Palmistry 

The Tg of the angle is equal to the ratio. Right triangle. Complete illustrated guide (2019)

One of the branches of mathematics with which schoolchildren cope with the greatest difficulties is trigonometry. No wonder: in order to freely master this area of ​​knowledge, you need spatial thinking, the ability to find sines, cosines, tangents, cotangents using formulas, simplify expressions, and be able to use the number pi in calculations. In addition, you need to be able to apply trigonometry when proving theorems, and this requires either a developed mathematical memory or the ability to deduce complex logical chains.

Origins of trigonometry

Acquaintance with this science should begin with the definition of the sine, cosine and tangent of the angle, but first you need to figure out what trigonometry does in general.

Historically, right triangles have been the main object of study in this section of mathematical science. The presence of an angle of 90 degrees makes it possible to carry out various operations that allow one to determine the values ​​of all parameters of the figure under consideration using two sides and one angle or two angles and one side. In the past, people noticed this pattern and began to actively use it in the construction of buildings, navigation, astronomy, and even art.

First stage

Initially, people talked about the relationship of angles and sides exclusively on the example of right triangles. Then special formulas were discovered that made it possible to expand the boundaries of use in everyday life of this section of mathematics.

The study of trigonometry at school today begins with right-angled triangles, after which the acquired knowledge is used by students in physics and solving abstract trigonometric equations, work with which begins in high school.

Spherical trigonometry

Later, when science reached the next level of development, formulas with sine, cosine, tangent, cotangent began to be used in spherical geometry, where different rules apply, and the sum of the angles in a triangle is always more than 180 degrees. This section is not studied at school, but it is necessary to know about its existence, at least because the earth's surface, and the surface of any other planet, is convex, which means that any surface marking will be "arc-shaped" in three-dimensional space.

Take the globe and thread. Attach the thread to any two points on the globe so that it is taut. Pay attention - it has acquired the shape of an arc. It is with such forms that spherical geometry, which is used in geodesy, astronomy, and other theoretical and applied fields, deals.

Right triangle

Having learned a little about the ways of using trigonometry, let's return to basic trigonometry in order to further understand what sine, cosine, tangent are, what calculations can be performed with their help and what formulas to use.

The first step is to understand the concepts related to a right triangle. First, the hypotenuse is the side opposite the 90 degree angle. She is the longest. We remember that, according to the Pythagorean theorem, its numerical value is equal to the root of the sum of the squares of the other two sides.

For example, if two sides are 3 and 4 centimeters respectively, the length of the hypotenuse will be 5 centimeters. By the way, the ancient Egyptians knew about this about four and a half thousand years ago.

The two remaining sides that form a right angle are called legs. In addition, we must remember that the sum of the angles in a triangle in a rectangular coordinate system is 180 degrees.

Definition

Finally, with a solid understanding of the geometric base, we can turn to the definition of the sine, cosine and tangent of an angle.

The sine of an angle is the ratio of the opposite leg (i.e., the side opposite the desired angle) to the hypotenuse. The cosine of an angle is the ratio of the adjacent leg to the hypotenuse.

Remember that neither sine nor cosine can be greater than one! Why? Because the hypotenuse is by default the longest. No matter how long the leg is, it will be shorter than the hypotenuse, which means that their ratio will always be less than one. Thus, if you get a sine or cosine with a value greater than 1 in the answer to the problem, look for an error in calculations or reasoning. This answer is clearly wrong.

Finally, the tangent of an angle is the ratio of the opposite side to the adjacent side. The same result will give the division of the sine by the cosine. Look: in accordance with the formula, we divide the length of the side by the hypotenuse, after which we divide by the length of the second side and multiply by the hypotenuse. Thus, we get the same ratio as in the definition of tangent.

The cotangent, respectively, is the ratio of the side adjacent to the corner to the opposite side. We get the same result by dividing the unit by the tangent.

So, we have considered the definitions of what sine, cosine, tangent and cotangent are, and we can deal with formulas.

The simplest formulas

In trigonometry, one cannot do without formulas - how to find sine, cosine, tangent, cotangent without them? And this is exactly what is required when solving problems.

The first formula that you need to know when starting to study trigonometry says that the sum of the squares of the sine and cosine of an angle is equal to one. This formula is a direct consequence of the Pythagorean theorem, but it saves time if you want to know the value of the angle, not the side.

