Analysis of trigonometric rivnyan ege on stake. Razv'azannya trigonometric rivnyan on promizhku. Comparison of two methods

⁰

a) Razv'yazhit rivnyannya:.

b) Find out the mustache of the roots of the equal, which lies in the vіdrіzku.

Solution of the task

At this level, the butt of the development of trigonometric alignment is seen, which can be victorious as a butt for the development of problems of type C1 for an hour of preparation to ЄDI s of mathematics.

Nasampered, the scope of the assigned function is determined - all valid values of the argument. Then, in the course of the solution, the transformation of the trigonometric function of the sine into the cosine from the formula of the reduction takes place. Dali all the members of the equal are transferred to the yogo left part, where the guilt is carried out for the arms of the head multiplier. The skin multiplier is equal to zero, which allows you to determine the root equal. Then, by the method of turns, the roots are determined, which should lie with the given vіrіzku. For which, on a single occasion, a coil is assigned from the left between the given winding to the right. Dalі znaydenі roots on a single kolі z'єdnuyutsya vіrіzki z її tsії і vynachayutsya points, yіѕ і і і іії ії іїії peretinayut coil. Give points to cross the line and shukanoyu vіdpovіddu on another part of the problem.

meta lesson:

a) close in mint rozvyazuvat simple trigonometric alignment;

b) learn to choose the root of trigonometric equalities from a given gap

Hid lesson.

1. Actualization of knowledge.

a) Rechecking the homework: the class is given a superior homework - virishity equal and know the way to choose the root of your homework.

1) cos x= -0.5 de xI [-]. Suggestion:.

2) sin x=, de xI. Suggestion: ; .

3) cos2 x= - de xI. Suggestion:

Learn to write down the solution on the doshtsі htos from the auxiliary graph, htos by the selection method.

What time is class practice orally.

Find out the meaning of virus:

a) tg - sin + cos + sin. Suggestion: 1.

b) 2 arccos 0 + 3 arccos 1. Suggestion: ?

c) arcsin + arcsin. Suggestion:.

d) 5 arctg (-) - arccos (-). Suggestion:–.

– Let’s reconsider homework, open up your sewing from home robots.

Deyakі from you knew the solution by the method of selection, and dehto for the help of the schedule.

2. Visnovok about the ways of accomplishing these goals and posing the problem, to help them teach the lesson.

- a) For help picking up, it’s easy to win, like a great gap in tasks.

- b) The graphical method does not give accurate results, it will require rechecking and it takes a lot of time.

– There can be at least one way, the most universal one – we will try to know it. Father, what do we do today at the lesson? (Learn to choose the root of the trigonometric alignment for a given gap.)

- Butt 1. (Learn to go to the board)

cos x= -0.5 de xI [-].

Inquiry: Why lay waste on the plant? (Vіd zagalnogo rіshennya vnyannya. Let's write down the solution in zagalny vglyadі). The solution is to sign up on the board

x = + 2?k, de k R.

- Let's write down the decision at the sight of the totality:

- How do you care, for which record of the decision to manually select the root for the prom? (From another entry). Ale, I’ll redo the method of selection. What do we need to know in order to take the correct advice? (You need to know the value of k).

(A foldable mathematical model for the value of k).



scalars kI Z, then k = 0, stars X= =	From the point of unevenness, it is clear that there are no integer values of k.

Visnovok: To choose a root from a given gap when arranging a trigonometric alignment requirement:

for the perfection of the mind sin x = a, cos x = a it’s easier to write down the root equal, like two series of the root.
for the sake of perfection tan x = a, ctg x = a write down the formula of the root.
put together a mathematical model for a skin solution that looks like underwire unevenness and know the purpose of the value of the parameter k or n.
Substitute the values of the formula of the roots and calculate them.

3. Fixed.

Butt No. 2 and No. 3 from the home task of virishity, vikoristovuyuchi deleting the algorithm. At the same time, two students work on the doshka with a further reverberation.

In my article I will try to explain 2 ways selection of roots in trigonometric alignment: for the help of irregularities, that for the help of a trigonometric stake. Let's move on to the last butt and let's do the right thing.

A) Razv'yazati equalization sqrt(2)cos^2x=sin(Pi/2+x)
b) Find out the mustache of the root of the river, which lies before the gap [-7Pi / 2; -2Pi]

Let's untie point a.

Accelerated by the reduction formula for the sine sin(Pi/2+x) = cos(x)

Sqrt(2)cos^2x = cosx

Sqrt(2)cos^2x - cosx = 0

Cosx(sqrt(2)cosx - 1) = 0

X1 = Pi/2 + Pin, n ∈ Z

Sqrt(2)cos - 1 = 0

cox = 1/sqrt(2)

Cox = sqrt(2)/2

X2 = arccos(sqrt(2)/2) + 2Pin, n ∈ Z
x3 = -arccos(sqrt(2)/2) + 2Pin, n ∈ Z

X2 = Pi/4 + 2Pin, n ∈ Z
x3 = -Pi/4 + 2Pin, n ∈ Z

Virishima paragraph b.

1) Vіdbіr korіnіv for pomomogoyu nerіvnosti

Here everything is easy to fight, the otrimane of the root is presented in the tasks of the interval [-7Pi / 2; -2Pi], we know the number of values for n.

