Testing with instant verification. Learning - with interest!)

you can get acquainted with functions and derivatives.

As you can see, this circle is built in a Cartesian coordinate system. The radius of the circle is unity, while the center of the circle lies at the origin, the initial position of the radius vector is fixed along the positive direction of the axis (in our example, this is the radius).

Each point of the circle corresponds to two numbers: the coordinate along the axis and the coordinate along the axis. And what are these numbers-coordinates? And in general, what do they have to do with the topic under discussion? To do this, you need to remember about the considered right-angled triangle. In the figure above, you can notice as many as two right-angled triangles. Consider a triangle. It is rectangular as it is perpendicular to the axis.

What is triangle equal to? That's all right. In addition, we know that - is the radius of the unit circle, and therefore,. Let's substitute this value in our formula for cosine. Here's what happens:

And what is equal to from the triangle? Well, of course, ! Substitute the value of the radius into this formula and get:

So, can you tell what coordinates the point belonging to the circle has? Well, no way? And if you think that and are just numbers? What coordinate does it correspond to? Well, of course, the coordinate! And what coordinate does it correspond to? That's right, coordinate! So the point.

And what then are equal to and? That's right, we use the corresponding definitions of tangent and cotangent and we get that, a.

But what if the angle is larger? Here, for example, as in this figure:

What has changed in this example? Let's figure it out. To do this, again turn to a right-angled triangle. Consider a right-angled triangle: corner (as adjacent to the corner). What is the value of sine, cosine, tangent and cotangent for an angle? That's right, we adhere to the corresponding definitions of trigonometric functions:

Well, as you can see, the value of the sine of the angle still corresponds to the coordinate; the cosine of the angle is the coordinate; and the values \u200b\u200bof the tangent and cotangent to the corresponding ratios. Thus, these relationships apply to any rotations of the radius vector.

It has already been mentioned that the initial position of the radius vector is along the positive direction of the axis. So far, we have rotated this vector counterclockwise, but what happens if you turn it clockwise? Nothing extraordinary, an angle of a certain magnitude will also turn out, but only it will be negative. Thus, when rotating the radius vector counterclockwise, you get positive angles, and when rotating clockwise - negative.

So, we know that the whole revolution of the radius vector in a circle is or. Is it possible to rotate the radius vector by or by? Of course you can! In the first case, thus, the radius vector will make one full revolution and stop at or.

In the second case, that is, the radius vector will make three full turns and stop at or.

Thus, from the above examples, we can conclude that angles that differ by or (where is any integer) correspond to the same position of the radius vector.

The picture below shows the angle. The same image corresponds to a corner, etc. This list goes on and on. All these angles can be written as a general formula or (where - any integer)

Now, knowing the definitions of the basic trigonometric functions and using the unit circle, try to answer what the values \u200b\u200bare equal to:

Here's a unit circle to help you:

Having trouble? Then let's figure it out. So, we know that:

From here, we determine the coordinates of the points corresponding to certain measures of the angle. Well, let's start in order: the point at coordinates corresponds to the corner at, therefore:

Does not exist;

Further, adhering to the same logic, we find out that the corners in correspond to the points with coordinates, respectively. Knowing this, it is easy to determine the values \u200b\u200bof trigonometric functions at the corresponding points. First try it yourself, and then check the answers.

Answers:

Does not exist

Thus, we can make the following plate:

It is not necessary to remember all these values. It is enough to remember the correspondence of the coordinates of the points on the unit circle and the values \u200b\u200bof trigonometric functions:

But the values \u200b\u200bof the trigonometric functions of the angles in and, given in the table below, need to remember:

Do not be scared, now we will show one example quite simple memorization of the corresponding values:

To use this method, it is vital to remember the values \u200b\u200bof the sine for all three measures of the angle (), as well as the value of the tangent of the angle b. Knowing these values, it is quite simple to restore the entire table as a whole — cosine values \u200b\u200bare transferred in accordance with the arrows, that is:

Knowing this, you can restore the values \u200b\u200bfor. The numerator "" will match, and the denominator "" will match. The cotangent values \u200b\u200bare carried over according to the arrows shown in the figure. If you understand this and remember the diagram with arrows, then it will be enough to remember all the values \u200b\u200bfrom the table.

Point coordinates on a circle

Is it possible to find a point (its coordinates) on a circle, knowing the coordinates of the center of the circle, its radius and angle of rotation?

Well, of course you can! Let's bring general formula for finding the coordinates of a point.

Here, for example, we have such a circle:

We are given that the point is the center of the circle. The radius of the circle is equal. It is necessary to find the coordinates of the point obtained by turning the point by degrees.

