1 appointed stake. Trigonometric Col. Detailed theory with applications. An example of a task to understand the inscribed and described stake

Vzagali, tse pitanya merits for special respect, but here everything is simple: there are degrees and sine and cosine are positive (wondering little ones), then we take the plus sign.

Now try to know the sine and cosine of cutiv on the basis of what has been added: i

You can cheat: zocrema for kuta degrees. Shards of one cut of a straight-cut tricutnik are closer to degrees, then the other - to degrees. Now nabuvayut chivalry know you formulas:

Then so, then th. So, then th. With degrees, everything is still simpler: so if one of the kutiv of a straight-cut tricot is closer to degrees, then the other one is closer to degrees, and also such a tricot of equal thighs.

So, yoga catheti is equal. And also, equal yogo sine and cosine.

Now find out for yourself for new appointments (through іks and іgorok!) the sine and cosine of cutіv in degrees and degrees. There are no more tricks to paint here! Already they will stink flat!

You don't have much to see:

You can know the tangent and cotangent ti independently using the formulas:

Catch your respect, you can't go to zero!

Now you can add numbers to the table:

Here the value of sine, cosine, tangent and cotangent cutiv 1st quarter. For the sake of clarity, you should point both in degrees and in radians (but now you know the connection between them!). Return respect for 2 dashes in the table: and for the cotangent of zero and the tangent of degrees. Not for nothing!

Zokrema:

Now let's understand the sine and cosine to a certain cut. I will look at two vipadas here:

1. Kut lie at the borders of the view to degrees
2. Kut more degrees

Seemingly vzagali, I twisted my soul, talking about "the arc of everything" kuti. The stench is so negative! Ale cei vpadok me with you will be looked at in another article. On the back of the hand, we are on the first step.

If you lie at the 1st quarter, then everything became clear, my view has already been looked at and painted on the tables.

Now let our kut be bigger for degrees and no more lower. Tse means that the vіn raztashovaniya either in 2, or in 3 or in 4 quarters.

How do we do it? But just like that!

Let's take a look replace the axis of such a vipadku ...

... axis like this:

So we can look at the kut, which lies at the other quarter. What can we say about the new one?

At the point, like a point, the crossbar changes and stakes, like before, there are 2 coordinates (nothing supernatural, right?). Tse coordinates in.

Moreover, the first coordinate is negative, and the other is positive! Tse means what at the end of the other quarter, the cosine is negative, and the sine is positive!

Wonderful, right? Until now, we have never stuck with a negative cosine.

The same principle of tsgogo boulo, if we looked at the trigonometric functions as seen from the blue sides of the tricutnik. Before the speech, think about it, do some kutahs have a cosine dorivnyu? What about equal sine?

Similarly, you can look at the kuti in all other quarters. I can hardly guess what is the case against the year's arrow! (as shown in the rest of the little one!).

Obviously, it is possible to wind up in the next day, but it will be even worse until such times.

Vhodyachi s pointing more mirkuvan, you can put signs at the sine, cosine, tangent (like dividing sine by cosine) and cotangent (like dividing cosine by sine) for all quarters.

And again, I repeat, there is no sense to remember these little ones. Everything you need to know:

Come on, let's work out a little bit with you. Forgive me the task:

Z'yasuvati, which sign can have such values:

Is it correct?

1. degrees - tse kut, greater and lesser, and also, lie at the 3rd quarter. Paint be-like a kut at 3 quarters and marvel, like a gravel at a new one. Vіn appear negative. Todi.
   degrees - cut 2 hours. The sine is positive and the cosine is negative. Plus dility by minus - will be minus. To mean.
   degrees - kut, greater and lesser. So, the vin lies at the 4th quarter. Whether there is a kuta of the fourth quarter, the “iks” will be positive,
2. With radians, it is practically the same: the other quarter is the same (the sine of the other quarter is positive).
   .
   , Tse kut fourth quarter. There cosine is positive.
   - I'll call you on the fourth quarter. The cosine is positive and the sine is negative. Then the tangent will be less than zero:

Maybe it's hard to guess the number of quarters by radians. At such a time, you can always go up to degrees. See, I realized, I’ll be exactly the same.