7Pi/2 less or one Pi/2 + Pin less or more -2Pi

Everything is immediately divisible on Pi

7/2 less or one 1/2 + n less or more -2

7/2 - 1/2 less or more n less or more -2 - 1/2

4 less or less n less or less -5/2

Tsіlі n at tsmu promіzhku tse -4 and -3. Mean the root, which should be given to this gap, will be Pi/2 + Pi(-4) = -7Pi/2, Pi/2 + Pi(-3) = -5Pi/2

Similarly, two more nervousness

7Pi/2 less or more Pi/4 + 2Pin less or more -2Pi
-15/8 less or one n less or one -9/8

Tsilih n for whom there is no space

7Pi/2 less or more -Pi/4 + 2Pin less or more -2Pi
-13/8 less or one n less or one -7/8

One goal n for each interim tse -1. To mean the selection of the root of this interval -Pi/4 + 2Pi*(-1) = -9Pi/4.

Mean value for point b: -7Pi/2, -5Pi/2, -9Pi/4

2) Selection of roots for the help of a trigonometric stake

In order to be curly in this way, you need to understand how to practice as a whole. I am trying to explain with my simple mind how I am convinced. I think that in schools for an hour of algebra lessons, the topic was explained in a rich way with the sensible words of the teacher, and in handymen, it was folded formulary. In particular, I understand it as a colo, as it is possible to remember an infinite number of times, it is explained by the fact that the functions sine and cosine are periodic.

Let's do it again against the year's arrow

Obidemo 2 times against the arrows of the year

Obidemo 1 time for the annual arrow (the value will be negative)

Let's turn to our food, we need to choose the root for the interweaving [-7Pi/2; -2Pi]

To spend up to the number -7Pi / 2 and -2Pi, you need to go around the counter of the year's arrow. In order to know the root of the line for your future, you need to estimate and represent.

Look at x = Pi/2 + Pin. What is approximately responsible for buti n, so that the value of x was here for whom? It is possible, permissible -2, acceptable Pi / 2 - 2Pi = -3Pi / 2, obviously not included in our gap, it means less than -3, Pi / 2 - 3Pi = -5Pi / 2, it’s worth it, we can try -4 , Pi/2 - 4Pi = -7Pi/2, also suitable.

Razmirkovuyuchi similarly for Pi / 4 + 2Pin and -Pi / 4 + 2Pin, we know one more root -9Pi / 4.

Comparison of two methods.

The first way (for the help of inconsistencies) is richly superior and richly simple for reasoning, but even more seriously, if you seriously deal with a trigonometric stake and with another method of selection, then the selection of roots will be richer, you can save close to 15 spililin.

Manager No. 1

The logic is simple: let's fix it the way it was done before, regardless of those that now have trigonometric functions with a folding argument!

Yakby virishuvali look:

Then we would write down the axis like this:

Abo (shards)

But now the role we play is such a viraz:

Todi can be written:

Our meta is with you - robiti so that the lion-hander standing simply, without the usual "houses"!

Let's step by step call them!

On the back of the head, we’ll take away the banner at: for whom we multiply our jealousy by:

Now let's wake up, dividing into new offending parts:

Now let's take a break:

Otrimane Viraz can be written as two series of solutions

We need to know the biggest negative root! It dawned on me that I needed to sort it out.

Let's take a look back at the beginning of the series:

It is clear that we should take positive numbers as a result, but the stink does not stink us.

Otzhe, you need to take the negative. Come on.

At the root will be already:

And we need to know the most negative! There is no sense here to go with a negative beak. І the most negative root for the tsієї series is more expensive.

Now we look at a friend of the series:

I re-submit: , Todi:

Do not click!

Todi zbіshuvati more no sense! Changememo! Come on then:

Go!

Come on. Todi

Todi is the biggest negative root!

Suggestion:

Manager No. 2

I’m victorious again, regardless of the collapsible argument of the cosine:

Now I’m speaking again with a levoruch:

We multiply the offending parties on

Dilimo offending parties on

All that is left out is to transfer the right hand, changing the sign from minus to plus.

We are re-entering 2 series of roots, one, and another with.

We need to know the biggest negative root. Let's take a look at the first series:

It is clear that the first negative root is taken away, it will be more expensive and will be the largest negative root in series 1.

For another series

The first negative root will be taken away and will be stronger. So, that is the most negative root of equalization.

Suggestion: .

Manager No. 3

Virishuemo, regardless of the folding argument of the tangent.

Axis, there’s nothing foldable, isn’t it?

Like before, it can be seen in the left part:

Well, it’s a miracle, there’s only one series of roots here! I know the biggest negative.

It is clear what to go out, what to put down. I root is dear.

Suggestion:

Now try to independently sing such a task.

Home robot or 3 tasks for an independent achievement.

Untie the fiery.
Untie the fiery.
The vіd-vі-tі on-pi-shi-te has the smallest in-lo-zhi-tel-ny root.
Untie the fiery.
The vіd-vі-tі on-pi-shi-te has the smallest in-lo-zhi-tel-ny root.

Ready? We revise. I will not describe the entire algorithm of the solution in a report, I’m sure, I’ve been given enough respect to you.

Well, is everything correct? Oh, already, these are fluid sinuses, with them, for sure, it’s dashing!

Well, now you've got the simplest trigonometric alignment!

Look at the decisions and opinions:

Manager No. 1

Vislovimo

The smallest positive root is viide, as if to put it, to that

Suggestion:

Manager No. 2

The smallest positive root is viide.

Vіn dorivnyuvatime.

Suggestion: .

Manager No. 3

With otrimuemo, maєmo.

Suggestion: .

This knowledge will help you to see richly in advance, with which you are hunkered down in sleep.

