MBOU "Mordovsko-Paevskaya secondary school" of the Insarsky 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) Basic types of trigonometric equations and methods for solving them:
1. Equations reduced to the simplest. …………………………………..5
2. Equations reducing to quadratic……………………………….5
3. Homogeneous equations acosx + b sin x = 0………………………………...6
4. Equations of the form acosx + b sin x = 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 on the Unified State Exam of previous years…………...……10-14
Conclusion………………………………………………………………………………….14
Appendix………………………………………………………..……………………….15-17
Literature………………………………………………………………………………………..18
Introduction
“The only path leading to knowledge is activity...”
Bernard Show
Relevance of the work.
In a few months I graduate from school.
To avoid problems with further choice life path, necessary obtain a school certificate, and in order to obtain a school certificate, you must pass two mandatory exams in the form of the Unified State Examination - and one of themmathematics. What can we 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 Exam is an important test before moving to new life and admission to a university or college. It is especially important to pass it with good scores.The Unified State Examination in mathematics is a serious test and without a good foundation, a 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 the tasks well. I do not claim the maximum score, but nevertheless I prepare diligently. 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 them is to sequentially simplify them in order to reduce them to one or several simplest 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, I set myself the followingtarget:
Systematize and expand knowledge and skills related to the use of methods for solving trigonometric equations.
Object of study is the study of trigonometric equations in Unified State Examination tasks.
Subject of study- is the solution to trigonometric equations
Thus, main goal writing this course work is the study of trigonometric equations and their systems, methods of solving them.
In accordance with the goals, object and subject of the study, the following are defined: tasks:
1). Study all the tasks related to solving trigonometric equations offered at Unified State Examination works previous years and when performing diagnostic work;
2) Study methods for solving trigonometric equations.
3). Identify the main possible mistakes when solving such equations;
4). Find out the reason for making such mistakes.
6). Draw conclusions.
In my work I will solve several trigonometric equations, show possible errors in solving them and try to answer the following questions:
1). Is it possible to avoid mistakes when performing type C1 tasks?
2) If I practice solving equations of this type, then I can
Is it possible to carry out such tasks without errors?
For this purpose, I studied all the demonstration and training tasks carried out with us, Unified State Exam materials previous years;
studied reference sources;
independently solved tasks from the Internet;
consulted her teacher in case of difficulty;
I learned to analyze and correctly format the results.
Chapter I. About trigonometric equations.
1) Definition 1. A trigonometric equation is an equation containing a variable under the sign 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.
Solving a trigonometric equation consists of two stages: transforming the equation to obtain its simplest form and solving the resulting simplest trigonometric equation.
2) Basic types of trigonometric equations.
Equations reduced to the simplest.
Solve the equation
Solution:
Answer:
Equations reducing 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. This means cosx ≠ 0 and we can divide the equation by cosx.
We get
Answer:
Example: Solve the equation
Solution: 
Answer:
Equations solved by factorization.
Priper: 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:
Let's look at the functions
Chapter II. Basic concepts and formulas of trigonometry.
Trigonometric equations are a required topic in any mathematics exam.
ABOUTx, how much agony learning trigonometry causes students.
Certain difficulties arise even if there is a teacher nearbymathematics and explains every little detail. This is understandable; there are more than twenty basic formulas alone. And if we count their derivatives... The student gets confused in the calculations and cannot remember the mechanisms by which these formulas allow one 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 don’t just need to know the formula, but you need to know where it can be applied, how to open it, and what the essence of the formula is, and for this you need to solve examples specifically for those problems that are difficult to solve.
At first it seemed to metrigonometry is a boring set of formulas and graphs. However, as I got acquainted with new concepts of trigonometry and methods for solving trigonometric equations, I became convinced every time how interesting and fascinating the world of trigonometry is.
Firstly, to successfully solve trigonometric equations, you need to know trigonometric formulas well, not only the basic ones, but also additional ones (converting the sum of trigonometric functions into a product and products into a sum, formulas for reducing degrees, etc.),since the use of cheat sheets and mobile phones prohibited
(Annex 1)
Secondly , we must know clearly standard formulas roots of the simplest trigonometric equations (useful to remember or be able to obtain simplified formulas for the roots of equations using the trigonometric circle)
Each of these equations is solved using formulas that you should know. These are the formulas:
a) Functiony= sinx. The function is limited: it is within [-1; 1]. This means that when solving equations likesinx=2 orsinxsinx
1) sinx =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 . The function is limited: it is within [-1; 1]. This means that when solving equations likecosx=2 orcosx=-5 the answer turns out to be: no roots. Formulas for the function y=cosx:
1. cosx =a, X=± arccos a+2n,n Z
2.cos x=-a, X=±( - arccos a)+2n,n Z
Special cases: 1.
cosx =-1,
X =
+2
n,
n Z
2.
cosx =0,
3. cosx =1, X= 2n,n Z
3. Functiony= tgx.
There is only one formula, without special cases:tgx = ± a .