As can be seen from the figure, the coordinate of the point corresponds to the length of the segment. The length of the segment corresponds to the coordinate of the center of the circle, that is, equal to. The length of the segment can be expressed using the definition of cosine:

Then we have that for the point the coordinate.

By the same logic, we find the y coordinate value for the point. Thus,

So, in general terms, the coordinates of the points are determined by the formulas:

Circle center coordinates,

Circle radius,

The angle of rotation of the radius of the vector.

As you can see, for the unit circle we are considering, these formulas are significantly reduced, since the center coordinates are zero and the radius is unity:

Well, let's taste these formulas by practicing finding points on a circle?

1. Find the coordinates of a point on the unit circle obtained by turning the point on.

2. Find the coordinates of a point on the unit circle obtained by turning the point by.

3. Find the coordinates of a point on the unit circle obtained by turning the point by.

4. The point is the center of the circle. The radius of the circle is. It is necessary to find the coordinates of the point obtained by turning the initial radius vector by.

5. Point is the center of the circle. The radius of the circle is equal. It is necessary to find the coordinates of the point obtained by rotating the initial radius vector by.

Having trouble finding the coordinates of a point on a circle?

Solve these five examples (or figure it out well in the solution) and you will learn how to find them!

You may notice that. But we know that it corresponds to the full revolution of the starting point. Thus, the desired point will be in the same position as when turning on. Knowing this, we find the desired coordinates of the point:

2. The unit circle is centered on a point, which means we can use simplified formulas:

You may notice that. We know what corresponds to two complete revolutions of the starting point. Thus, the desired point will be in the same position as when turning on. Knowing this, we find the desired coordinates of the point:

Sine and cosine are tabular values. Recall their values \u200b\u200band get:

Thus, the desired point has coordinates.

3. The unit circle is centered on a point, which means we can use simplified formulas:

You may notice that. We will depict the considered example in the figure:

The radius makes angles with the axis equal to and. Knowing that the tabular values \u200b\u200bof the cosine and sine are equal, and determining that the cosine here takes a negative value, and the sine is positive, we have:

Such examples are analyzed in more detail when studying the formulas for the reduction of trigonometric functions in the topic.

Thus, the desired point has coordinates.

Angle of rotation of the radius of the vector (by condition)

To determine the corresponding signs of the sine and cosine, we construct the unit circle and angle:

As you can see, the meaning, that is, is positive, and the meaning, that is, is negative. Knowing the tabular values \u200b\u200bof the corresponding trigonometric functions, we obtain that:

Let's substitute the obtained values \u200b\u200binto our formula and find the coordinates:

Thus, the desired point has coordinates.

5. To solve this problem, we will use formulas in general form, where

The coordinates of the center of the circle (in our example,

Radius of a circle (by condition)

The angle of rotation of the radius of the vector (by condition,).

Substitute all the values \u200b\u200bin the formula and get:

and are tabular values. Remember and substitute them in the formula:

Thus, the desired point has coordinates.

SUMMARY AND BASIC FORMULAS

The sine of the angle is the ratio of the opposite (far) leg to the hypotenuse.

The cosine of the angle is the ratio of the adjacent (close) leg to the hypotenuse.

The tangent of the angle is the ratio of the opposite (far) leg to the adjacent (close).

The cotangent of the angle is the ratio of the adjacent (close) leg to the opposite (far) leg.

Trigonometry, as a science, originated in the Ancient East. The first trigonometric ratios were derived by astronomers to create an accurate calendar and star orientation. These calculations related to spherical trigonometry, while in school course study the aspect ratio and angle of a flat 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 of the 1st millennium AD, knowledge spread from the Ancient East to Greece. But the main discoveries of trigonometry are the merit of the husbands of the Arab caliphate. In particular, the Turkmen scientist al-Marazvi introduced such functions as tangent and cotangent, compiled the first tables of values \u200b\u200bfor sines, tangents and cotangents. The concept of sine and cosine was introduced by Indian scientists. A lot of attention is devoted to trigonometry in the works of such great figures of antiquity as Euclid, Archimedes and Eratosthenes.

Basic quantities of trigonometry

The main trigonometric functions of a numerical argument are sine, cosine, tangent, and cotangent. Each of them has its own graph: sine wave, cosine wave, tangent wave and cotangent wave.

The formulas for calculating the values \u200b\u200bof these quantities are based on the Pythagorean theorem. Pupils know it better in the wording: “ Pythagorean Pants, in all directions are equal ", since the proof is given by the example of an isosceles right triangle.

