Trigonometric equations are not an easy topic. They are too diverse.) For example, these:
sin 2 x + cos3x = ctg5x
sin(5x+π /4) = cot(2x-π /3)
sinx + cos2x + tg3x = ctg4x
Etc...
But these (and all other) trigonometric monsters have two common and obligatory features. First - you won’t believe it - there are trigonometric functions in the equations.) Second: all expressions with x are found within these same functions. And only there! If X appears somewhere outside, For example, sin2x + 3x = 3, this will already be an equation of mixed type. Such equations require an individual approach. We will not consider them here.
We will not solve evil equations in this lesson either.) Here we will deal with the simplest trigonometric equations. Why? Yes because the solution any trigonometric equations consists of two stages. At the first stage, the evil equation is reduced to a simple one through a variety of transformations. On the second, this simplest equation is solved. No other way.
So, if you have problems at the second stage, the first stage does not make much sense.)
What do elementary trigonometric equations look like?
sinx = a
cosx = a
tgx = a
ctgx = a
Here A stands for any number. Any.
By the way, inside a function there may not be a pure X, but some kind of expression, like:
cos(3x+π /3) = 1/2
etc. This complicates life, but does not affect the method of solving a trigonometric equation.
How to solve trigonometric equations?
Trigonometric equations can be solved in two ways. The first way: using logic and the trigonometric circle. We will look at this path here. The second way - using memory and formulas - will be discussed in the next lesson.
The first way is clear, reliable, and difficult to forget.) It is good for solving trigonometric equations, inequalities, and all sorts of tricky non-standard examples. Logic is stronger than memory!)
Solving equations using a trigonometric circle.
We include elementary logic and the ability to use the trigonometric circle. Don't you know how? However... You will have a hard time in trigonometry...) But it doesn’t matter. Take a look at the lessons "Trigonometric circle...... What is it?" and "Measuring angles on a trigonometric circle." Everything is simple there. Unlike textbooks...)
Oh, you know!? And even mastered “Practical work with the trigonometric circle”!? Congratulations. This topic will be close and understandable to you.) What is especially pleasing is that the trigonometric circle does not care what equation you solve. Sine, cosine, tangent, cotangent - everything is the same for him. There is only one solution principle.
So we take any elementary trigonometric equation. At least this:
cosx = 0.5
We need to find X. Speaking in human language, you need find the angle (x) whose cosine is 0.5.
How did we previously use the circle? We drew an angle on it. In degrees or radians. And right away saw trigonometric functions of this angle. Now let's do the opposite. Let's draw a cosine on the circle equal to 0.5 and immediately we'll see corner. All that remains is to write down the answer.) Yes, yes!
Draw a circle and mark the cosine equal to 0.5. On the cosine axis, of course. Like this:
Now let's draw the angle that this cosine gives us. Hover your mouse over the picture (or touch the picture on your tablet), and you'll see this very corner X.
The cosine of which angle is 0.5?
x = π /3
cos 60°= cos( π /3) = 0,5
Some people will chuckle skeptically, yes... Like, was it worth making a circle when everything is already clear... You can, of course, chuckle...) But the fact is that this is an erroneous answer. Or rather, insufficient. Circle connoisseurs understand that there are a whole bunch of other angles here that also give a cosine of 0.5.
If you turn the moving side OA full turn, point A will return to its original position. With the same cosine equal to 0.5. Those. the angle will change by 360° or 2π radians, and cosine - no. The new angle 60° + 360° = 420° will also be a solution to our equation, because
An infinite number of such complete revolutions can be made... And all these new angles will be solutions to our trigonometric equation. And they all need to be written down somehow in response. All. Otherwise, the decision does not count, yes...)
Mathematics can do this simply and elegantly. Write down in one short answer infinite set decisions. Here's what it looks like for our equation:
x = π /3 + 2π n, n ∈ Z
I'll decipher it. Still write meaningfully It’s more pleasant than stupidly drawing some mysterious letters, right?)
π /3 - this is the same corner that we saw on the circle and determined according to the cosine table.
2π is one complete revolution in radians.
n - this is the number of complete ones, i.e. whole rpm It is clear that n can be equal to 0, ±1, ±2, ±3.... and so on. As indicated by the short entry:
n ∈ Z
n belongs ( ∈ ) set of integers ( Z ). By the way, instead of the letter n letters may well be used k, m, t etc.
