MBOU "Mordovsko-Paevskaya secondary school" of the Insar district of the Republic of Moldova
Completed by: Pantileikina Nadezhda,
11th grade student
Head: Kadyshkina N.V.,
mathematic teacher
Table of contents
Introduction…………………………………………………………………………….
Chapter I. About trigonometric equations……………………………………..…5
1) The main types of trigonometric equations and methods for their solution:
1. Equations Reducing to the Simplest. …………………………………..5
2. Equations that reduce to squares…………………………………….5
3. Homogeneous equations acosx + b sin x = 0………………………………...6
4. Equations of the form acosx + b sin x \u003d c, c≠ 0……………………………………7
5. Equations solved by factorization…………………...….7
6. Non-standard equations………………………………………………….8
Chapter II. Basic concepts and formulas of trigonometry…………………….8-10
Chapter II I. Equations offered at the Unified State Examination of past years…………...……10-14
Conclusion…………………………………………………………………………….14
Application……………………………………………..……………………….15-17
Literature…………………………………………………………………………..18
Introduction
“The only way leading to knowledge is activity...”
Bernard Show
The relevance of the work.
I'm finishing school in a few months.
In order to avoid problems with the further choice of a life path, it is necessary get a school certificate, and in order to get a school certificate, you must pass two mandatory exams in the form of the Unified State Examination - and one of themmaths. What can I say, final exams are a crucial period in the life of any student, on which not only the final grade in the certificate depends, but also his professional future, income and career.
The Unified State Examination is an important test before moving on to a new life and entering a university or college. It is especially important to pass it with good scores.The USE in mathematics is a serious test, and without a good base, the student will not be able to claim a decent result.
How to avoid failing the exam and get good scores? To do this, you need to solve problems well. I do not pretend to the maximum score, nevertheless, I am diligently preparing. And I noticed that even on the first task of part C, namely, on solving trigonometric equations and their systems, I make mistakes.At first glance, problem C1 is a relatively simple equation or system of equations that may contain trigonometric functions,one of the main approaches to solving which consists in their successive simplification in order to reduce them to one or more simple ones.So why am I wrong?
Relevance of the topic is determined by the fact that students must understand certain methods of solving trigonometric equations.
Therefore, in front of me I put the followinggoal:
Systematize, expand knowledge and skills related to the use of methods for solving trigonometric equations.
Object of study is the study of trigonometric equations in the tasks of the exam.
Subject of study- is the solution of trigonometric equations
In this way, main goal writing this term paper is the study of trigonometric equations and their systems, ways to solve them.
In accordance with the goals, object and subject of the study, the following tasks:
one). To study all the tasks related to the solution of trigonometric equations offered at the Unified State Examination of works of previous years and when performing diagnostic work;
2) Study methods for solving trigonometric equations.
3). Identify the main possible errors in solving such equations;
four). Find out the reason for such errors.
6). To conclude.
In my work, I will solve several trigonometric equations, show possible errors in solving them, and try to answer the following questions:
one). Is it possible to avoid mistakes when performing tasks of type C1
2) If I practice solving equations of this type, then I can
Is it possible to perform such tasks without error?
For this purpose, I studied all the demonstration and training tasks conducted with us, the USE materials of previous years;
studied reference sources;
independently solved tasks from the Internet;
consulted with her teacher in case of difficulty;
I learned how to analyze and correctly draw up the results.
Chapter I. On trigonometric equations.
1) Definition 1. A trigonometric equation is an equation containing a variable under the sign of trigonometric functions.
The simplest trigonometric equations are equations of the form sin x = a,
cos x=a, tg x=a, ctg x = a.
In such equations, the variable is under the sign of the trigonometric function, and is the given number.
The solution of the trigonometric equation consists of two stages: the transformation of the equation to obtain its simplest form and the solution of the resulting simplest trigonometric equation.
2) Basic types of trigonometric equations.
Equations reduced to the simplest.
solve the equation
Solution:
Answer:
Equations that reduce to quadratic.
1) Solve the equation 2 sin 2 x - cosx -1 = 0.
Answer:
Homogeneous equations: asinx + bcosx = 0
a sin 2 x + b sinxcosx + c cos 2 x = 0.
Solve the equation 2sinx - 3cosx = 0
Solution: Let cosx = 0, then 2sinx = 0 and sinx = 0 - a contradiction with
that sin 2 x + cos 2 x = 1. So cosx ≠ 0 and we can divide the equation by cosx.