Many students cannot remember the second formula, which is also very popular when solving school problems: the sum of one and the square of the tangent of an angle is equal to one divided by the square of the cosine of the angle. Take a closer look: after all, this is the same statement as in the first formula, only both sides of the identity were divided by the square of the cosine. It turns out that a simple mathematical operation makes the trigonometric formula completely unrecognizable. Remember: knowing what sine, cosine, tangent and cotangent are, the conversion rules and a few basic formulas, you can at any time independently derive the required more complex formulas on a sheet of paper.

Double angle formulas and addition of arguments

Two more formulas that you need to learn are related to the values ​​\u200b\u200bof the sine and cosine for the sum and difference of the angles. They are shown in the figure below. Please note that in the first case, the sine and cosine are multiplied both times, and in the second, the pairwise product of the sine and cosine is added.

There are also formulas associated with double angle arguments. They are completely derived from the previous ones - as a practice, try to get them yourself, taking the angle of alpha equal to the angle of beta.

Finally, note that the double angle formulas can be converted to lower the degree of sine, cosine, tangent alpha.

Theorems

The two main theorems in basic trigonometry are the sine theorem and the cosine theorem. With the help of these theorems, you can easily understand how to find the sine, cosine and tangent, and therefore the area of ​​\u200b\u200bthe figure, and the size of each side, etc.

The sine theorem states that as a result of dividing the length of each of the sides of the triangle by the value of the opposite angle, we get the same number. Moreover, this number will be equal to two radii of the circumscribed circle, that is, the circle containing all points of the given triangle.

The cosine theorem generalizes the Pythagorean theorem, projecting it onto any triangles. It turns out that from the sum of the squares of the two sides, subtract their product multiplied by the double cosine of the angle adjacent to them - the resulting value will be equal to the square of the third side. Thus, the Pythagorean theorem turns out to be a special case of the cosine theorem.

Mistakes due to inattention

Even knowing what sine, cosine and tangent are, it is easy to make a mistake due to absent-mindedness or an error in the simplest calculations. To avoid such mistakes, let's get acquainted with the most popular of them.

First, you should not convert ordinary fractions to decimals until the final result is obtained - you can leave the answer as an ordinary fraction, unless the condition states otherwise. Such a transformation cannot be called a mistake, but it should be remembered that at each stage of the problem new roots may appear, which, according to the author's idea, should be reduced. In this case, you will waste time on unnecessary mathematical operations. This is especially true for values ​​such as the root of three or two, because they occur in tasks at every step. The same applies to rounding "ugly" numbers.

Further, note that the cosine theorem applies to any triangle, but not the Pythagorean theorem! If you mistakenly forget to subtract twice the product of the sides multiplied by the cosine of the angle between them, you will not only get a completely wrong result, but also demonstrate a complete misunderstanding of the subject. This is worse than a careless mistake.

Thirdly, do not confuse the values ​​​​for angles of 30 and 60 degrees for sines, cosines, tangents, cotangents. Remember these values, because the sine of 30 degrees is equal to the cosine of 60, and vice versa. It is easy to mix them up, as a result of which you will inevitably get an erroneous result.

Application

Many students are in no hurry to start studying trigonometry, because they do not understand its applied meaning. What is sine, cosine, tangent for an engineer or astronomer? These are concepts thanks to which you can calculate the distance to distant stars, predict the fall of a meteorite, send a research probe to another planet. Without them, it is impossible to build a building, design a car, calculate the load on the surface or the trajectory of an object. And these are just the most obvious examples! After all, trigonometry in one form or another is used everywhere, from music to medicine.

Finally

So you are sine, cosine, tangent. You can use them in calculations and successfully solve school problems.

The whole essence of trigonometry boils down to the fact that unknown parameters must be calculated from the known parameters of the triangle. There are six parameters in total: the lengths of three sides and the magnitudes of three angles. The whole difference in the tasks lies in the fact that different input data are given.

How to find the sine, cosine, tangent based on the known lengths of the legs or the hypotenuse, you now know. Since these terms mean nothing more than a ratio, and a ratio is a fraction, the main goal of the trigonometric problem is to find the roots of an ordinary equation or a system of equations. And here you will be helped by ordinary school mathematics.