This notation means you can take any integer n . At least -3, at least 0, at least +55. Whatever you want. If you substitute this number into the answer, you will get a specific angle, which will definitely be the solution to our harsh equation.)
Or, in other words, x = π /3 is the only root of an infinite set. To get all the other roots, it is enough to add any number of full revolutions to π /3 ( n ) in radians. Those. 2πn radian.
All? No. I deliberately prolong the pleasure. To remember better.) We received only part of the answers to our equation. I will write this first part of the solution like this:
x 1 = π /3 + 2π n, n ∈ Z
x 1 - not just one root, but a whole series of roots, written down in a short form.
But there are also angles that also give a cosine of 0.5!
Let's return to our picture from which we wrote down the answer. Here she is:
Hover your mouse over the image and we see another angle that also gives a cosine of 0.5. What do you think it is equal to? The triangles are the same... Yes! He equal to angle X , only delayed in the negative direction. This is the corner -X. But we have already calculated x. π /3 or 60°. Therefore, we can safely write:
x 2 = - π /3
Well, of course, we add all the angles that are obtained through full revolutions:
x 2 = - π /3 + 2π n, n ∈ Z
That's all now.) On the trigonometric circle we saw(who understands, of course)) All angles that give a cosine of 0.5. And wrote down these angles in short mathematical form. The answer resulted in two infinite series of roots:
x 1 = π /3 + 2π n, n ∈ Z
x 2 = - π /3 + 2π n, n ∈ Z
This is the correct answer.
Hope, general principle for solving trigonometric equations using a circle is clear. We mark the cosine (sine, tangent, cotangent) from the given equation on a circle, draw the angles corresponding to it and write down the answer. Of course, we need to figure out what corners we are saw on the circle. Sometimes it's not so obvious. Well, I said that logic is required here.)
For example, let's look at another trigonometric equation:
Please take into account that the number 0.5 is not the only possible number in equations!) It’s just more convenient for me to write it than roots and fractions.
We work according to the general principle. We draw a circle, mark (on the sine axis, of course!) 0.5. We draw all the angles corresponding to this sine at once. We get this picture:
Let's deal with the angle first X in the first quarter. We recall the table of sines and determine the value of this angle. It's a simple matter:
x = π /6
We remember about full turns and, with a clear conscience, write down the first series of answers:
x 1 = π /6 + 2π n, n ∈ Z
Half the job is done. But now we need to determine second corner... It's trickier than using cosines, yes... But logic will save us! How to determine the second angle through x? Yes Easy! The triangles in the picture are the same, and the red corner X equal to angle X . Only it is counted from the angle π in the negative direction. That’s why it’s red.) And for the answer we need an angle, measured correctly, from the positive semi-axis OX, i.e. from an angle of 0 degrees.
We hover the cursor over the drawing and see everything. I removed the first corner so as not to complicate the picture. The angle we are interested in (drawn in green) will be equal to:
π - x
X we know this π /6 . Therefore, the second angle will be:
π - π /6 = 5π /6
Again we remember about adding full revolutions and write down the second series of answers:
x 2 = 5π /6 + 2π n, n ∈ Z
That's all. A complete answer consists of two series of roots:
x 1 = π /6 + 2π n, n ∈ Z
x 2 = 5π /6 + 2π n, n ∈ Z
Tangent and cotangent equations can be easily solved using the same general principle for solving trigonometric equations. If, of course, you know how to draw tangent and cotangent on a trigonometric circle.
In the examples above, I used the table value of sine and cosine: 0.5. Those. one of those meanings that the student knows must. Now let's expand our capabilities to all other values. Decide, so decide!)
So, let's say we need to solve this trigonometric equation:
Such a cosine value in brief tables No. We coldly ignore this terrible fact. Draw a circle, mark 2/3 on the cosine axis and draw the corresponding angles. We get this picture.
Let's look, first, at the angle in the first quarter. If only we knew what x is equal to, we would immediately write down the answer! We don’t know... Failure!? Calm! Mathematics does not leave its own people in trouble! She came up with arc cosines for this case. Do not know? In vain. Find out, It's a lot easier than you think. There is not a single tricky spell about “inverse trigonometric functions” on this link... This is superfluous in this topic.