Get
Answer:
Example: solve the equation
Solution: 
Answer:
Equations Solved by Factoring.
Pryper: Solve the equation sin2x - sinx = 0.
Solution: Using the formula sin2x = 2sinxcosx, we get
2sinxcosx – sinx = 0,
sinx(2cosx - 1) = 0.
The product is equal to zero if at least one of the factors is equal to zero.
Answer:
Non-standard equations.
Solve the equation cosx = X 2 + 1.
Solution:
Consider the functions
Chapter II. Basic concepts and formulas of trigonometry.
Trigonometric equations are a required topic for any math exam.
Ox, how much torment the study of trigonometry gives students.
Certain difficulties arise even if a teacher is nearbymathematics and explains every little thing. This is understandable, there are more than twenty basic formulas alone. And if we consider their derivatives ... The student gets confused in the calculations and cannot remember the mechanisms by which these formulas make it possible to find, for example, .
You know the formulas - it's easy for you to decide. If you don't know, you won't understand, even if they give you the formula.You need to know the formula not just stupidly, but to know where it can be applied, how to reveal and what the essence of the formula is, and for this you need to solve examples for precisely those tasks that are difficult.
At first it seemed to metrigonometry is a boring set of formulas and graphs. However, getting acquainted with new concepts of trigonometry and methods for solving trigonometric equations, each time I was convinced how interesting and fascinating the world of trigonometry is.
Firstly, to successfully solve trigonometric equations, you need to know trigonometric formulas well, and not only basic, but also additional ones (converting the sum of trigonometric functions to a product and products to a sum, formulas for lowering degrees, and others),since the use of cheat sheets and mobile phones on the exam is prohibited
(Attachment 1)
Secondly , we must clearly know the standard formulas for the roots of the simplest trigonometric equations (it is useful to remember or be able to obtain simplified formulas for the roots of equations using a trigonometric circle)
Each of these equations is solved by formulas that you should know. Here are the formulas:
a) Functiony= sinx. Limited function: within [-1; one]. This means that when solving equations of the typesinx=2 orsinxsinx
1) sinx \u003d a,x= (-1) n arcsin a +n,n 
Z
2) sinx = - a,x= (-1) n+1 arcsin a +n,n Z
Also, you need to know special cases: 1)
sinx =- 1, 
2)sinx =0,
3)sinx =
a,
You also need to be able to solvein the form of two series of roots
2. Function y = cos x . Limited function: within [-1; one]. This means that when solving equations of the typecosx=2 orcosx=-5 in the answer it turns out: there are no roots. Formulas for the function y=cosx:
1. cosx=a, X=± arccos a+2n,n Z
2.cosx=-a, X=±( - arccos a)+2n,n Z
Special cases: 1.
cosx =-1,
X=
+2
n,
n Z
2.
cox=0,
3. cosx=1, X= 2n,n Z
3. Functiony= tgx.
There is only one formula, without special cases:tgx = ± a .
X = ±
arctg a+n,n Z
Thirdly, one must know the values of trigonometric functions;
(Annex 2)
Fourth, If in the equation the trigonometric function is under the sign of the radical, then such a trigonometric equation will be irrational. In such equations, all the rules that are used in solving ordinary irrational equations should be observed (the range of permissible values \u200b\u200bof both the equation itself and when freeing from the root of an even degree is taken into account).
V. Equations proposed at the Unified State Examination of past years.
"A method of solution is good if from the very beginning we can foresee - and subsequently confirm this - that by following this method we will reach the goal."
Leibniz
1. Equations that reduce to a quadratic.
C1. Solve the equation: 
Solution: Using the basic trigonometric identity,rewrite the equation in the form
replacementcos=
tthe equation is reduced to a quadratic:2t 2
+ 9
t-5 =0 which has rootst 1
= ½ andt 2
= -5. Returning to the variable x, we get 
,
The second equation has no roots since |cosx |≥1, and from the first x =± 
+6k , k Z
Answer: =± +6k , k Z
Conclusion: when introducing a new variable, you need to take into account that the values of sin x and cos x are limited by the interval 
, otherwise extraneous roots will appear.
2. Equations Solved by Factoring
Task C1 (2011)
a) Solve the equation
b) Indicate the roots of the equation belonging to the segment 
Solution: a) solve by factoring the left side:
group and put the common factor out of brackets, we get
Equation 1) has no solutions.