The line y \u003d f (x) will be tangent to the graph shown in the figure at the point x0 if it passes through the point with coordinates (x0; f (x0)) and has a slope f "(x0). Find such a coefficient, knowing the features of the tangent, it is not difficult.

You will need

  • - mathematical reference book;
  • - a simple pencil;
  • - notebook;
  • - protractor;
  • - compass;
  • - pen.

Instruction

If the value f‘(x0) does not exist, then either there is no tangent, or it passes vertically. In view of this, the presence of the derivative of the function at the point x0 is due to the existence of a non-vertical tangent that is in contact with the graph of the function at the point (x0, f(x0)). In this case, the slope of the tangent will be equal to f "(x0). Thus, the geometric meaning of the derivative becomes clear - the calculation of the slope of the tangent.

Draw on additional tangents that would be in contact with the function graph at points x1, x2 and x3, and also mark the angles formed by these tangents with the abscissa axis (such an angle is counted in the positive direction from the axis to the tangent line). For example, the angle, that is, α1, will be acute, the second (α2) is obtuse, and the third (α3) is zero, since the tangent line is parallel to the OX axis. In this case, the tangent of an obtuse angle is negative, the tangent of an acute angle is positive, and for tg0 the result is zero.

note

Correctly determine the angle formed by the tangent. To do this, use a protractor.

Useful advice

Two oblique lines will be parallel if their slopes are equal to each other; perpendicular if the product of the slopes of these tangents is -1.

Sources:

  • Tangent to function graph

Cosine, like sine, is referred to as "direct" trigonometric functions. The tangent (together with the cotangent) is added to another pair called "derivatives". There are several definitions of these functions, which make it possible to find the tangent of the cosine given by the known value of the same value.

Instruction

Subtract the quotient from unity by the cosine of the given angle raised to the value, and extract the square root from the result - this will be the value of the tangent from the angle, expressed by its cosine: tg (α) \u003d √ (1-1 / (cos (α)) ²) . At the same time, pay attention to the fact that in the formula the cosine is in the denominator of the fraction. The impossibility of dividing by zero excludes the use of this expression for angles equal to 90°, as well as differing from this value by multiples of 180° (270°, 450°, -90°, etc.).

There is an alternative way to calculate the tangent from a known cosine value. It can be used if there is no restriction on the use of other . To implement this method, first determine the angle value from a known cosine value - this can be done using the arccosine function. Then simply calculate the tangent for the angle of the resulting value. In general, this algorithm can be written as follows: tg(α)=tg(arccos(cos(α))).

There is another exotic option using the definition of cosine and tangent through the acute angles of a right triangle. The cosine in this definition corresponds to the ratio of the length of the leg adjacent to the considered angle to the length of the hypotenuse. Knowing the value of the cosine, you can choose the lengths of these two sides corresponding to it. For example, if cos(α)=0.5, then the adjacent can be taken equal to 10 cm, and the hypotenuse - 20 cm. Specific numbers do not matter here - you will get the same and correct with any values ​​\u200b\u200bthat have the same. Then, using the Pythagorean theorem, determine the length of the missing side - the opposite leg. It will be equal to the square root of the difference between the lengths of the squared hypotenuse and the known leg: √(20²-10²)=√300. By definition, the tangent corresponds to the ratio of the lengths of the opposite and adjacent legs (√300/10) - calculate it and get the tangent value found using the classical definition of cosine.

Sources:

  • cosine through tangent formula

One of the trigonometric functions, most often denoted by the letters tg, although the notation tan is also found. The easiest way is to represent the tangent as the ratio of the sine angle to its cosine. This is an odd periodic and not continuous function, each cycle of which is equal to the number Pi, and the break point corresponds to the mark at half this number.

Where the tasks for solving a right-angled triangle were considered, I promised to present a technique for memorizing the definitions of sine and cosine. Using it, you will always quickly remember which leg belongs to the hypotenuse (adjacent or opposite). I decided not to put it off indefinitely, the necessary material is below, please read it 😉

The fact is that I have repeatedly observed how students in grades 10-11 have difficulty remembering these definitions. They remember very well that the leg refers to the hypotenuse, but which one- forget and confused. The price of a mistake, as you know in the exam, is a lost score.