If you are in the know, just say to yourself: “X is an angle whose cosine is equal to 2/3.” And immediately, purely by the definition of arc cosine, we can write:
We remember about the additional revolutions and calmly write down the first series of roots of our trigonometric equation:
x 1 = arccos 2/3 + 2π n, n ∈ Z
The second series of roots for the second angle is almost automatically written down. Everything is the same, only X (arccos 2/3) will be with a minus:
x 2 = - arccos 2/3 + 2π n, n ∈ Z
And that's it! This is the correct answer. Even easier than with table values. There is no need to remember anything.) By the way, the most attentive will notice that this picture shows the solution through the arc cosine in essence, no different from the picture for the equation cosx = 0.5.
Exactly! General principle That's why it's common! I deliberately drew two almost identical pictures. The circle shows us the angle X by its cosine. Whether it is a tabular cosine or not is unknown to everyone. What kind of angle this is, π /3, or what arc cosine is - that’s up to us to decide.
Same song with sine. For example:
Draw a circle again, mark the sine equal to 1/3, draw the angles. This is the picture we get:
And again the picture is almost the same as for the equation sinx = 0.5. Again we start from the corner in the first quarter. What is X equal to if its sine is 1/3? No problem!
Now the first pack of roots is ready:
x 1 = arcsin 1/3 + 2π n, n ∈ Z
Let's deal with the second angle. In the example with a table value of 0.5, it was equal to:
π - x
It will be exactly the same here too! Only x is different, arcsin 1/3. So what!? You can safely write down the second pack of roots:
x 2 = π - arcsin 1/3 + 2π n, n ∈ Z
This is a completely correct answer. Although it doesn't look very familiar. But it’s clear, I hope.)
This is how trigonometric equations are solved using a circle. This path is clear and understandable. It is he who saves in trigonometric equations with the selection of roots on a given interval, in trigonometric inequalities - they are generally solved almost always in a circle. In short, in any tasks that are a little more difficult than standard ones.
Let's apply knowledge in practice?)
Solve trigonometric equations:
First, simpler, straight from this lesson.
Now it's more complicated.
Hint: here you will have to think about the circle. Personally.)
And now they are outwardly simple... They are also called special cases.
sinx = 0
sinx = 1
cosx = 0
cosx = -1
Hint: here you need to figure out in a circle where there are two series of answers and where there is one... And how to write one instead of two series of answers. Yes, so that not a single root from an infinite number is lost!)
Well, very simple):
sinx = 0,3
cosx = π
tgx = 1,2
ctgx = 3,7
Hint: here you need to know what arcsine and arccosine are? What is arctangent, arccotangent? The most simple definitions. But you don’t need to remember any table values!)
The answers are, of course, a mess):
x 1= arcsin0,3 + 2π n, n ∈ Z
x 2= π - arcsin0.3 + 2
Not everything works out? Happens. Read the lesson again. Only thoughtfully(there is such obsolete word...) And follow the links. The main links are about the circle. Without it, trigonometry is like crossing the road blindfolded. Sometimes it works.)
If you like this site...
By the way, I have a couple more interesting sites for you.)
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Lesson and presentation on the topic: "Solving simple trigonometric equations"
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Software environment "1C: Mathematical Constructor 6.1"
What we will study:
1. What are trigonometric equations?
3. Two main methods for solving trigonometric equations.
4. Homogeneous trigonometric equations.
5. Examples.
What are trigonometric equations?
Guys, we have already studied arcsine, arccosine, arctangent and arccotangent. Now let's look at trigonometric equations in general.
Trigonometric equations are equations in which a variable is contained under the sign of a trigonometric function.
Let us repeat the form of solving the simplest trigonometric equations:
1)If |a|≤ 1, then the equation cos(x) = a has a solution:
X= ± arccos(a) + 2πk
2) If |a|≤ 1, then the equation sin(x) = a has a solution:
3) If |a| > 1, then the equation sin(x) = a and cos(x) = a have no solutions 4) The equation tg(x)=a has a solution: x=arctg(a)+ πk
5) The equation ctg(x)=a has a solution: x=arcctg(a)+ πk
For all formulas k is an integer
The simplest trigonometric equations have the form: T(kx+m)=a, T is some trigonometric function.