The second equation is homogeneous, it is solved by dividing term by term by cosx ≠0, we get 
, where 
b) 
Answer: a) 
b) 
Conclusion:
1. When solving an equation of this kind, firstly, you need to know that |sin x|≤1 and |cosx |≤1, and the equation sinx =-2 has no solutions;
2. Secondly, justify the division by cosx ≠o (because if cosx = 0, then sin x = 0, and this is impossible;
thirdly, it is reasonable to make a selection of roots belonging to this interval
3
.Equation for the application of reduction formulas
C1 (2010) Equation given
a) solve the equation;
b 
) Specify the roots belonging to the segment
Solution: Using the reduction formulas, we get:
sin 2 x - cos x \u003d 0,
2 sinx cosx - cosx =0,
With osx (2 sinx -1)=0, whence cosx= 0 or sinx =½,
b) Find the values of k for which the roots will belong to
the specified interval. To pick roots. belonging to a given interval, the solution can be represented as:
b 
) Find the values of k for which the roots will belong to the specified interval.
2)
Solving this inequality, the whole
we will not get the value of k.
Answer: a)
b) 
Conclusion:
When solving an equation of this kind, it is necessary to know the formulas of the above equation and apply it correctly; be able to present a solution 
into two series of roots; correctly select the roots belonging to a given segment.
4. Systems of trigonometric equations
C1 (2010).
Solve a system of equations 
Solution: O.D.Z 
A fraction is zero if the numerator is 0 and the denominator is not 0.
From the equation 2sin 2 x - 3 sinx +1 \u003d 0, solving by the method of introducing a new variable, we find
or sinx=1.
1) Let , then 
and y = cos x = 
›0 (using the basic trigonometric identity)
or 
and 
- there is no decision.
2) Let sinx \u003d 1, then y \u003d cos x \u003d 0 - there is no solution.
Answer: and y =
Conclusion: 1) it is necessary to take into account the limitedness of trigonometric
functions
2) Record and take into account O.D.Z.
5. C1 (USE 2011) Solve the equation:
O.D.Z. - cos x ≥ 0, sin x ≤ 0.
4sin 2 x + 12 sinx + 5 = 0 or cos x =0
sinx=t 
4 t 2 + 12 t + 5=0, whence t 1 \u003d -½, t 2 \u003d -
sinx = -½ sinx=- - has no solution
x = 
x = 
taking into account O.D.Z. x =
Answer: x =
Conclusion: Write down the answer taking into account O.D.Z.
CONCLUSION
In my work, the solutions of trigonometric equations were studied, recommendations for solving trigonometric equations, methods for solving trigonometric equations were considered, and errors that are possible when solving them were considered.
I came to the following conclusions:
1. Tasks of type C1 test the ability to solve trigonometric equations. These tasks are really simple, which gives extra self-confidence and lulls attentiveness. The only difficulty of these tasks is that, having solved an equation or a system of equations, it is necessary to discard extraneous roots.
2.
Problem C1 is the simplest problem of group C. When solving it, cumbersome transformations and complex calculations should not occur. If they appear, you need to stop immediately, check the solution and try to understand what is wrong here.
3. Ultimately,the main requirement is that the solution must be mathematically literate, the course of reasoning must be clear from it.You need to try to write down your decision briefly and clearly, but most importantly - correctly!
4. And most importantly - to learn how to solve equations without errors, you need to solve them! After all, as Poya said, “If you want to learn how to swim, then boldly dive into the water, and if you want to learn how to solve problems, you need to solve them!”
Appendix 1 (basic formulas of trigonometry)
1) basic trigonometric identitysin 2 α + cos 2 α= 1,
Dividing this equation by the square of the cosine and sine, respectively, we have
2) double argument formulassin2α =2sinα cos α,
cos 2α = cos 2 α -sin 2 α ,
Cos 2α = 1- 2sin 2 α,
3) lowering formulas: 
4) formulas for the sum and difference of two arguments:
sin(α+ β )= sinα cosβ + cos α sinβ
sin(α- β )= sinα cos β - cos α sin β
cos(α+ β )= cosα cos β + sin α sin β
cos(α- β )= sinα cos β + sinα sin β
5) Cast formulas
Reduction formulas are called formulas of the following form:
Sums sums and differences of trigonometric equations
Parity
Cosine-even, sine, tangent and cotangent, that is:
Continuity
Sine and cosine -
. Tangent and has
,cotangent 0; ±π; ±2π;…
Periodicity
Functionsy = cosx, y = sinx -
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You may be asked to provide your personal information at any time when you contact us.
The following are some examples of the types of personal information we may collect and how we may use such information.
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