If you are applying for a rating of "5", then it is simply necessary to go to the reading of the article for average level, yak will be assigned to the folded trigonometric alignments (task C1).

MIDDLE RIVEN

In my article I will describe rozvyazannya trigonometric rivnyan folded type and how to conduct a selection of their roots. Here I rush to the next by those:

Trigonometric rіvnyannya for cob rіvnya (div vishche).

Greater folding trigonometric alignment is the basis for advanced folding. The stench is necessary, as if you yourself were equal to the infamous looker, and to know the root of that equal, to lie with the given given intercourse.

Razvyazannya trigonometric equals to be created up to two days:

Virishennya Rivnyannia
Vidbіr koreniv

Significantly, what is not necessary for a friend, but all the same, in most cases, it is necessary to carry out a vote. And if you don’t need wine, then you can do it better - it doesn’t mean that it’s equal to do it on your own.

My report to the head of C1 shows that the stench sounds like this category.

Chotiri category of advanced folding (earlier C1)

Rіvnyannya, scho zvodatsya until the laying out of the multipliers.
Rivnyannya, what to look at.
Rivnyannya, yakі vyrishyuyutsya zamіnoy zminnoy.
Rivnyannya, which requires an additional selection of roots through irrationality or a banner.

Speaking in a simple way: what happened to you one of the first three types, then vvazhay, scho you spared. For them, sound additionally necessary to pick up the root, which should lie with the deaky intermediary.

If you were trapilos equal to type 4, then you were less fortunate: you need to tinker with it more and more respectfully, then it’s often not necessary to dodatkovo to pick up the root. I will analyze the proteo type of equals in the offensive article, and I will assign the highest equal of the first three types.

Rіvnyannya, scho zvodatsya before laying out for multipliers

Most importantly, what do you need to remember, so that you can see

As practice shows, sound your knowledge enough. Let's go back to butts:

Butt 1

Re-shi-te ur-no-nya
Find out all the roots of that river, at-above-le-zha-schі vіd-rіz-ku

Here, as I said, the formulas are given:

Then my jealousy will look like this:

Todi my jealousy in the future of the offensive form:

A short-sighted scholar could say in a moment: and now I will shorten the insults to the parts that I take away in the simplest way, and I am quiet of life! I will have a warm mercy!

REMEMBER: IT IS NOT POSSIBLE TO SHORTLY LOOK AT THE PARTS OF THE TRIGONOMETRIC LEVEL ON A FUNCTION WHAT YOU ARE INVISIBLE! THUS, YOU'RE SPINING THE ROOT!

What is work? Everything is simple, transfer everything into one book and add a double multiplier:

Well, axis, they split it into multipliers, cheers! Now we see:

First equal may root:

And to another:

On the first part of the task, it has been written. Now it is necessary to choose the root:

The promo is like this:

Otherwise, you can write the axis like this:

Well, let's pick the root:

On the back of the head, it’s better with the first series (that one is simpler than that, what else to say!)

Since our promozhok is entirely negative, then there is no need to take non-negative ones, all the same stench will give non-negative roots.

Vіzmemo, todi - funny, don't get it.

Let it go - I didn’t spend it again.

One more try - todi, є, after eating! First root found!

I shoot more times: Todi - having drank again!

Well, one more time: - it’s already flown.

So, from the first series, there are 2 roots: .

Pratsiyuemo with another series (reduced at the steps behind the rule):

Not a share!

I don't get it again!

I'm sorry again!

Vluchiv!

Flight!

In such a rank, my promiscuity lies on the axis of such roots:

The axis behind such an algorithm is mi and virishuvatememo all other applications. Let's work out at once on one butt.

Example 2

Untie Rivnyannia

Solution:

I know the bitter formulas given:

Znov do not think fast!

First equal may root:

And to another:

Now I'm going to search again for the root.

I'll start from another series, I already know everything about it from the front butt! Look at and perekonaysya, what is the root, what is to lay a gap, step on:

Now the first series and won't be simpler:

Yakshcho - go

Yakshcho - tezh fit

Yakshcho - already overflowing.

Todi root will come:

Self-supporting robot. 3 rows.

Well, did the technology understand you? Razv'yazannya trigonometric equals already not zdaetsya so collapsible? Sodі shvidenko vyrіshuy vіrishu vіshі vіdnі zavdannya independently, and then we'll virіshuvatimemo with you іnіshі applied:

Untie Rivnyannia
Find out all the roots of that river, at-above-the-le-zha-shchi promizhka.
Re-shi-te ur-no-nya
Indicate the root of the river, at-above-le-zha-shchі vіd-rіz-ku
Re-shi-te ur-no-nya
Find out all the roots of this river, at-above-le-zha-shchie about-m-zhut-ku.

Rivnyanya 1.

I know the reduced formula:

First series of roots:

Another series of roots:

We fix the fee for the interim

Suggestion: , .

Rivnyannia 2. Rechecking independent work.

To finish the cunning grouping into multipliers (the stagnant formula of the sine of the podvyy kuta):

same chi

Tse spіlne solution. Now you need to choose the root. Bida y tsomu, as we can say exactly the meaning of kuta, the cosine of some old one quarter. That's why I can't just pozbutisu arccosine - the axis is such a crunch!

What can I do, then think about it, what is it like that, those.