Example.Solve the equations: a) sin(3x)= √3/2
Solution:
A) Let us denote 3x=t, then we will rewrite our equation in the form:
The solution to this equation will be: t=((-1)^n)arcsin(√3 /2)+ πn.
From the table of values we get: t=((-1)^n)×π/3+ πn.
Let's return to our variable: 3x =((-1)^n)×π/3+ πn,
Then x= ((-1)^n)×π/9+ πn/3
Answer: x= ((-1)^n)×π/9+ πn/3, where n is an integer. (-1)^n – minus one to the power of n.
More examples of trigonometric equations.
Solve the equations: a) cos(x/5)=1 b)tg(3x- π/3)= √3Solution:
A) This time let’s move directly to calculating the roots of the equation right away:
X/5= ± arccos(1) + 2πk. Then x/5= πk => x=5πk
Answer: x=5πk, where k is an integer.
B) We write it in the form: 3x- π/3=arctg(√3)+ πk. We know that: arctan(√3)= π/3
3x- π/3= π/3+ πk => 3x=2π/3 + πk => x=2π/9 + πk/3
Answer: x=2π/9 + πk/3, where k is an integer.
Solve the equations: cos(4x)= √2/2. And find all the roots on the segment.
Solution:
We'll decide in general view our equation: 4x= ± arccos(√2/2) + 2πk
4x= ± π/4 + 2πk;
X= ± π/16+ πk/2;
Now let's see what roots fall on our segment. At k At k=0, x= π/16, we are in the given segment.
With k=1, x= π/16+ π/2=9π/16, we hit again.
For k=2, x= π/16+ π=17π/16, but here we didn’t hit, which means that for large k we also obviously won’t hit.
Answer: x= π/16, x= 9π/16
Two main solution methods.
We looked at the simplest trigonometric equations, but there are also more complex ones. To solve them, the method of introducing a new variable and the method of factorization are used. Let's look at examples.Let's solve the equation:
Solution:
To solve our equation, we will use the method of introducing a new variable, denoting: t=tg(x).
As a result of the replacement we get: t 2 + 2t -1 = 0
Let's find the roots quadratic equation: t=-1 and t=1/3
Then tg(x)=-1 and tg(x)=1/3, we get the simplest trigonometric equation, let’s find its roots.
X=arctg(-1) +πk= -π/4+πk; x=arctg(1/3) + πk.
Answer: x= -π/4+πk; x=arctg(1/3) + πk.
An example of solving an equation
Solve equations: 2sin 2 (x) + 3 cos(x) = 0
Solution:
Let's use the identity: sin 2 (x) + cos 2 (x)=1
Our equation will take the form: 2-2cos 2 (x) + 3 cos (x) = 0
2 cos 2 (x) - 3 cos(x) -2 = 0
Let us introduce the replacement t=cos(x): 2t 2 -3t - 2 = 0
The solution to our quadratic equation is the roots: t=2 and t=-1/2
Then cos(x)=2 and cos(x)=-1/2.
Because cosine cannot take values greater than one, then cos(x)=2 has no roots.
For cos(x)=-1/2: x= ± arccos(-1/2) + 2πk; x= ±2π/3 + 2πk
Answer: x= ±2π/3 + 2πk
Homogeneous trigonometric equations.
Definition: Equations of the form a sin(x)+b cos(x) are called homogeneous trigonometric equations of the first degree.Equations of the form
homogeneous trigonometric equations of the second degree.
To solve a homogeneous trigonometric equation of the first degree, divide it by cos(x): 
You cannot divide by the cosine if it is equal to zero, let's make sure that this is not the case:
Let cos(x)=0, then asin(x)+0=0 => sin(x)=0, but sine and cosine are not equal to zero at the same time, we get a contradiction, so we can safely divide by zero.
Solve the equation:
Example: cos 2 (x) + sin(x) cos(x) = 0
Solution:
Let's take out the common factor: cos(x)(c0s(x) + sin (x)) = 0
Then we need to solve two equations:
Cos(x)=0 and cos(x)+sin(x)=0
Cos(x)=0 at x= π/2 + πk;
Consider the equation cos(x)+sin(x)=0 Divide our equation by cos(x):
1+tg(x)=0 => tg(x)=-1 => x=arctg(-1) +πk= -π/4+πk
Answer: x= π/2 + πk and x= -π/4+πk
How to solve homogeneous trigonometric equations of the second degree?