Now I want to buzz briefly for some moment. Let's guess the basic trigonometric identity again.

As I have already shown, we can now hang the sine through the cosine of the chi in the same way:

On the vibir of the sign, it’s less than that quarter, in which our kut alpha is known. On the rest of the two formulas, there is a mass task for EDI, for example, such:

manager

Know, yakscho i.

That's right, it's a quarter! Look how you see it:

Solution

So, then let's put this value, then. Now on the right behind the malim: rozіbratisya zі sign. What do we need for whom? To know, our kut is known to the yakіy quarter. For the mental task:. What a quarter? Fourth. What is the sign of the cosine of the fourth quarter? The cosine of the fourth quarter is positive. Then we have to choose the plus sign before. same.

I will not immediately report on such tasks, you can find their reports in the article "News". I don't want to point out to you the importance of what kind of sign this chi is a trigonometric function in the fall.

Cool more degrees

Remain, what would I want to mean in this article - why should I be with kutami, greater than degrees?

What is it and why can you eat, so as not to be pressed in? I’ll take, I’ll say, cut off degrees (radian) and follow the anti-year arrow.

I painted a spiral on a little one, but you understand, that in fact we don’t have any good spirals: we only have a few.

Then where can we go, how will we start from the singing kut and we will go through all around (degrees chi radians)?

Where shall we come? And let's come to the same kut!

Well, obviously, it is true for any other kut:

Having taken a certain kut and passed all around, we will turn to that same kut.

What can we give? And the axis sho: yaksho, then

The remaining sounds are taken away:

For whatever whole. Tse means what sine and cosine are periodic functions with a period.

In this order, there are many problems in knowing the sign of the now already full kuta: it’s enough for us to throw out all the “strong stakes”, as they fit in our vugill and z’yasuvati, to lie in the kut’s quarter, which is over.

For example, know the sign:

Verify:

1. Degrees decrease by degrees (degrees):
   out of degrees. Tse kut 4 quarters. There is a negative sine
2. . degrees Tse kut 3 quarters. There is a negative cosine. Todi
3. . . Bo, then - cut the first quarter. There cosine is positive. Todi cos
4. . . Oskіlki our kut lie at the other side, de sine is positive.

We can use a similar order for tangent and cotangent. However, they are actually simpler: they are also periodic functions, only the axis period is 2 times smaller:

Otzhe, ty ozumіv scho such a trigonometric number and for what it is needed.

Ale, we have lost more than a rich food:

1. And what are the negative kuti?
2. How to calculate the values ​​of trigonometric functions in circles
3. Like for the given values ​​of trigonometric functions 1 quarter, look for the values ​​of functions in the other quarters (shouldn't you need to cram the table?!)
4. How to ask for the help of a stake for the decoupling of trigonometric equals?

MIDDLE RIVEN

Well, well, at this article, you can continue to plant a trigonometric stake and discuss the upcoming moments:

1. What are negative kuti?
2. How to calculate the values ​​of trigonometric functions in these circles?
3. How to find the values ​​of trigonometric functions in four quarters?
4. What are all tangents and all cotangents?

We don’t need any additional knowledge, around the basic skills of a robot with a single stake (front article). Well, let's get down to the first meal: what are the negatives?

Negative Kuti

Negative cuts in trigonometry they are placed on a trigonometric stake down into the cob, at the right turn of the year's arrow:

Let's guess, as we used to put kuti on the trigonometric stake: We went in the direction of a positive axis anti-year arrow:

Todі on a little urge kut, equal. Similarly, we were all kuti.

However, nothing prevents us from going in the direction of positive axis straightening. according to the anniversary shooter.

We will also take away the different kuti, but the stench will be already negative:

On the next picture, two kuti are depicted, equal behind the absolute value, and parallel behind the sign:

The rule can be summarized as follows:

• Idemo against Godinnikov's arrow - an extremely positive cut
• Idemo for the year's arrow - take away the negative kuti

Schematically, the rule is depicted on this little one:

In a moment, put me a whole reasonable supply: well, we need it in order to change their values ​​of sine, cosine, tangent and cotangent.