Let's make a table: promizhok:

Well, well, with the help of more poshukivs, we shared a nevtish vysnovka about those that our zealous maє one root for the ordered interlude: \displaystyle arccos\frac(1)(4)-5\pi

3. Revalidation of independent work.

Equal to the mind that you are lying. However, it is difficult to simply write down the formula of the sine of the subway kuta:

Fast forward to 2:

Grouped first dodanok with another and third with a quarter and blameless zagalnі multipliers:

It is clear that the first equal root cannot, but now we can look at a friend:

As soon as I got up, I climbed a little bit later to rise to the heights of such equals, but as soon as it turned up, then do nothing, I need to vierish.

Equal to mind:

Dane rivnyannya is broken by rozpodilom both parts on:

In this rank, our zeal can have a single series of roots:

It is necessary to know tі, yakі to lie between: .

I will renew the sign, as I have been timid and earlier:

Suggestion: .

Rivnyannya, what to look at:

Well, axis, now it’s time to move on to another portion of equals, it’s bigger, so I already let it slip into what I say about the decoupling of trigonometric equals of a new type. Ale, we will not say, we will repeat what is equal to the mind

Virishuetsya rozpodіlom both parts by cosine:

Re-shi-te ur-no-nya
Indicate the root of the river, at-above-le-zha-schі vіd-rіz-ku.
Re-shi-te ur-no-nya
Indicate the root of the river, on-d-le-zha-shchi-shchi pro-m-zhut-ku.

example 1.

First - well, let's just say. Transferring to the right and stating the formula of the cosine of the subvertical kuta:

Aha! Equal to mind: . I share the insults on

Robimo vidsiv root:

Promіzhok:

Suggestion:

butt 2.

All the same, it is trivial to do it: opening the arms of the right-hander:

The basic trigonometric totality:

Sinus of the subway kut:

Remaining taken:

Root note: promіzhok.

Suggestion: .

Well, how is that technique, isn’t it foldable? I agree that no. You can change your mind at once: with a clean looking equal, you can grow up to a equal tangent, and rarely do it. As a rule, this transition (divided by the cosine) is only a part of the folding task. The axis of you is butt, so that you can instantly adjust:

Re-shi-te ur-no-nya
Know-di-those all the root of this river, at-above-le-zha-schі vіd-rіz-ku.

Let's chime in:

Rivnyannya vіrіshuєtsya vіrazu well, it is enough to add insults to parts of:

Root entries:

Suggestion: .

So, otherwise, we may be familiar with the equals of that kind, which we have chosen so well. It’s too early for us to round off: one more “layer” of equals has been left, and they didn’t pick it out for us. Father:

Razv'azannya trigonometric equals by replacing the change

Everything is transparent here: marveling respectfully at the equal, as much as possible yoga, timidly replace, virishuemo, timidly return the change! In words, everything is easy. Let's marvel at the dili:

butt.

Rozvyazati rivnyannya: .
Know-di-those all the root of this river, at-above-le-zha-schі vіd-rіz-ku.

Well, then, the replacement itself will be asked to reach us at hand!

Then our rivnyannya pretend to be like this:

First equal may root:

And the other axis is like this:

Now we know the root, what to lay down

Suggestion: .

Let's take a look at the folded butt at once:

Re-shi-te ur-no-nya
Indicate the root of this rivnyannya, on-d-le-zh-shchi pro-m-zhut-ku.

Here the change is not immediately visible, moreover, it is not obvious. Let's think for a moment: what can we do?

Maybe, for example, reveal

And contagion

Then I will see my jealousy in the future:

And now respect, focus:

Let's divide the insults of the equal part into:

Raptom with you, we got a square equal amount of money! I’ll change it, then take it away:

Rivnyannya may come the root:

Unacceptable other series is root, but you can’t see anything! We carry out selection of roots for the interim.

We should also be told that

So like i, then

Suggestion:

For fixing, first of all, you yourself virishuvatimesh zavdannya, the axis is right:

Re-shi-te ur-no-nya
Know-di-those all the roots of this river, at-above-le-zha-schі-shі pro-mі-zhut-ku.

Here the need for trimati vuho gostro: we have appeared bannermen, yakі can be zero! For this, you need to be especially respectful to the root!

Nasampered, it is necessary for me to remake the equal so that I can instantly make a change. I can’t come up with anything short at once, I can’t rewrite the tangent through sine and cosine:

Now I will pass from the cosine to the sine for the basic trigonometric totonism:

I, nareshti, I will bring everything to the sleeping banner:

Now I can go to equal:

Ale for (tobto for).

Now everything is ready for replacement:

Todi chi

However, beastly respect, what if, then with whom!

Who is suffering? Bida s tangent, vin no assignments, if the cosine reaches zero (it goes to zero).

Otzhe, the root is like this:

Now it is possible to add roots to the field:


	- walk
	- sorted out

In this rank, our rivnyannya can be a single root for the prom, and vіn dorіvnyuє.

Bachish: the appearance of a bannerman (likewise, like a tangent, to bring to singing difficulties with roots! Here we need to be more respectful!).

Well, with you, we may have completed the analysis of trigonometric equals, we have lost our good fortune - independently write two tasks. Axis stink.

Untie Rivnyannia
Know-di-those all the root of this river, at-above-le-zha-schі vіd-rіz-ku.
Re-shi-te ur-no-nya
Indicate the root of which river, at-above-le-zha-schі vіd-rіz-ku.