The information that I will present directly to mathematics has nothing to do. It is associated with figurative thinking, and with the methods of verbal-logical connection. That's right, I myself, once and for all remembereddefinition data. If you still forget them, then with the help of the presented techniques it is always easy to remember.

Let me remind you the definitions of sine and cosine in a right triangle:

Cosine acute angle in a right triangle is the ratio of the adjacent leg to the hypotenuse:

Sinus acute angle in a right triangle is the ratio of the opposite leg to the hypotenuse:

So, what associations does the word cosine evoke in you?

Probably everyone has their ownRemember the link:

Thus, you will immediately have an expression in your memory -

«… ratio of ADJACENT leg to hypotenuse».

The problem with the definition of cosine is solved.

If you need to remember the definition of the sine in a right triangle, then remembering the definition of the cosine, you can easily establish that the sine of an acute angle in a right triangle is the ratio of the opposite leg to the hypotenuse. After all, there are only two legs, if the adjacent leg is “occupied” by the cosine, then only the opposite side remains for the sine.

What about tangent and cotangent? Same confusion. Students know that this is the ratio of legs, but the problem is to remember which one refers to which - either opposite to adjacent, or vice versa.

Definitions:

Tangent an acute angle in a right triangle is the ratio of the opposite leg to the adjacent one:

Cotangent acute angle in a right triangle is the ratio of the adjacent leg to the opposite:

How to remember? There are two ways. One also uses a verbal-logical connection, the other - a mathematical one.

MATHEMATICAL METHOD

There is such a definition - the tangent of an acute angle is the ratio of the sine of an angle to its cosine:

* Remembering the formula, you can always determine that the tangent of an acute angle in a right triangle is the ratio of the opposite leg to the adjacent one.

Likewise.The cotangent of an acute angle is the ratio of the cosine of an angle to its sine:

So! Remembering these formulas, you can always determine that:

- the tangent of an acute angle in a right triangle is the ratio of the opposite leg to the adjacent

- the cotangent of an acute angle in a right triangle is the ratio of the adjacent leg to the opposite one.

VERBAL-LOGICAL METHOD

About tangent. Remember the link:

That is, if you need to remember the definition of the tangent, using this logical connection, you can easily remember what it is

"... the ratio of the opposite leg to the adjacent"

If it comes to cotangent, then remembering the definition of tangent, you can easily voice the definition of cotangent -

"... the ratio of the adjacent leg to the opposite"

There is an interesting technique for memorizing tangent and cotangent on the site " Mathematical tandem " , look.

METHOD UNIVERSAL

You can just grind.But as practice shows, thanks to verbal-logical connections, a person remembers information for a long time, and not only mathematical.

I hope the material was useful to you.

Sincerely, Alexander Krutitskikh

P.S: I would be grateful if you tell about the site in social networks.

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". Here's how it sounds:

Let's say Achilles runs ten times faster than the tortoise and is a thousand paces behind it. During the time during which Achilles runs this distance, the tortoise crawls a hundred steps in the same direction. When Achilles has run a hundred steps, the tortoise will crawl another ten steps, and so on. The process will continue indefinitely, Achilles will never catch up with the tortoise.

This reasoning became a logical shock for all subsequent generations. Aristotle, Diogenes, Kant, Hegel, Gilbert... All of them, in one way or another, considered Zeno's aporias. The shock was so strong that " ... 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 became a universally accepted solution to the problem ..."[Wikipedia," Zeno's Aporias "]. Everyone understands that they are being fooled, but no one understands what the deception is.

From the point of view of mathematics, Zeno in his aporia clearly demonstrated the transition from the value to. This transition implies 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. The application of our usual logic leads us into a trap. We, by the inertia of thinking, apply constant units of time to the reciprocal. From a physical point of view, it looks like time slowing down to a complete stop at the moment when Achilles catches up with the tortoise. If time stops, Achilles can no longer overtake the tortoise.

If we turn the logic we are used to, everything falls into place. Achilles runs at a constant speed. Each subsequent segment of its 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 overtake the tortoise."

How to avoid this logical trap? Remain in constant units of time and do not switch to reciprocal values. In Zeno's language, it looks like this:

In the time it takes Achilles to run a thousand steps, the tortoise crawls 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 one hundred steps. Now Achilles is eight hundred paces ahead of the tortoise.