Sine, cosine and other dependencies establish a connection between sharp angles and the sides of any right triangle. We give formulas for calculating these quantities for angle A and trace the relationship of trigonometric functions:

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

Trigonometric circle

Graphically, the ratio of these quantities can be represented as follows:

The circle, in this case, represents all possible values \u200b\u200bof the angle α - from 0 ° to 360 °. As you can see from the figure, each function takes a negative or positive value depending on the angle. For example, sin α will be with a "+" sign if α belongs to I and II quarters of a circle, that is, is in the range from 0 ° to 180 °. When α is from 180 ° to 360 ° (III and IV quarters), sin α can only be negative.

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

The values \u200b\u200bof α equal to 30 °, 45 °, 60 °, 90 °, 180 ° and so on are called special cases. The values \u200b\u200bof trigonometric functions for them are calculated and presented in the form of special tables.

These angles were not chosen by chance. The designation π in the tables stands for radians. Rad is the angle at which the length of a circular arc corresponds to its radius. This value was introduced in order to establish a universal dependence; in calculations in radians, the actual radius length in cm does not matter.

The angles in the tables for trigonometric functions correspond to the values \u200b\u200bof radians:

So, it's not hard to guess that 2π is a full 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 a comparison table of properties for a sine wave and a cosine wave:

Sine wave	Cosine wave
y \u003d sin x	y \u003d cos x
ODZ [-1; 1]	ODZ [-1; 1]
sin x \u003d 0, for x \u003d πk, where k ϵ Z	cos x \u003d 0, for x \u003d π / 2 + πk, where k ϵ Z
sin x \u003d 1, for x \u003d π / 2 + 2πk, where k ϵ Z	cos x \u003d 1, for x \u003d 2πk, where k ϵ Z
sin x \u003d - 1, for x \u003d 3π / 2 + 2πk, where k ϵ Z	cos x \u003d - 1, for x \u003d π + 2πk, where k ϵ Z
sin (-x) \u003d - sin x, i.e. the function is odd	cos (-x) \u003d cos x, i.e. the function is even
	periodic function, shortest period - 2π
sin x ›0, for x belonging to I and II quarters or from 0 ° to 180 ° (2πk, π + 2πk)	cos x ›0, for x belonging to I and IV quarters or from 270 ° to 90 ° (- π / 2 + 2πk, π / 2 + 2πk)
sin x ‹0, for x belonging to the III and IV quarters or from 180 ° to 360 ° (π + 2πk, 2π + 2πk)	cos x ‹0, for x belonging to quarters II and III or from 90 ° to 270 ° (π / 2 + 2πk, 3π / 2 + 2πk)
increases on the interval [- π / 2 + 2πk, π / 2 + 2πk]	increases on the interval [-π + 2πk, 2πk]
decreases on the intervals [π / 2 + 2πk, 3π / 2 + 2πk]	decreases in intervals
derivative (sin x) ’\u003d cos x	derivative (cos x) ’\u003d - sin x

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

The introduction of radians and a listing of the basic properties of a sinusoid and cosine can lead to the following pattern:

It is very easy to make sure that the formula is correct. For example, for x \u003d π / 2 the sine is 1, as is the cosine x \u003d 0. The check can be carried out by referring to tables or by tracing the curves of functions for given values.

Properties of tangentoids and cotangentoids

Plots of tangent and cotangent functions differ significantly from sine and cosine. The tg and ctg values \u200b\u200bare inverse to each other.

Y \u003d tg x.
The tangensoid tends to the y-values \u200b\u200bat x \u003d π / 2 + πk, but never reaches them.
The smallest positive period of the tangentoid is π.
Tg (- x) \u003d - tg x, that is, the function is odd.
Tg x \u003d 0, for x \u003d πk.
The function is increasing.
Tg x\u003e 0, for x ϵ (πk, π / 2 + πk).
Tg x ‹0, for x ϵ (- π / 2 + πk, πk).
Derivative (tg x) ’\u003d 1 / cos 2 \u2061x.

Consider a graphical representation of a cotangentoid below in the text.

The main properties of a cotangensoid:

Y \u003d ctg x.
Unlike the sine and cosine functions, in the tangentoid Y can take values \u200b\u200bof the set of all real numbers.
The cotangentoid tends to y values \u200b\u200bat x \u003d πk, but never reaches them.
The smallest positive period of cotangentoids is π.
Ctg (- x) \u003d - ctg x, that is, the function is odd.
Ctg x \u003d 0, for x \u003d 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) ’\u003d - 1 / sin 2 \u2061x Correct