What is the difference, if we have a positive kut, and if it is negative? I swear to you: as a rule, є.

However, you can always calculate the calculation of the trigonometric function in the form of a negative cut to the calculation of the function in the round positive.

Take a look at this picture:

I prompted two kuti, stench equal to absolute values, but to make an opposite sign. The sine and cosine on the axes are significant for the skin.

What are we doing with you? And what about the axis:

• Sinuses bіlya kutіv і protilezhnі behind the sign! Just the same
• Cosine bіlya kutіv i zbіgayutsya! Just the same
• Because:
• Because:

In this rank, we can always get rid of negative sign in the middle, be like a trigonometric function: either simply by missing yogo, like a cosine, or putting yogo in front of the function, like a sine, tangent and cotangent.

Before the speech, guess what the function is called, in which, for whatever is acceptable, it is vykonuetsya: ?

Such a function is called unpaired.

And how for any admissible vykonuєtsya:? Then in such a time the function is called a steam room.

In this rank, we have shown you well that:

In such a manner, as if you were wise, there is no other difference, we are talking about a sine in a positive cut or a negative one: it’s easy to turn around with a minus. Also, we do not need tables for negative cuts.

From the other side, wait a minute, it would be better to know only the trigonometric functions of the first quarter, to calculate the analogous functions for the other quarters. Chi can tse zrobiti? Obviously you can! You have to take two paths: the first one is to make a tricot and to put the Pythagorean theorem (this is how we figured out the values ​​of trigonometric functions for the main cuts of the first quarter), and the other - remembering the values ​​of the functions for the quarters of the first quarter and, as a clumsy rule, remember to calculate the trigonometric functions for all the other quarters. Another way to spare you long-term troubles with tricots and with Pythagoras, so I can look forward to promising ones:

Later, the Danish method (or the rule) is called the formula of the reduction.

Guidance formulas

Roughly seeming, these formulas will help you not to remember such a table (you won’t have to miss 98 numbers!):

how do you remember the qiu axis (for a total of 20 numbers):

So you can not fill your head with uncommon 78 numbers! Come on, for example, we need virahuvati. I realized that a small table does not have such a thing. What are we supposed to do? And what about the axis:

First, we need such knowledge:

1. Sine and cosine determine the period (degrees) tobto

   Tangent (cotangent) of period (degrees)

   Whether it's a whole number

2. Sine and tangent are unpaired functions, and cosine is paired:

We have already brought the first firmness to you, and the justice of the other was established not long ago.

The direct reduction rule looks like this:

1. However, we can calculate the value of the trigonometric function in the form of a negative kuta - we can also use the positive one for the additional group of formulas (2). For example:
2. It can be seen for the sine and cosine of the yogo period: (in degrees), and for the tangent - (degrees). For example:
3. Like a “little piece”, which is overflowing, less degrees, then the task is higher: it’s joking yogo in the “small table”.
4. Otherwise, it’s safe to say that our kut lies at some quarter: it will be 2, 3 o’clock 4 quarter. We wonder what sign may need a function in the fourth. Forgotten this sign!
5. We represent a kut in one of the following forms:

   (like the other quarter)
   (like the other quarter)
   (like in the third quarter)
   (like in the third quarter)
   
   (like the fourth quarter)

   so, shob kut, scho zalishivsya, more than zero and less degrees. For example:

   In principle, it doesn’t matter, in which of the two alternative forms for the skin, you will find it. I'm sorry not to be identified.

6. Now we are wondering what we have done: if you select the record through either degrees plus minus, then the sign of the function will not change: you just pick up or write down the sine, cosine or tangent of the kuta, which is gone. If you select the record through degrees or degrees, then the sine is changed to cosine, cosine to sine, tangent to cotangent, cotangent to tangent.
7. We put the sign і from point 4 in front of the viraz.