Virishiv? Chi is not too complicated? Let's chime in:

Practice for the formulas given:
Submitted in equal:
Let's rewrite everything through cosines, so that it would be easier to work the change:
Now it's easy to make a change:
It dawned on me that a third-party root, no shards of equal solution. Todi:
Shukaemo we need a root for the interlude
Suggestion: .
Here the change can be clearly seen:
Todi chi

- Come on! - Come on!
- Come on! - Come on!
- Bagato! - tezh rich!
Suggestion:

Well, that's all for now! Ale, the solution of trigonometric equals does not end in any way, overboard we have lost the most foldable twists: if in equals there is irrationality of a different kind of “folding banners”. How to write similar orders, we can look at the article for a slipped equal.

PROSUNUTIY RIVEN

In addition to looking at the front two articles of trigonometric equals, we look at one more class of equals, which will require even more respectful analysis. Apply trigonometric data to either irrationality, or a standard, so that we can analyze them in a folded way.. Tim is not less, but you can stay close to these peers in part of the examination work. However, there is nothing nasty without good: for such equals, as a rule, no food is given about those who, as a rule, belong to a given gap. Let's not go sideways and sideways, but to the right trigonometric butt.

example 1.

Virishiti equal and know those roots, like laying down the wind.

Solution:

We have a banner, which is not guilty of reaching zero! Todі virіshiti tse rivnyannya - tse everything is the same, scho virіshiti system

Rozv'yazhemo kozhne z rivnyan:

And now a friend:

Now let's look at the series:

It’s clear that we don’t have the option, that at which we reset the banner (divine formula of the root of another equal)

Well, then everything is good, and the banner is not equal to zero! Todi root equal to the next:,.

Now it is carried out the selection of roots, which lie down until the intercourse.


	- do not come	- walk
	- walk	- walk
	sorted out	sorted out

Same root coming:

Bachish, having appeared a small change at the sight of the flagman, it was clearly marked on the top of the line: we saw a series of roots that nullified the banner. More folding can be on the right, so that you can trap trigonometrically, but you can use irrationality.

butt 2.

Untie the river:

Solution:

Well, if you don't want to pick up the root and that's good! Let's talk about rozvjazhemo ryvnyanya, regardless of irrationality:

What is everything? No, sorry, it would have been so easy! It is necessary to remember that under the root can stand more than an unknown number. Todi:

Vision of nervousness:

Now there is no more z'yasuvati, which was not inadvertently consumed by a part of the root of the first jealousy of the place, where the nervousness does not win.

For whom I can again speed up the table:


	: , ale	Hi!
		So!
		So!

In this rank, I have “vipav” one of the roots! Win to go out, to put it down. You can write the same thing in such a look:

Suggestion:

Bachish, root for more respect! Convenience: now let me have a trigonometric function under the roots.

Example 3.

Like before: we’ll open the cob of the skin, and then we’ll think about what we’ve done.

Now another equal:

Now it’s more convenient - z’yasuvati, so that there are no negative values under the arithmetic root, as we can imagine there the root of the first equal:

The number you need to understand is like a radian. The radian scales are about degrees, the radians are close degrees. Tse kut other quarter. Cosine of the other quarter, what is the sign? Minus. What about sine? Plus. So what can we say about Viraz:

Won less than zero!

And that means - not the root of jealousy.

Now damn.

The same number is equal to zero.

Cotangent - the function is recessive in 1 quarter (the smaller the argument, the greater the cotangent). radiani is about degrees. At the same hour

so, then, and mean i
,

Suggestion: .

Can you be more foldable? Please! It will be more important, just like the root, like before, the trigonometric function, and the other part is equal - the new trigonometric function.

The more trigonometric applications, the more marvelous the distance:

Example 4.

The root is no good, through the exchange of the cosine

Now friend:

Vodnochas for the appointment of the root:

It is required to guess a single call: the same number of quarters, de sine less than zero. What are the quarters? Third and fourth. That same decision of the first equal will call us, like lying at the third or fourth quarter.

The first series gives the root, which lies on the third and fourth quarters. Another series - їй diametrically protilezhna - і porodzhuє korіnnya, scho to lie between the first and the other quarter. So this series is not suitable for us.

Suggestion: ,

I renew trigonometric applications with "important irrationality". Not only that, the trigonometric function is new in us under the roots, but now it’s more like a banner!

Example 5.

Well, don't worry about anything - work like before.

Now pratsyuemo with a banner:

I don’t want to correct the trigonometric unevenness, but I’ll do it cunningly: I’ll take it and put it into the unevenness of my series of roots:

Yakshcho - a guy, then maybe:

the shards of all sight lie at the fourth quarter. I again sacred food: what is the sign of the sine at the fourth quarter? Negative. Same unevenness

Well unpaired, then:

In what quarter to lie kut? Tse kut other quarter. Todі all kuti - I will rekindle the kuti of the other quarter. The sine is positive. The very ones you need! So the series:

Go!

So it goes without saying with another series of roots:

Substituting for our nervousness:

Yakscho - guy, then

Kuti first quarter. The sine is positive there, so the series is coming. Now yakscho is unpaired, then:

you can come!

Well, the axis is now recorded vіdpovіd!

Suggestion:

Well, axis, tse bov, mabut, naivazhchy vipadok. Now I pronounce you a task for an independent celebration.

Training

To untie and know the mustache of the roots of the river, which lie in the wind.

Solution:

First alignment:
or
Root ODZ:
Other equals:
Vіdbіr korenіv, scho to lie until promіzhku
Suggestion:

Abo
or
ale

Let's take a look: . Yakscho - guy, then
- do not come!
Yakshcho - unpaired - go!
Otzhe, our zeal may have such a series of roots:
or
Selection of roots for the interim:


	- do not come	- walk
	- walk	- rich
	- walk	rich

Suggestion: , .