X = ±
arctan a+n,n Z
Thirdly, you need to know the values of trigonometric functions;
(Appendix 2)
Fourthly, If in an equation the trigonometric function is under the radical sign, then such a trigonometric equation will be irrational. In such equations, one must follow all the rules that are used when solving ordinary irrational equations (the range of permissible values of both the equation itself and when removing the root of an even degree is taken into account).
V. Equations offered on the Unified State Exam of previous 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 achieve the goal.”
Leibniz
1. Equations that reduce to quadratic.
C1. Solve the equation: 
Solution: Using the basic trigonometric identity,we rewrite the equation in the form
Replacementcos=
tthe equation reduces to 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 segment 
, otherwise extraneous roots will appear.
2. Equations solved by factorization
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:
we group and put the common factor out of brackets, we get
Equation 1) has no solutions.
The second equation is homogeneous, can be solved by dividing term by term by cosx ≠0, we get 
, where 
b) 
Answer: a) 
b) 
Conclusion:
1. When solving an equation of this type, firstly, you need to know that |sin x|≤1 and |cosx |≤1, and the equation sinx =-2 has no solutions;
2. Secondly, justify division by cosx ≠о (since if cosx = 0, then sin x = 0, but this is impossible;
thirdly, it is reasonable to select roots belonging to a given interval
3
.Equation for applying reduction formulas
C1 (2010) Given the equation
a) solve the equation;
b 
) Indicate the roots belonging to the segment
Solution: Using the reduction formulas, we get:
sin 2 x – cos x =0,
2 sinx cosx- cosx =0,
With osx (2 sinx -1)=0, whence cosx= 0 or sinx =½,
b) Find the values of k at which the roots will belong to
the specified interval. To select the roots. belonging to a given interval, we present the solution in the form:
b 
) Find the values of k at which the roots will belong to the specified interval.
2)
Solving this inequality, the whole
we won’t get values for k.
Answer: a)
b) 
Conclusion:
When solving an equation of this type, it is necessary to know the formulas of the given equation and apply it correctly; be able to present a solution 
into two series of roots; choose the correct roots belonging to a given segment.
4. Systems of trigonometric equations
C1 (2010).
Solve system of equations 
Solution: O.D.Z 
A fraction is equal to zero if the numerator is 0 and the denominator is not 0.
From the equation 2sin 2 x – 3 sinx +1 =0, solving by introducing a new variable, we find
or sin x =1.
1) Let , Then 
and y = cos x = 
›0 (using the basic trigonometric identity)
or 
And 
- there is no decision.
2) Let sinx = 1, then y = cos x = 0 – there is no solution.
Answer: and y =
Conclusion: 1) it is necessary to take into account the limitations 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, from where t 1 = -½, t 2 = -
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 the work I did, I studied solutions to trigonometric equations, considered recommendations for solving trigonometric equations, methods for solving trigonometric equations, and considered errors that are possible when solving them.
I came to the following conclusions:
1. Type C1 tasks test the ability to solve trigonometric equations. These tasks are, indeed, simple, which gives excess self-confidence and lulls attention. The only difficulty of these tasks is that, having solved an equation or system of equations, discard extraneous roots.
2.
Problem C1 is the simplest problem of group C. When solving it, cumbersome transformations and complex calculations should not arise. If they appear, you immediately need to stop, check the solution and try to understand what is wrong here.
3. Ultimately,The main requirement is that the solution must be mathematically literate, and 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, in order to learn how to solve equations without errors, you need to solve them! After all, as Polya said, “If you want to learn how to swim, then feel free to 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) formulas for reducing the degree: 
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) Reduction formulas
Reduction formulas are called formulas the following type:
Sums and differences of trigonometric equations
Cosine-even, sine, tangent and cotangent, that is:
Sine and cosine -
. Tangent and has
,cotangent 0; ±π; ±2π;…
Functionsy = cosx, y = sinx -
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Maintaining your privacy is important to us. For this reason, we have developed a Privacy Policy that describes how we use and store your information. Please review our privacy practices and let us know if you have any questions.
Personal information refers to data that can be used to identify or contact a specific person.
You may be asked to provide your personal information at any time when you contact us.
Below are some examples of the types of personal information we may collect and how we may use such information.
What personal information do we collect:
How we use your personal information:
We do not disclose the information received from you to third parties.
Exceptions:
We take precautions - including administrative, technical and physical - to protect your personal information from loss, theft, and misuse, as well as unauthorized access, disclosure, alteration and destruction.
To ensure that your personal information is secure, we communicate privacy and security standards to our employees and strictly enforce privacy practices.