Let's demonstrate all the above on butts:

1. Calculate
2. Calculate
3. Know-dі-te know-che-nya vi-ra-same-nya:

Let's start in order:

1. Diemo zgidno z our algorithm. We see the number of kilos for:

   Zagalom, robimo wisnovok, what should I go to the kut again 5 times, and the skilki is gone? Lost. Todi

   Well, we got kicked out of the zave. Now we know each other. lie at the 4th quarter. The sine of the fourth quarter may be a minus sign, and I am not guilty of forgetting to put it in vіdpovіdі. Dalі, podaєmo zgіdno z dnієyu z two formulas to paragraph 5 of the guidance rules. I choose:

   Now we wonder what happened: we have a drop in degrees, then we see that the sine is changed to the cosine. I put a minus sign in front of it!

   degrees - cut at the first quarter. We know (those are the names of the small table !!) of this meaning:

   Then we take away the residual evidence:

   Suggestion:

2. all the same, but instead of degrees - radians. Nothing terrible. Golovne remembrance, scho

   Ale, you can not change radians to degrees. Tse food for your relish. I am not changing anything. I’ll start again after seeing the number of kil:

   Looks like two whole stakes. Too much to count. Tsey kut is found at the third quarter. The cosine of the third quarter is negative. Don't forget to put a minus sign next to vodpovidi. you can show yak. Let's make a new rule: we have a fall of an integer number (otherwise), then the function does not change:

   Todi.
   Suggestion: .

3. . It is necessary to work all the same, but also with two functions. I will be three short: and degrees - cut another quarter. The cosine of the other quarter has a “minus” sign, and the sine is a “plus”. you can show it like:, but yak, then

   Obidva vipadki - "halves of the whole". Then the sine changes to a cosine, and the cosine changes to a sine. Moreover, there is a minus sign in front of the cosinus:

Suggestion: .

Now practice on your own on these stocks:

Axis i solution:

1. 
   Let's get rid of the minus on the back, vinish yoga before the sinus (sinus oscillators - the function is unpaired !!!). Let's take a look at Kuti:

   Vіdkidaєmo tsіlu kіlkіst kіl - that's three stakes ().
   Required to calculate: .
   So we fix it ourselves and with another kut:

   We can see the number of kіl - 3 cola () todi:

   Now we think: in what quarter to lie kut, what is left? Vіn "does not reach" to everything. Todi yak a quarter? Fourth. What is the sign of the cosine of the fourth quarter? Positive. Now we see. If the stones are visible to me from a whole number, then the sign of the cosine is not changed:

   Submitting all subtractive data to the formula:

   Suggestion: .

2. 
   Standard: minus is taken from the cosine, peeling tim, scho.
   The cosine of degrees has been lost. Let's take the number of stakes: . Todi

   Todi.
   Suggestion: .

3. Diemo, like a front butt.

   Don’t forget that the period of the tangent is (or) the difference between the cosine and the sine, for some wines it is 2 times greater, then we can see the number of times.

   degrees - cut at the other quarter. The tangent of the other quarter is negative, but let's not forget about the "minus"! you can write it down. The tangent is changed by the cotangent. Remaining taken:

   Todi.
   Suggestion: .

Well, well, I lost my good fortune!

All tangents and all cotangents

Stop, on what I would like to be here zupinitis - tse two additional axes. As we have already discussed, we may have two axes:

1. Віс - віс cosines
2. All - all sinuses

We ran out of real coordinate axes, why not? How about using tangents and cotangents?

Is there no other graphic interpretation for them?

Really won є, її you can work on the axis of this picture:

Zokrema, behind these pictures you can say the axis of what:

1. Tangent and cotangent may have the same signs on quarters
2. The stench is positive in the 1st and 3rd quarters
3. The stench is negative for 2 and 4 quarters
4. Tangent of inconsistencies at cutahs
5. Cotangent of inconsistencies in kutahs

Need more pictures? Finding out on a slipped equal, de I will tell you, as for the help of a trigonometric stake, you can ask for the derivation of trigonometric equals!

PROSUNUTIY RIVEN

In my article I will describe how Single Colo (Trigonometric Colo) can become in good luck with the virishenni trigonometric rivnyan.