Guys, always follow these rules!
1. See what the coefficient a is equal to, if a=0 then our equation will take the form cos(x)(bsin(x)+ccos(x)), an example of the solution of which is on the previous slide
2. If a≠0, then you need to divide both sides of the equation by the cosine squared, we get:
We change the variable t=tg(x) and get the equation:
Solve example No.:3
Solve the equation:Solution:
Let's divide both sides of the equation by the cosine square: 
We change the variable t=tg(x): t 2 + 2 t - 3 = 0
Let's find the roots of the quadratic equation: t=-3 and t=1
Then: tg(x)=-3 => x=arctg(-3) + πk=-arctg(3) + πk
Tg(x)=1 => x= π/4+ πk
Answer: x=-arctg(3) + πk and x= π/4+ πk
Solve example No.:4
Solve the equation:Solution:
Let's transform our expression:
We can solve such equations: x= - π/4 + 2πk and x=5π/4 + 2πk
Answer: x= - π/4 + 2πk and x=5π/4 + 2πk
Solve example no.:5
Solve the equation:Solution:
Let's transform our expression:
Let us introduce the replacement tg(2x)=t:2 2 - 5t + 2 = 0
The solution to our quadratic equation will be the roots: t=-2 and t=1/2
Then we get: tg(2x)=-2 and tg(2x)=1/2
2x=-arctg(2)+ πk => x=-arctg(2)/2 + πk/2
2x= arctg(1/2) + πk => x=arctg(1/2)/2+ πk/2
Answer: x=-arctg(2)/2 + πk/2 and x=arctg(1/2)/2+ πk/2
Problems for independent solution.
1) Solve the equationA) sin(7x)= 1/2 b) cos(3x)= √3/2 c) cos(-x) = -1 d) tg(4x) = √3 d) ctg(0.5x) = -1.7
2) Solve the equations: sin(3x)= √3/2. And find all the roots on the segment [π/2; π].
3) Solve the equation: cot 2 (x) + 2 cot (x) + 1 =0
4) Solve the equation: 3 sin 2 (x) + √3sin (x) cos(x) = 0
5) Solve the equation: 3sin 2 (3x) + 10 sin(3x)cos(3x) + 3 cos 2 (3x) =0
6) Solve the equation: cos 2 (2x) -1 - cos(x) =√3/2 -sin 2 (2x)
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The relationships between the basic trigonometric functions - sine, cosine, tangent and cotangent - are given trigonometric formulas. And since there are quite a lot of connections between trigonometric functions, this explains the abundance of trigonometric formulas. Some formulas connect trigonometric functions of the same angle, others - functions of a multiple angle, others - allow you to reduce the degree, fourth - express all functions through the tangent of a half angle, etc.
In this article we will list in order all the main trigonometric formulas, which are sufficient to solve the vast majority of trigonometry problems. For ease of memorization and use, we will group them by purpose and enter them into tables.
Page navigation.
Basic trigonometric identities
Basic trigonometric identities define the relationship between sine, cosine, tangent and cotangent of one angle. They follow from the definition of sine, cosine, tangent and cotangent, as well as the concept of the unit circle. They allow you to express one trigonometric function in terms of any other.
For a detailed description of these trigonometry formulas, their derivation and examples of application, see the article.
Reduction formulas
Reduction formulas follow from the properties of sine, cosine, tangent and cotangent, that is, they reflect the property of periodicity of trigonometric functions, the property of symmetry, as well as the property of shift by a given angle. These trigonometric formulas allow you to move from working with arbitrary angles to working with angles ranging from zero to 90 degrees.
The rationale for these formulas, a mnemonic rule for memorizing them and examples of their application can be studied in the article.
Addition formulas
Trigonometric addition formulas show how trigonometric functions of the sum or difference of two angles are expressed in terms of trigonometric functions of those angles. These formulas serve as the basis for deriving the following trigonometric formulas.
Formulas for double, triple, etc. angle
Formulas for double, triple, etc. angle (they are also called multiple angle formulas) show how trigonometric functions of double, triple, etc. angles () are expressed in terms of trigonometric functions of a single angle. Their derivation is based on addition formulas.
More detailed information is collected in the article formulas for double, triple, etc. angle
Half angle formulas
Half angle formulas show how trigonometric functions of a half angle are expressed in terms of the cosine of a whole angle. These trigonometric formulas follow from the double angle formulas.