Abo
Oskіlki, then with tangent it is not indicated. We see a series of roots!

Other part:

At the same time, according to the ODZ, it is necessary

It is verified that the first equal root is found:

Yakscho sign:

Cut the first quarter, de tangent is positive. Don't go!
Yakscho sign:

Kut fourth quarter. There is a negative tangent. Come. We write down the note:

Suggestion: , .

At the same time, we picked out folding trigonometric stocks from our stats, and then we varto virishuvati ourselves.

SHORT VICLAD AND BASIC FORMULA

Trigonometrically equal - tse equal, at some unknown home, it is safe to change under the sign of the trigonometric function.

There are two ways to untie trigonometric alignments:

The first way is to use different formulas.

Another way is through trigonometric colo.

Allowing you to win the kuti, know their sines, cosines and more.

Obov'yazkovy minimum knowledge

sin x \u003d a, -1 a 1 (a 1)
x = arcsin a + 2 n, n Z
x = - arcsin a + 2 n, n Z
or
x = (-1) k arcsin a + k, k Z
arcsin (- a) = - arcsin a
sin x = 1
x = /2 + 2 k, k Z
sin x = 0
x = k, kZ
sin x = - 1
x = - /2 + 2 k, k Z
y
y
x
y
x
x

Obov'yazkovy minimum knowledge

cos x = a, -1 a 1 (a 1)
x = arccos a + 2 n, n Z
arccos (-a) = - arccos a
cos x = 1
x = 2 k, k Z
cos x = 0
x = /2 + k, k Z
y
y
x
cos x = - 1
x = + 2 k, k Z
y
x
x

Obov'yazkovy minimum knowledge

tg x = a, a R
x = arctg a + n, n Z
ctg x = a, a R
x = arcctg a + n, n Z
arctg (-a) = - arctg a
arctg (- a) = - arctg a Match up to one function
Call up to one argument
Deyakі methods of cherry
trigonometric equals
Fixing trigonometric formulas
Vikoristannya formulas of shortened multiplication
Multiplication
Reduction to square alignment sin x, cos x, tg x
Introduction of an additional argument
Podіlom both parts of the same level of the first step
(asin x +bcosx = 0) to cos x
Podіlom both parts of the same level of the other step
(a sin2 x +bsin x cos x + c cos2x = 0) to cos2 x

Use the right Calculate

arcsin½
arcsin(-√2/2)
arccos √3/2
arccos (-1/2)
arctan √3
arctan (-√3/3)
= /6
= - /4
= /6
= - arccos ½ = - /3 = 2 /3
= /3
= - /6

(for the help of a trigonometric stake)
cos2x=?,x[-/2; 3/2]
2x = ± arccos ½ + 2 n, n Z
2x = ± /3 + 2n, nZ
x = ± /6 + n, n Z
We take root for the help of a trigonometric stake
Vidpovid: - / 6; /6; 5/6; 7/6

Different ways to select the root

To know the roots of equalness, what should this intercourse lie with
sin 3x = √3/2, x[-/2; /2]
3x = (-1)k /3 + k, k Z
x = (– 1)k /9 + k/3, k Z
Let's take a root for additional enumeration of the value of k:
k \u003d 0, x \u003d / 9 - overlap
k \u003d 1, x \u003d - / 9 + / 3 \u003d 2 / 9 - lay a gap
k \u003d 2, x \u003d / 9 + 2 / 3 \u003d 7 / 9 - do not overlap
k \u003d - 1, x \u003d - / 9 - / 3 \u003d - 4 / 9 - lay a gap
k \u003d - 2, x \u003d / 9 - 2 / 3 \u003d - 5 / 9 - do not overlap
Vidpovid: -4/9; /nine; 2/9

Different ways to select the root

To know the roots of equalness, what should this intercourse lie with
(for additional nervousness)
tan 3x = - 1, x (-/2;)
3x = - / 4 + n, nZ
x = - /12 + n/3, n Z
We take the root for additional nervousness:
– /2 < – /12 + n/3 < ,
– 1/2 < – 1/12 + n/3 < 1,
– 1/2 + 1/12 < n/3 < 1+ 1/12,
– 5/12 < n/3 < 13/12,
– 5/4 < n < 13/4, n Z,
n = - 1; 0; one; 2; 3
n = - 1, x = - / 12 - / 3 = - 5 / 12
n=0, x=-/12
n = 1, x = - / 12 + / 3 = / 4
n = 2, x = - / 12 + 2 / 3 = 7 / 12
n = 3, x = - / 12 + = 11 / 12
Vidpovid: - 5/12; - / 12; /4; 7/12; 11/12

10. Different ways to select the root

To know the roots of equalness, what should this intercourse lie with
(for additional graphics)
cos x = - √2/2, x [-4; 5/4]
x = arccos (–√2/2) + 2n, nZ
x = 3/4 + 2n, nZ
We take the root for the help schedule:
x = - / 2 - / 4 = - 3 / 4; x = - - / 4 = - 5 / 4
Vidpovid: 5/4; 3/4

11. 1. Draw the line 72cosx \u003d 49sin2x and put the yogo root on the line [; 5/2]