I can see two drops, if you can see the brown one:

1. We don’t have a “garniy” kut in our country, but all the same, it’s necessary to carry out a selection of roots
2. At the vіdpovіdі come out too rich a series of roots

You don’t need any specific knowledge of the day, but the knowledge of those:

Topic " trigonometric alignment”I tried to write, without going into the stake. Bagato who wouldn’t praise me for such a pidkhid.

Alemeni nicer formula, scho porobish here. However, there are few formulas in such vipadkas. Write this article for me, motivating me with a stepping butt:

Untie the river:

Well, what. Virishiti itself is awkward.

Return replacement:

Zvіdsi our vihіdne rіvnyannja rіvnosnіnja аlѕо thе simplest ріvnannyа! It will not be necessary for us to write down 4 series of roots:

At the principle, on which one can b zupinitis. But I don’t just read the articles, as I pretend to be “complicated”!

On the back of the hand, I look at the root series. Otzhe, take a single kolo, now let's put the same root on the kolo (okremo for and for):

Strike respect: what kind of kut viyshov mizh kutami ta? Tse cut. Now zrobimo those same for the series: .

Mіzh korіnnyam rivnyannya znovu viyshov kut. And now summіsny qі two pictures:

What are we doing? And then all kuti mіzh our roots are equal. And what does it mean?

As soon as we start from the kuta and brotherly kuti, equal (for whatever whole), then we will always be consumed in one of the four points on the upper stake! In this order, 2 series of roots:

Can be combined into one:

Unfortunately, for the series root:

Qi mirkuvannya will no longer be fair. Zrob the chair and understand why it is so. However, they can be united by an offensive rank:

Todi vihіdne equal maє root:

What to do with a short and concise reply. And what about succinctness and conciseness? About the level of your mathematical literacy.

This was the first butt, for which the victorious trigonometric stake gave corny fruits.

Another butt is rivnyannya, yakі mayut "negarne korіnnya".

For example:

1. Untie the fiery.
2. Find out the root, what to lay down.

The first part is not anything folding.

If you already know the topic, then I will allow myself to be short in my tabs.

same chi

So we knew the root of our jealousy. Nothing complicated.

Folding the other part of the task, not knowing why exactly the arccosine of the minus one quarter (not tabular value) is correct.

However, we can depict the known series of roots on a single stake:

What are we doing? First, the little ones gave us a sense of understanding, in such a boundary lies the arccosine:

Tsya visual interpretation will help us to know the root, what to lie down: .

First, until the next one, we spend the number itself, then (div. Fig.).

so lie down.

In such a rank, alone, helping to make a difference, in between, they spend “negarni” kuti.

You have little to spare for one more food: and how can we use tangents and cotangents?

In fact, for them, they have their own axis, it’s true, the stench may have a specific look:

In another way, they will be the same way, like with sine and cosine.

butt

Rivnyanya is given.

• Untie the given equal.
• Enter root this equal, scho lie down to promіzhku.

Solution:

Slightly alone, and indicative of our new decision:

From the little one you can understand that:

Abo navit more: so like, then

Todi we know the root, what to lie in a vіdrіzku.

, (so yak)

I hope that you will independently change, that there is no other root, that there is no need for intercourse, our jealousy cannot.

SHORT VICLAD AND BASIC FORMULA

The main tool of trigonometry - ce trigonometric colo, it allows you to win the kuti, know their sines, cosines and more.

There are two ways to win kuti.

1. Through degrees
2. Through radio

І navpaki: from radians to degrees:

To know the sine and cosine of the kuta is necessary:

1. Draw a single column from the center, which runs from the top of the kut.
2. Find the point of the crossbar of the th kut with the stake.
3. Її "іksova" coordinate - tse cosine of a shuk kut.
4. Її "іgrekova" coordinate - tse sine of a shukany kut.

Guidance formulas

Tse formulas that allow easy folding of trigonometric functions.

Qi formulas will help you not to remember the following table:

Delivery of bags

You learned how to work a universal spur with trigonometry.