This approach adequately describes reality without any logical paradoxes. But this is not a complete solution to the problem. Einstein's statement about the insurmountability of the speed of light is very similar to Zeno's aporia "Achilles and the tortoise". We have yet 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 of a flying arrow:

A flying arrow is motionless, since at each 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 movement. There is another point 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 the fact of the movement of the car, two photographs taken from the same point at different points in time are needed, but they cannot be used to determine the distance. To determine the distance to the car, you need two photographs taken from different points in space at the same time, but you cannot determine the fact of movement from them (of course, you still need additional data for calculations, trigonometry will help you). What I want to point out in particular is that two points in time and two points in space are two different things that should not be confused as they provide different opportunities for exploration.

Wednesday, July 4, 2018

Very well the differences between set and multiset are described in Wikipedia. We look.

As you can see, "the set cannot have two identical elements", but if there are identical elements in the set, such a set is called a "multiset". Reasonable beings will never understand such logic of absurdity. This is the level of talking parrots and trained monkeys, in which the mind is absent from the word "completely." Mathematicians act as ordinary trainers, preaching their absurd ideas to us.

Once upon a time, the engineers who built the bridge were in a boat under the bridge during the tests of 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 desk, paying salaries. Here a mathematician comes to us for his money. We count the whole amount to him and lay it out on our table into different piles, in which we put bills of the same denomination. Then we take one bill from each pile and give the mathematician his "mathematical salary set". We explain the mathematics that he will receive the rest of the bills only when he proves that the set without identical elements is not equal to the set with identical elements. This is where the fun begins.

First of all, the deputies' logic will work: "you can apply it to others, but not to me!" Further, assurances will begin that there are different banknote numbers on banknotes of the same denomination, which means that they cannot be considered identical elements. Well, we count the salary in coins - there are no numbers on the coins. Here the mathematician will frantically recall physics: different coins have different amounts of dirt, the crystal structure and arrangement of atoms for each coin is unique ...

And now I have the most interesting question: where is the boundary beyond which 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 here is not even close.

Look here. We select football stadiums with the same field area. The area of ​​the fields is the same, which means we have a multiset. But if we consider the names of the same stadiums, we get a lot, because the names are different. As you can see, the same set of elements is both a set and a multiset at the same time. How right? And here the mathematician-shaman-shuller takes out a trump ace from his sleeve and begins to tell us about either 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 will 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 they are shamans for that, 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 "Sum of Digits of a Number" page. She doesn't exist. There is no formula in mathematics by which you can find the sum of the digits of any number. After all, numbers are graphic symbols with which we write numbers, and in the language of mathematics, the task sounds like this: "Find the sum of graphic symbols representing any number." Mathematicians cannot solve this problem, but shamans can do it elementarily.

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's say we 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 to a number graphic symbol. This is not a mathematical operation.

2. We cut one received picture into several pictures containing separate numbers. Cutting a picture is not a mathematical operation.

3. Convert individual graphic characters to numbers. This is not a mathematical operation.

4. Add up the resulting numbers. Now that's mathematics.

The sum of the digits of the number 12345 is 15. These are the "cutting and sewing courses" from shamans used by mathematicians. But that is not all.

From the point of view of mathematics, it does not matter in which number system we write the number. So, in different 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 of 12345, I don’t want to fool my head, consider the number 26 from the article about. Let's write this number in binary, octal, decimal and hexadecimal number systems. We will not consider each step under a microscope, we have 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 like finding the area of ​​a rectangle in meters and centimeters would give you completely different results.

Zero in all number systems looks the same and has no sum of digits. This is another argument in favor of the fact that . A question for mathematicians: how is it denoted in mathematics that which is not a number? What, for mathematicians, nothing but numbers exists? For shamans, I can allow this, but for scientists, no. Reality is not just about numbers.

The result obtained should be considered as proof that number systems are units of measurement of numbers. After all, we cannot compare numbers with different units of measurement. If the same actions with different units of measurement of the same quantity lead to different results after comparing them, then this has nothing to do with mathematics.

What is real mathematics? This is when the result of a mathematical action does not depend on the value of the number, the unit of measure used, and on who performs this action.

Sign on the door Opens the door and says:

Ouch! Isn't this the women's restroom?
- Young woman! This is a laboratory for studying the indefinite holiness of souls upon ascension to heaven! Nimbus on top and arrow up. What other toilet?