1. Razvyazati equalization 72cosx = 49sin2x
say that root to the vіdrіzka [ ; 5/2]
Rozv'yazhemo rivnyannya:
72cosx = 49sin2x,
72cosx = 72sin2x,
2cos x = 2sin 2x,
cos x - 2 sinx cosx = 0,
cosx(1 - 2sinx) = 0,
cos x = 0
x = /2 + k, k Z
or
1 - 2 sinx = 0,
sin x = ½,
x = (-1)n /6 + n, n Z
We will conduct a selection of roots for help
trigonometric stake:
x = 2 + /6 = 13/6
Suggestion:
a) /2+k, kZ, (-1)n/6+n,nZ
b) 3/2; 5/2; 13/6

12. 2. Razvyazat rivnyanna 4cos2 x + 8 cos (x - 3/2) +1 = 0

2. Razvyazati equalization 4cos2 x + 8 cos (x - 3/2) +1 = 0
Know yoga root on vіdrіzku
4cos2 x + 8 cos (x - 3/2) +1 = 0
4cos2x + 8 cos (3/2 - x) +1 = 0,
4cos2x - 8 sin x +1 = 0,
4 - 4sin2 x - 8sin x +1 = 0,
4sin 2x + 8sin x - 5 = 0,
D/4 = 16 + 20 = 36,
sin x = - 2.5
or
sin x = ½
x = (-1) k /6 + k, k Z

13. We will conduct a selection of roots for a replacement (for additional schedules)

We will carry out the selection of roots for the vіdrіzka
(for help schedule)
sin x = ½
Let's get the function graphs y = sin x and y = ½
x = 4 + / 6 = 25 / 6
Vidpovid: a) (-1)k/6+k, kZ; b) 25/6

14. 3. Razvyazati ryvnyanya

4 - cos2 2x = 3 sin2 2x + 2 sin 4x
4 (sin2 2x + cos2 2x) – cos2 2x = 3 sin2 2x + 4 sin 2x cos 2x,
sin2 2x + 3 cos2 2x - 4 sin 2x cos 2x = 0
If cos2 2x = 0, then sin2 2x = 0, which is impossible, then
cos2 2x 0 that offending part of the equation can be divided into cos2 2x.
tg22x + 3 – 4 tg2x = 0,
tg22x - 4tg 2x + 3 = 0,
tg 2x = 1,
2x = /4 + n, n Z
x = /8 + n/2, n Z
or
tg 2x = 3,
2x = arctg 3 + k, k Z
x \u003d ½ arctan 3 + k / 2, k Z

15.

4 - cos2 2x = 3 sin2 2x + 2 sin 4x
x = /8 + n/2, n Z or x = ½ arctan 3 + k/2, k Z
Oskilki 0< arctg 3< /2,
0 < ½ arctg 3< /4, то ½ arctg 3
є decisions
Oskilki 0< /8 < /4 < 1,значит /8
also є decisions
Other decisions not to spend time with
promіzhok, skіlki stench
come out of numbers? arctg 3 and /8
adding numbers that are multiples of /2.
Case: a) /8 + n/2, n Z; ½ arctan 3 + k/2, k Z
b) / 8; ½ arctan 3

16. 4. Razvyazat equalization log5(cos x - sin 2x + 25) = 2

4. Razvyazati equalization log5(cos x - sin 2x + 25) = 2
Know yoga root on vіdrіzku
Rozv'yazhemo rivnyannya:
log5(cos x - sin 2x + 25) = 2
ODZ: cos x - sin 2x + 25 > 0,
cos x - sin 2x + 25 = 25, 25> 0,
cos x – 2sin x cos x = 0,
cosx(1 - 2sinx) = 0,
cos x = 0,
x = /2 + n, n Z
or
1 - 2 sinx = 0,
sin x = 1/2
x = (-1) k /6 + k, k Z

17.

We will carry out the selection of roots for the vіdrіzka
Let's conduct a selection of roots for a vіdrіzka:
1) x = /2 + n, n Z
2/2 + n 7/2, n Z
2 1/2 + n 7/2, n Z
2 – ½ n 7/2 – ½, n Z
1.5 n 3, n Z
n = 2; 3
x = / 2 + 2 = 5 / 2
x = / 2 + 3 = 7 / 2
2) sin x = 1/2
x = 2 + /6 = 13/6
x = 3 - / 6 = 17 / 6
Suggestions: a) /2 + n, n Z; (-1)k/6+k, kZ
b) 13/6; 5/2; 7/2; 17/6

18. 5. Draw the line 1/sin2x + 1/sin x = 2 Find the second root of the line [-5/2; -3/2]

5. Razvyazati equalization 1/sin2x + 1/sin x = 2
Know the yogo root on the vіdrіzku [-5/2; -3/2]
Rozv'yazhemo rivnyannya:
1/sin2x + 1/sinx = 2
x k
Change 1/sin x = t,
t2 + t = 2,
t2 + t - 2 = 0,
t1 = - 2, t2 = 1
1/sin x = - 2,
sin x = - ½,
x = - /6 + 2n, n Z
or
x = – 5/6 + 2n, nZ
1/sin x = 1,
sin x = 1,
x = /2 + 2n, nZ
The root series is excluded, because -150º+ 360ºn
set interval [-450 º; -270º]

19.