Their conclusion and examples of application can be found in the article.
Degree reduction formulas
Trigonometric formulas for reducing degrees are intended to facilitate the transition from natural degrees trigonometric functions to sines and cosines to the first degree, but multiple angles. In other words, they allow you to reduce the powers of trigonometric functions to the first.
Formulas for the sum and difference of trigonometric functions
The main purpose formulas for the sum and difference of trigonometric functions is to go to the product of functions, which is very useful when simplifying trigonometric expressions. These formulas are also widely used in solving trigonometric equations, as they allow you to factor the sum and difference of sines and cosines.
Formulas for the product of sines, cosines and sine by cosine
The transition from the product of trigonometric functions to a sum or difference is carried out using the formulas for the product of sines, cosines and sine by cosine.
Universal trigonometric substitution
We complete our review of the basic formulas of trigonometry with formulas expressing trigonometric functions in terms of the tangent of a half angle. This replacement was called universal trigonometric substitution. Its convenience lies in the fact that all trigonometric functions are expressed in terms of the tangent of a half angle rationally without roots.
Bibliography.
- Algebra: Textbook for 9th grade. avg. school/Yu. N. Makarychev, N. G. Mindyuk, K. I. Neshkov, S. B. Suvorova; Ed. S. A. Telyakovsky. - M.: Education, 1990. - 272 pp.: ill. - ISBN 5-09-002727-7
- Bashmakov M. I. Algebra and the beginnings of analysis: Textbook. for 10-11 grades. avg. school - 3rd ed. - M.: Education, 1993. - 351 p.: ill. - ISBN 5-09-004617-4.
- Algebra and the beginning of analysis: Proc. for 10-11 grades. general education institutions / A. N. Kolmogorov, A. M. Abramov, Yu. P. Dudnitsyn and others; Ed. A. N. Kolmogorov. - 14th ed. - M.: Education, 2004. - 384 pp.: ill. - ISBN 5-09-013651-3.
- Gusev V. A., Mordkovich A. G. Mathematics (a manual for those entering technical schools): Proc. allowance.- M.; Higher school, 1984.-351 p., ill.
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The simplest trigonometric equations are solved, as a rule, using formulas. Let me remind you that the simplest trigonometric equations are:
sinx = a
cosx = a
tgx = a
ctgx = a
x is the angle to be found,
a is any number.
And here are the formulas with which you can immediately write down the solutions to these simplest equations.
For sine:
For cosine:
x = ± arccos a + 2π n, n ∈ Z
For tangent:
x = arctan a + π n, n ∈ Z
For cotangent:
x = arcctg a + π n, n ∈ Z
Actually, this is the theoretical part of solving the simplest trigonometric equations. Moreover, everything!) Nothing at all. However, the number of errors on this topic is simply off the charts. Especially if the example deviates slightly from the template. Why?
Yes, because a lot of people write down these letters, without understanding their meaning at all! He writes down with caution, lest something happen...) This needs to be sorted out. Trigonometry for people, or people for trigonometry, after all!?)
Let's figure it out?
One angle will be equal to arccos a, second: -arccos a.
And it will always work out this way. For any A.
If you don’t believe me, hover your mouse over the picture, or touch the picture on your tablet.) I changed the number A to something negative. Anyway, we got one corner arccos a, second: -arccos a.
Therefore, the answer can always be written as two series of roots:
x 1 = arccos a + 2π n, n ∈ Z
x 2 = - arccos a + 2π n, n ∈ Z
Let's combine these two series into one:
x= ± arccos a + 2π n, n ∈ Z
And that's all. We have obtained a general formula for solving the simplest trigonometric equation with cosine.
If you understand that this is not some kind of superscientific wisdom, but just a shortened version of two series of answers, You will also be able to handle tasks “C”. With inequalities, with selecting roots from a given interval... There the answer with a plus/minus does not work. But if you treat the answer in a businesslike manner and break it down into two separate answers, everything will be resolved.) Actually, that’s why we’re looking into it. What, how and where.
In the simplest trigonometric equation
sinx = a
we also get two series of roots. Always. And these two series can also be recorded in one line. Only this line will be trickier:
x = (-1) n arcsin a + π n, n ∈ Z
But the essence remains the same. Mathematicians simply designed a formula to make one instead of two entries for series of roots. That's all!
Let's check the mathematicians? And you never know...)
In the previous lesson, the solution (without any formulas) of a trigonometric equation with sine was discussed in detail:
The answer resulted in two series of roots:
x 1 = π /6 + 2π n, n ∈ Z
x 2 = 5π /6 + 2π n, n ∈ Z
If we solve the same equation using the formula, we get the answer:
x = (-1) n arcsin 0.5 + π n, n ∈ Z
Actually, this is an unfinished answer.) The student must know that arcsin 0.5 = π /6. The complete answer would be:
x = (-1)n π /6+ π n, n ∈ Z
Here it arises interest Ask. Reply via x 1; x 2 (this is the correct answer!) and through lonely X (and this is the correct answer!) - are they the same thing or not? We'll find out now.)
We substitute in the answer with x 1 values n =0; 1; 2; etc., we count, we get a series of roots:
x 1 = π/6; 13π/6; 25π/6 and so on.
With the same substitution in response with x 2 , we get:
x 2 = 5π/6; 17π/6; 29π/6 and so on.
Now let's substitute the values n (0; 1; 2; 3; 4...) into the general formula for single X . That is, we raise minus one to the zero power, then to the first, second, etc. Well, of course, we substitute 0 into the second term; 1; 2 3; 4, etc. And we count. We get the series:
x = π/6; 5π/6; 13π/6; 17π/6; 25π/6 and so on.
That's all you can see.) The general formula gives us exactly the same results as are the two answers separately. Just everything at once, in order. The mathematicians were not fooled.)
Formulas for solving trigonometric equations with tangent and cotangent can also be checked. But we won’t.) They are already simple.
I wrote out all this substitution and checking specifically. It is important to understand one thing here simple thing: there are formulas for solving elementary trigonometric equations, just a short summary of the answers. For this brevity, we had to insert plus/minus into the cosine solution and (-1) n into the sine solution.
These inserts do not interfere in any way in tasks where you just need to write down the answer to an elementary equation. But if you need to solve an inequality, or then you need to do something with the answer: select roots on an interval, check for ODZ, etc., these insertions can easily unsettle a person.
So what should I do? Yes, either write the answer in two series, or solve the equation/inequality using the trigonometric circle. Then these insertions disappear and life becomes easier.)
We can summarize.
To solve the simplest trigonometric equations, there are ready-made answer formulas. Four pieces. They are good for instantly writing down the solution to an equation. For example, you need to solve the equations:
sinx = 0.3
Easily: x = (-1) n arcsin 0.3 + π n, n ∈ Z
cosx = 0.2
No problem: x = ± arccos 0.2 + 2π n, n ∈ Z
tgx = 1.2
Easily: x = arctan 1,2 + π n, n ∈ Z
ctgx = 3.7
One left: x= arcctg3,7 + π n, n ∈ Z
cos x = 1.8
If you, shining with knowledge, instantly write the answer:
x= ± arccos 1.8 + 2π n, n ∈ Z
then you are already shining, this... that... from a puddle.) Correct answer: there are no solutions. Don't understand why? Read what arc cosine is. In addition, if on the right side of the original equation there are tabular values of sine, cosine, tangent, cotangent, - 1; 0; √3; 1/2; √3/2 and so on. - the answer through the arches will be unfinished. Arches must be converted to radians.
And if you come across inequality, like
then the answer is:
x πn, n ∈ Z
there is rare nonsense, yes...) Here you need to solve using the trigonometric circle. What we will do in the corresponding topic.
For those who heroically read to these lines. I simply cannot help but appreciate your titanic efforts. Bonus for you.)
Bonus:
When writing down formulas in an alarming combat situation, even seasoned nerds often get confused about where πn, And where 2π n. Here's a simple trick for you. In everyone formulas worth πn. Except for the only formula with arc cosine. It stands there 2πn. Two peen. Keyword - two. In this same formula there are two sign at the beginning. Plus and minus. Here and there - two.
So if you wrote two sign before the arc cosine, it’s easier to remember what will happen at the end two peen. And it also happens the other way around. The person will miss the sign ± , gets to the end, writes correctly two Pien, and he’ll come to his senses. There's something ahead two sign! The person will return to the beginning and correct the mistake! Like this.)
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