Female... A halo on top and an arrow down is male.

If you have such a work of design art flashing before your eyes several times a day,

Then it is not surprising that you suddenly find a strange icon in your car:

Personally, I make an effort on myself to see minus four degrees in a pooping person (one picture) (composition of several pictures: minus sign, number four, degrees designation). And I do not consider this girl a fool who does not know physics. She just has an arc stereotype of perception of graphic images. And mathematicians teach us this all the time. Here is an example.

1A is not "minus four degrees" or "one a". This is "pooping man" or the number "twenty-six" in the hexadecimal number system. Those people who constantly work in this number system automatically perceive the number and letter as one graphic symbol.

Average level

Right triangle. Complete illustrated guide (2019)

RIGHT TRIANGLE. FIRST LEVEL.

In problems, a right angle is not at all necessary - the lower left one, so you need to learn how to recognize a right triangle in this form,

and in such

and in such

What is good about a right triangle? Well... first of all, there are special beautiful names for his parties.

Attention to the drawing!

Remember and do not confuse: legs - two, and the hypotenuse - only one(the only, unique and longest)!

Well, we discussed the names, now the most important thing: the Pythagorean Theorem.

Pythagorean theorem.

This theorem is the key to solving many problems involving a right triangle. It was proved by Pythagoras in completely immemorial times, and since then it has brought many benefits to those who know it. And the best thing about her is that she is simple.

So, Pythagorean theorem:

Do you remember the joke: “Pythagorean pants are equal on all sides!”?

Let's draw these very Pythagorean pants and look at them.

Does it really look like shorts? Well, on which sides and where are they equal? Why and where did the joke come from? And this joke is connected precisely with the Pythagorean theorem, more precisely with the way Pythagoras himself formulated his theorem. And he formulated it like this:

"Sum area of ​​squares, built on the legs, is equal to square area built on the hypotenuse.

Doesn't it sound a little different, doesn't it? And so, when Pythagoras drew the statement of his theorem, just such a picture turned out.


In this picture, the sum of the areas of the small squares is equal to the area of ​​the large square. And so that the children better remember that the sum of the squares of the legs is equal to the square of the hypotenuse, someone witty invented this joke about Pythagorean pants.

Why are we now formulating the Pythagorean theorem

Did Pythagoras suffer and talk about squares?

You see, in ancient times there was no ... algebra! There were no signs and so on. There were no inscriptions. Can you imagine how terrible it was for the poor ancient students to memorize everything with words??! And we can be glad that we have a simple formulation of the Pythagorean theorem. Let's repeat it again to better remember:

Now it should be easy:

The square of the hypotenuse is equal to the sum of the squares of the legs.

Well, the most important theorem about a right triangle was discussed. If you are interested in how it is proved, read the next levels of theory, and now let's move on ... into the dark forest ... of trigonometry! To the terrible words sine, cosine, tangent and cotangent.

Sine, cosine, tangent, cotangent in a right triangle.

In fact, everything is not so scary at all. Of course, the "real" definition of sine, cosine, tangent and cotangent should be looked at in the article. But you really don't want to, do you? We can rejoice: to solve problems about a right triangle, you can simply fill in the following simple things:

Why is it all about the corner? Where is the corner? In order to understand this, you need to know how statements 1 - 4 are written in words. Look, understand and remember!

1.
It actually sounds like this:

What about the angle? Is there a leg that is opposite the corner, that is, the opposite leg (for the corner)? Of course have! This is a cathet!

But what about the angle? Look closely. Which leg is adjacent to the corner? Of course, the cat. So, for the angle, the leg is adjacent, and

And now, attention! Look what we got:

See how great it is:

Now let's move on to tangent and cotangent.

How to put it into words now? What is the leg in relation to the corner? Opposite, of course - it "lies" opposite the corner. And the cathet? Adjacent to the corner. So what did we get?

See how the numerator and denominator are reversed?

And now again the corners and made the exchange:

Summary

Let's briefly write down what we have learned.

Pythagorean theorem:

The main right triangle theorem is the Pythagorean theorem.

Pythagorean theorem

By the way, do you remember well what the legs and hypotenuse are? If not, then look at the picture - refresh your knowledge

It is possible that you have already used the Pythagorean theorem many times, but have you ever wondered why such a theorem is true. How would you prove it? Let's do like the ancient Greeks. Let's draw a square with a side.