You have learned to virishuvati zavdannya richly easier and faster and more, more headily, without pardons.

You understand that you don’t need to cram the tables and you don’t need to cram!

Now I want to feel you!

How far away did you grow up from this collapsible topic?

What did you like? What did not fit?

Mozhlivo, do you know a pardon?

Write in the comments!

Good luck on your sleep!

Let's sort out the back of the head on the vіdmіnu between the stake and the circle. Schob pochit tsyu raznitsu, enough to look at, chim є offending figures. The number of points of the plane is indistinguishable, which are located on a straight line in a single central point. Ale, if it’s stacked up and out of the inner space, then it’s not enough to lie down. To come out, what is around and around, what is between the yogo (o-circle), and the indistinguishable number of points, what is in the middle of the stake.

For whether or not the point L, which lies on the stake, dіє rіvnіst OL=R. (Dovzhina vіdrіzka OL dorіvnyuє the radius of the stake).

Vіdrіzok, yakyi z'єdnuє dvі points of the stake, є її chordia.

Chord that goes straight through the center of the stake, є diameter which stake (D) . The diameter can be calculated using the following formula: D=2R

Dovzhina circumference calculated according to the formula: C=2\pi R

stake area: S=\pi R^(2)

arc stake that її part is called, as it is roztashovuєtsya between two її points. Two points indicate two arcs of the stake. The CD chord spans two arcs: CMD and CLD. However, the chords draw the same arcs.

central kutom such a cut is called, which is between two radii.

Dovzhina arc can be known by the formula:

1. Vikoristovuyuchi degree setting: CD = \frac(\pi R \alpha ^(\circ))(180^(\circ))
2. Vykoristovuyuchi radial zahіd: CD = \alpha R

The diameter, which is perpendicular to the chord, divides the chord and tightens the arcs with it.

If the chords AB and CD are to be crossed at the point N, then create two chords separated by the point N, equal between themselves.

AN\cdot NB = CN \cdot ND

Stosovno cola

Stosovno cola It is customary to name a straight line, for which there is one scorched dot for a ring.

Well, there are two doors dots, її name sichuchoi.

How to draw a radius at the point of the torsional, it will be perpendicular to the dot to the stake.

We will conduct two dotichni z tsієї points to our stake. Wiyde, that the dot-like branches are aligned one by one, and the center of the stake is spread out on the bisector of the kuta with the top at this point.

AC=CB

Now, to the stake from our point, we will draw it to the same point. Let's take it that the square of the dozhina vіdrіzka dotichї dоrіvnyuvatime dоbіtku аvsоgo vіrіzka sіchnі on the yogo ovnishnіshnyu part.

AC^(2) = CD \cdot BC

It is possible to make biscuits: to make a whole cut of the first siknoy on the yogo lower part of the door, to get a whole cut of the other siknoy to the yogo upper part.

AC \cdot BC = EC \cdot DC

Kuti in coli

Degrees come in the central kuta and arcs, like that one spirals, even.

\angle COD = \cup CD = \alpha ^(\circ)

Inscriptions cut- Tse kut, the top of which is known on a stake, and the sides are revenged by chords.

You can calculate it by knowing the size of the arc, the shards of the dorsal halves of the arc.

\angle AOB = 2 \angle ADB

Which spirals on the diameter, fits in the cut, straight.

\angle CBD = \angle CED = \angle CAD = 90^ (\circ)

Inscribed kuti, which spiral on one arc, are the same.

Relying on the same chord, inscribed kuti totozhnі chi yogo suma dorіvnyuє 180 ^ (\circ) .

\angle ADB + \angle AKB = 180 ^ (\circ)

\angle ADB = \angle AEB = \angle AFB

At one point, there are tops of trikutniks with the same kuts and a given basis.

The kut with the top in the middle of the stake and rotting between two chords is the same halves of the sum of the sum of the kutov values ​​of the stake arcs, as if they are the middle of this vertical kutiv.

\angle DMC = \angle ADM + \angle DAM = \frac(1)(2) \left (\cup DmC + \cup AlB \right)

The kut with the top of the posture is about і raztashovaniya міzh two sіchnimi the same halves of the variance of the kutov values ​​of the arches of the stake, yakі polagat the middle of the kuta.

\angle M = \angle CBD - \angle ACB = \frac(1)(2) \left (\cup DmC - \cup AlB \right)

inscribed colo

inscribed colo- Tse kolo, scho stuєtsya sides of the bagatokutnik.

At the point, de the bisections of the kutiv of the bagatokutnik are intertwined, the center is roztashovuetsya.

Kolo may be inscribed not in the leather bagatokutnik.

The area of ​​​​the bagatokutnik with an inscribed stake is known by the formula:

S = pr

p - napіvperimeter bagatokutnik,

r - radius of the inscribed stake.

You can see that the radius of the inscribed stake is correct:

r = \frac(S)(p)

The sumi of the dozhins of the opposite sides will be the same, as the colo is inscribed at the opukliy chotirikutnik. First of all: in opukliy chotirikutnik fit into a colo, as if in a new sumi dozhins of the opposite sides of the same.

AB+DC=AD+BC

You can write a colo on any of the trikutniks. Less than one unit. At the points, where the bisectors of the inner edges of the figures are intertwined, the center of the inscribed stake lies.

The radius of the inscribed stake is calculated using the following formula:

r = \frac(S)(p) ,

de p = \frac(a + b + c)(2)

Description

If it is possible for the colo to pass through the skin top of the bagatokutnik, then such a colo is accepted to be called description.

At the point of the crossbar of the middle perpendiculars of the sides of the figure, there will be the center of the described stake.

You can know the radius by counting it as the radius of the stake, as described by the tricot, which was designated by the tops of the mullet.

Є such a mind: how much can you describe a chotirikutnik more closely, as if the sum of the yogo of the adjacent kutіv is more expensive 180^(\circ) .

\angle A + \angle C = \angle B + \angle D = 180 (\circ)

Bіlya be-such a tricoutnik can be described as a colo, moreover, one-one. The center of such a stake will be ruffled at the points, where the middle perpendiculars of the sides of the tricutnik are intertwined.

The radius of the described stake can be calculated using the formulas:

R = \frac(a)(2 \sin A) = \frac(b)(2 \sin B) = \frac(c)(2 \sin C)

R = frac(abc)(4S)

a, b, c - Long sides of tricot,

S - trikutnik area.

Ptolemy's theorem

Nasamkinets, let's look at Ptolemy's theorem.

Ptolemy's theorem proves that the sum of the diagonals of the same sum of the opposite sides of the inscribed chotirikutnik is obtained.

AC \cdot BD = AB \cdot CD + BC \cdot AD

Lecture: Colo ta colo

circumference- the curve is closed, all points of which are on the same line in the center.

At everyday life You have repeatedly strummed the kolo. The very same time is described by that second arrow, the very shape of the stake can be a gymnastic hoop.

And now to show that you painted a colo on the arch paper and wanted to embellish it.

So the axis of the whole embellishment is spacious, the surroundings of the colo - ce and є colo.

The center is the whole point, which is equally distant from the other points of the stake. The center of the stake, that stake is denoted by the letter O.

Radius - tse v_dstan in the center to the stake (R).

Diameter - ce of the rails that pass through the center, which is the rear mustache of the point of the stake (d). Moreover, the diameter is equal to two radii: d = 2R.

The chord is a wedge, which happens to be two points on a stake. Diameter є let's call it a chordi.

In order to know the value of the stake, it is necessary to speed up with the formula:

l=2 πR

To raise respect, dovzhina stake, the area to lie only in the radius of the stake.

The area of ​​the stake can be known by the following formula:

S=πR 2 .

I would like to return your respect to the number "Pi". The meaning was found behind the help of the stake. For this її, the dovzhina was divided into two radii, and the number "Pі" became such a rank.

As soon as the two parts are divided into two parts by radii, such parts will be called sectors. The skin sector has its own degree world - the degree world of those arcs, spirals in a way.

In order to know the length of the arc, it is necessary to quickly use the formula:

2. Vikoristovuyuchi radial approach:

As the top of the deaky kut spirals onto the center of the stake, as if changing the stake, such a kut is called the central one.

Like two deacons, two chords intertwine at the singing points, their proportions are proportional:

І colo- geometric posts, mutually related to each other. є border laman line (curve) cola,

Appointment. The circle is a closed curve, the skin point is equal to the point called the center of the stake.

To encourage the stake, a sufficient point is selected, taken as the center of the stake, and a closed line is drawn behind the help of the compass.

As a point to the center of the stake with equal points on the stake, then all the other marks will be equal to each other, and such marks are called radii, shortly denoted by the Latin small or great letter "er" ( r or R). Radius at the time it is possible to draw styles w, skilki dot can dovzhina stake.

The diameter that connects two points of the stake and passes through the center is called the diameter. Diameter consists of two radii that lie on one straight line. The diameter is indicated by the Latin small or great letter "de" ( d or D).

Rule. Diameter kola dorivnuє two її radii.

d = 2r
D=2R

The length of the stake is calculated according to the formula and deposited according to the radius (diameter) of the stake. The formula has the number ¶, as it shows the number of times the number of times is larger, the lower one is the diameter. The number ¶ may be an infinite number of signs after Komi. For calculation it is accepted = 3.14.

Dovzhina stake is signified by the great Latin letter "ce" ( C). Dovzhina stake is proportional to the diameter. Formulas for rozrahunka dozhini stake for її radius and diameter:

C = ¶d
C = 2r

• Apply
• Given: d = 100 div.
• Dovzhina stake: C=3.14*100cm=314cm
• Given: d = 25 mm.
• Dovzhina stake: C \u003d 2 * 3.14 * 25 \u003d 157 mm

Sikna Cola That Arc Cola

Be-yaka sіchna (straight line) peretinaє kolo at two points and divide її into two arcs. The value of the arc of the stake lies in the middle of the center and the center of the stake and varies along a closed curve from the first point of the crossbar between the stake and the other.

arcs cola share sichuchoi to big and small, as if it didn’t fit with the diameter, and on two equal arcs, as if it were to pass through the diameter of the stake.

If it is possible to pass through the center of the stake, then її vіdrіzok, roztashovannyh between the points of the crossbar with the stake, є the diameter of the stake, or the largest chord of the stake.

What is the distance from the sіchna roztashovana in the center of the stake, the lesser degree of the world is the smaller arc of the stake and more - the greater arc of the stake, and the crown of the stake, ranks chordia, changing in the world of the distance from the center of the stake.

Appointment. It is called a part of the area that lies in the middle of the stake.

The center, radius, diameter of the stake is the same as the center, radius and diameter of the double stake.

So as a column - tse part of the area, then one of the yogo parameters is the area.

Rule. Stake area ( S) more square to the radius ( r2) to the number ¶.

• Apply
• Given: r = 100 cm
• Stake area:
• S \u003d 3.14 * 100 cm * 100 cm \u003d 31,400 cm 2 ≈ 3m 2
• Given: d = 50 mm
• Stake area:
• S \u003d ¼ * 3.14 * 50 mm * 50 mm \u003d 1 963 mm 2 ≈ 20 cm 2

If we draw two radii to the different points of the stake, then two parts of the stake are established, which are called sectors. If you draw a chord, then a part of the area between the arc and that chord is called stake segment.

What is so lonely. A single column - a column with a radius equal to 1, і with a center on the cob of coordinates. Guess what equal stake looks like x2+y2=1. Such a call can be used for the recognition of certain "special" trigonometric spivvіdnosheniya, as well as when prompting graphic images. For additional help, it is possible to estimate the numerical values ​​of trigonometric functions in the line.

Memorize 6 trigonometric ratios. Remember what

• sinθ=protean leg/hypotenuse
• cosθ=adjacent leg/hypotenuse
• tgθ=adjacent leg/adjacent leg
• cosecθ=1/sin
• secθ=1/cos
• ctgθ=1/tg.