We sell the choice of roots
We will look at other series of roots and we will conduct a selection of roots
on vіdrіzku [-5/2; -3 /2] ([-450º; -270º]):
1) x \u003d - / 6 + 2 n, n Z
2) x = /2 + 2n, n Z
-5 /2 - /6 + 2 n -3 /2, n Z
-5 /2 /2 + 2 n -3 /2, n Z
-5/2 -1/6 + 2n -3/2, n Z
-5/2 1/2 + 2n -3/2, n Z
-5/2 +1/6 2n -3/2 + 1/6, n Z
-5/2 - 1/2 2n -3/2 - 1/2, n Z
– 7/3 2n -4/3, n Z
– 3 2n -2, n Z
-7/6 n -2/3, n Z
-1.5 n -1, n Z
n=-1
n=-1
x = - /6 - 2 = -13 /6 (-390º)
x = /2 - 2 = -3 /2 (-270º)
Suggestions: a) /2 + 2 n, n Z; (-1)k+1 /6 + k, k Z
b) -13/6; -3/2

20. 6. Razvyazati equalization | sin x | / sin x + 2 = 2 cos x eight]

Rozv'yazhemo rivnyannya
|sinx|/sinx + 2 = 2cosx
1) If sin x >0, then | sinx| =sin x
Rivnyanna I will look in the future:
2 cosx=3,
cos x \u003d 1.5 - no root
2) Yakscho sin x<0, то |sin x| =-sin x
i look forward to seeing
2cosx=1, cosx=1/2,
x = ±π/3 +2πk, k Z
Vrahovoyuchi, sho sin x< 0, то
one series is left
x = - π/3 +2πk, k Z
Zrobimo vydbіr koreniv on
vіdrіzku [-1; eight]
k=0, x= - π/3 , - π< -3, - π/3 < -1,
-π / 3 do not lie on this
vіdrіzku
k=1, x = - π/3 +2π = 5π/3<8,
5 pi/3 [-1; eight]
k=2, x= - π/3 + 4π = 11π/3 > 8,
11π/3
vіdrіzku.
Proof: a) – π/3 +2πk, k Z
b) 5
π/3

21. 7. Untie the equal 4sin3x=3cos(x- π/2)

8. Razvyazati equalization √1-sin2x= sin x
Know the yoga root for the trade
Razv'yazhemo equal √1-sin2x= sin x.
sin x ≥ 0,
1-sin2x=sin2x;
sin x ≥ 0,
2sin2x = 1;
sinx≥0,
sin x =√2/2; sin x = - √2/2;
sin x =√2/2
x = (-1) k / 4 + k, k Z
sin x =√2/2

25. We will carry out the selection of roots for the plant

We will carry out the selection of roots for the vіdrіzka
x = (-1) k / 4 + k, k Z
sin x =√2/2
y \u003d sin x i y \u003d √2/2
5 /2 + /4 = 11 /4
Case: a) (-1) k / 4 + k, k Z; b) 11 / 4

26. 9. Razvyazat r_vnyanna (sin2x + 2 sin2x)/√-cos x =0 -7/2]

9. Razvyazati equalization (sin2x + 2 sin2x)/√-cos x =0
Know the yogo root for the middle [-5; -7/2]
Rozv'yazhemo rivnyannya
(sin2x + 2 sin2x)/√-cos x =0.
1) ODZ: cos x<0 ,
/2 +2n 2) sin2x + 2 sin2x = 0,
2 sinx∙cos x + 2 sin2x =0,
sin x (cos x + sin x) = 0,
sin x = 0, x = n, n Z
or
cos x + sin x = 0 | : cosx,
tg x = -1, x = - / 4 + n, n Z
Z urahuvannyam ODZ
x = n, n Z, x = +2 n, n Z;
x= - /4 + n, n Z,
x= 3 /4 + 2n, nZ

27. We choose the root on the given vіdrіzka

We choose the root on the given
vіdrіzku [-5; -7/2]
x= +2 n, n Z;
-5 ≤ +2 n ≤ -7 /2,
-5-1 ≤ 2n ≤ -7/2-1,
-3≤ n ≤ -9/4, n Z
n=-3, x=-6=-5
x= 3 /4 + 2n, nZ
-5 ≤ 3 /4 + 2n ≤ -7 /2
-23/8 ≤ n ≤ -17/8, there is no such thing
integer n.
Suggestion: a) +2 n, n Z;
3/4 + 2n, nZ;
b) -5.

28. 10. Razvyazat rіvnnja 2sin2x =4cos x –sinx+1 3/2]

10. Razvyazati equalization 2sin2x \u003d 4cos x -sinx + 1
Know the root for the prom_zhku [/2; 3/2]
Rozv'yazhemo rivnyannya
2sin2x = 4cosx - sinx+1
2sin2x \u003d 4cos x - sinx + 1,
4 sinx∙cos x - 4cos x + sin x -1 = 0,
4cos x(sin x - 1) + (sin x - 1) = 0,
(sin x - 1) (4cos x +1) = 0,
sin x – 1= 0, sin x = 1, x = /2+2 n, n Z
or
4cos x +1 = 0, cos x = -0.25
x = ±(-arccos(0.25)) + 2n,nZ
Let's write down the root of which equal otherwise
x = - arccos(0.25) + 2n,
x = -(- arccos(0.25)) + 2n, nZ

29. Let's take root for the help of the stake

x = /2+2 n, n Z, x = /2;
x = -arccos(0.25)+2n,
x \u003d - (-arccos (0.25)) +2 n, n Z,
x = - arccos (0.25),
x = + arccos(0.25)
Suggestion: a) /2+2 n,
-arccos(0.25)+2n,
-(-arccos(0,25)) +2 n, n Z;
b) /2;
- arccos(0.25); + arccos(0.25)


	- Come on!	- Come on!
	- Come on!	- Come on!
	- Bagato!	- tezh rich!