You see how cunningly we divided its sides into segments of lengths and!

Now let's connect the marked points

Here we, however, noted something else, but you yourself look at the picture and think about why.

What is the area of ​​the larger square? Right, . What about the smaller area? Certainly, . The total area of ​​the four corners remains. Imagine that we took two of them and leaned against each other with hypotenuses. What happened? Two rectangles. So, the area of ​​"cuttings" is equal.

Let's put it all together now.

Let's transform:

So we visited Pythagoras - we proved his theorem in an ancient way.

Right triangle and trigonometry

For a right triangle, the following relations hold:

The sine of an acute angle is equal to the ratio of the opposite leg to the hypotenuse

The cosine of an acute angle is equal to the ratio of the adjacent leg to the hypotenuse.

The tangent of an acute angle is equal to the ratio of the opposite leg to the adjacent leg.

The cotangent of an acute angle is equal to the ratio of the adjacent leg to the opposite leg.

And once again, all this in the form of a plate:

It is very convenient!

Signs of equality of right triangles

I. On two legs

II. By leg and hypotenuse

III. By hypotenuse and acute angle

IV. Along the leg and acute angle

a)

b)

Attention! Here it is very important that the legs are "corresponding". For example, if it goes like this:

THEN THE TRIANGLES ARE NOT EQUAL, despite the fact that they have one identical acute angle.

Need to in both triangles the leg was adjacent, or in both - opposite.

Have you noticed how the signs of equality of right triangles differ from the usual signs of equality of triangles? Look at the topic “and pay attention to the fact that for the equality of “ordinary” triangles, you need the equality of their three elements: two sides and an angle between them, two angles and a side between them, or three sides. But for the equality of right-angled triangles, only two corresponding elements are enough. It's great, right?

Approximately the same situation with signs of similarity of right triangles.

Signs of similarity of right triangles

I. Acute corner

II. On two legs

III. By leg and hypotenuse

Median in a right triangle

Why is it so?

Consider a whole rectangle instead of a right triangle.

Let's draw a diagonal and consider a point - the point of intersection of the diagonals. What do you know about the diagonals of a rectangle?

And what follows from this?

So it happened that

  1. - median:

Remember this fact! Helps a lot!

What is even more surprising is that the converse is also true.

What good can be gained from the fact that the median drawn to the hypotenuse is equal to half the hypotenuse? Let's look at the picture

Look closely. We have: , that is, the distances from the point to all three vertices of the triangle turned out to be equal. But in a triangle there is only one point, the distances from which about all three vertices of the triangle are equal, and this is the CENTER OF THE CIRCUM DEscribed. So what happened?

So let's start with this "besides...".

Let's look at i.

But in similar triangles all angles are equal!

The same can be said about and

Now let's draw it together:

What use can be drawn from this "triple" similarity.

Well, for example - two formulas for the height of a right triangle.

We write the relations of the corresponding parties:

To find the height, we solve the proportion and get first formula "Height in a right triangle":

So, let's apply the similarity: .

What will happen now?

Again we solve the proportion and get the second formula:

Both of these formulas must be remembered very well and the one that is more convenient to apply. Let's write them down again.

Pythagorean theorem:

In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the legs:.

Signs of equality of right triangles:

  • on two legs:
  • along the leg and hypotenuse: or
  • along the leg and the adjacent acute angle: or
  • along the leg and the opposite acute angle: or
  • by hypotenuse and acute angle: or.

Signs of similarity of right triangles:

  • one sharp corner: or
  • from the proportionality of the two legs:
  • from the proportionality of the leg and hypotenuse: or.

Sine, cosine, tangent, cotangent in a right triangle

  • The sine of an acute angle of a right triangle is the ratio of the opposite leg to the hypotenuse:
  • The cosine of an acute angle of a right triangle is the ratio of the adjacent leg to the hypotenuse:
  • The tangent of an acute angle of a right triangle is the ratio of the opposite leg to the adjacent one:
  • The cotangent of an acute angle of a right triangle is the ratio of the adjacent leg to the opposite:.

Height of a right triangle: or.

In a right triangle, the median drawn from the vertex of the right angle is equal to half the hypotenuse: .

Area of ​​a right triangle:

  • through the catheters: