The simplest trigonometric sine equations. Trigonometric equations

When solving many mathematical problems, especially those that occur before grade 10, the order of actions performed that will lead to the goal is clearly defined. Such problems include, for example, linear and quadratic equations, linear and quadratic inequalities, fractional equations and equations that reduce to quadratic ones. The principle of successfully solving each of the mentioned problems is as follows: you need to establish what type of problem you are solving, remember the necessary sequence of actions that will lead to the desired result, i.e. answer and follow these steps.

It is obvious that success or failure in solving a particular problem depends mainly on how correctly the type of equation being solved is determined, how correctly the sequence of all stages of its solution is reproduced. Of course, in this case it is necessary to have the skills to perform identical transformations and calculations.

The situation is different with trigonometric equations. It is not at all difficult to establish the fact that the equation is trigonometric. Difficulties arise when determining the sequence of actions that would lead to the correct answer.

It is sometimes difficult to determine its type based on the appearance of an equation. And without knowing the type of equation, it is almost impossible to choose the right one from several dozen trigonometric formulas.

To solve a trigonometric equation, you need to try:

1. bring all functions included in the equation to “the same angles”;
2. bring the equation to “identical functions”;
3. factor the left side of the equation, etc.

Let's consider basic methods for solving trigonometric equations.

I. Reduction to the simplest trigonometric equations

Solution diagram

Step 1. Express a trigonometric function in terms of known components.

Step 2. Find the function argument using the formulas:

cos x = a; x = ±arccos a + 2πn, n ЄZ.

sin x = a; x = (-1) n arcsin a + πn, n Є Z.

tan x = a; x = arctan a + πn, n Є Z.

ctg x = a; x = arcctg a + πn, n Є Z.

Step 3. Find the unknown variable.

Example.

2 cos(3x – π/4) = -√2.

Solution.

1) cos(3x – π/4) = -√2/2.

2) 3x – π/4 = ±(π – π/4) + 2πn, n Є Z;

3x – π/4 = ±3π/4 + 2πn, n Є Z.

3) 3x = ±3π/4 + π/4 + 2πn, n Є Z;

x = ±3π/12 + π/12 + 2πn/3, n Є Z;

x = ±π/4 + π/12 + 2πn/3, n Є Z.

Answer: ±π/4 + π/12 + 2πn/3, n Є Z.

II. Variable replacement

Solution diagram

Step 1. Reduce the equation to algebraic form with respect to one of the trigonometric functions.

Step 2. Denote the resulting function by the variable t (if necessary, introduce restrictions on t).

Step 3. Write down and solve the resulting algebraic equation.

Step 4. Make a reverse replacement.

Step 5. Solve the simplest trigonometric equation.

Example.

2cos 2 (x/2) – 5sin (x/2) – 5 = 0.

Solution.

1) 2(1 – sin 2 (x/2)) – 5sin (x/2) – 5 = 0;

2sin 2 (x/2) + 5sin (x/2) + 3 = 0.

2) Let sin (x/2) = t, where |t| ≤ 1.

3) 2t 2 + 5t + 3 = 0;

t = 1 or e = -3/2, does not satisfy the condition |t| ≤ 1.

4) sin(x/2) = 1.

5) x/2 = π/2 + 2πn, n Є Z;

x = π + 4πn, n Є Z.

Answer: x = π + 4πn, n Є Z.

III. Equation order reduction method

Solution diagram

Step 1. Replace this equation with a linear one, using the formula for reducing the degree:

sin 2 x = 1/2 · (1 – cos 2x);

cos 2 x = 1/2 · (1 + cos 2x);

tg 2 x = (1 – cos 2x) / (1 + cos 2x).

Step 2. Solve the resulting equation using methods I and II.

Example.

cos 2x + cos 2 x = 5/4.

Solution.

1) cos 2x + 1/2 · (1 + cos 2x) = 5/4.

2) cos 2x + 1/2 + 1/2 · cos 2x = 5/4;

3/2 cos 2x = 3/4;

2x = ±π/3 + 2πn, n Є Z;

x = ±π/6 + πn, n Є Z.

Answer: x = ±π/6 + πn, n Є Z.

IV. Homogeneous equations

Solution diagram

Step 1. Reduce this equation to the form

a) a sin x + b cos x = 0 (homogeneous equation of the first degree)

or to the view

b) a sin 2 x + b sin x · cos x + c cos 2 x = 0 (homogeneous equation of the second degree).

Step 2. Divide both sides of the equation by

a) cos x ≠ 0;

b) cos 2 x ≠ 0;

and get the equation for tan x:

a) a tan x + b = 0;

b) a tan 2 x + b arctan x + c = 0.

Step 3. Solve the equation using known methods.

Example.

5sin 2 x + 3sin x cos x – 4 = 0.

Solution.

1) 5sin 2 x + 3sin x · cos x – 4(sin 2 x + cos 2 x) = 0;

5sin 2 x + 3sin x · cos x – 4sin² x – 4cos 2 x = 0;

sin 2 x + 3sin x · cos x – 4cos 2 x = 0/cos 2 x ≠ 0.

2) tg 2 x + 3tg x – 4 = 0.

3) Let tg x = t, then

t 2 + 3t – 4 = 0;

t = 1 or t = -4, which means

tg x = 1 or tg x = -4.

From the first equation x = π/4 + πn, n Є Z; from the second equation x = -arctg 4 + πk, k Є Z.

Answer: x = π/4 + πn, n Є Z; x = -arctg 4 + πk, k Є Z.

V. Method of transforming an equation using trigonometric formulas

Solution diagram

Step 1. Using all possible trigonometric formulas, reduce this equation to an equation solved by methods I, II, III, IV.

Step 2. Solve the resulting equation using known methods.

Example.

sin x + sin 2x + sin 3x = 0.

Solution.

1) (sin x + sin 3x) + sin 2x = 0;

2sin 2x cos x + sin 2x = 0.

2) sin 2x (2cos x + 1) = 0;

sin 2x = 0 or 2cos x + 1 = 0;

From the first equation 2x = π/2 + πn, n Є Z; from the second equation cos x = -1/2.

We have x = π/4 + πn/2, n Є Z; from the second equation x = ±(π – π/3) + 2πk, k Є Z.

As a result, x = π/4 + πn/2, n Є Z; x = ±2π/3 + 2πk, k Є Z.

Answer: x = π/4 + πn/2, n Є Z; x = ±2π/3 + 2πk, k Є Z.

The ability and skill to solve trigonometric equations is very important, their development requires significant effort, both on the part of the student and on the part of the teacher.

Many problems of stereometry, physics, etc. are associated with the solution of trigonometric equations. The process of solving such problems embodies many of the knowledge and skills that are acquired by studying the elements of trigonometry.

Trigonometric equations occupy an important place in the process of learning mathematics and personal development in general.

Still have questions? Don't know how to solve trigonometric equations?
To get help from a tutor, register.
The first lesson is free!

website, when copying material in full or in part, a link to the source is required.

The video course “Get an A” includes all the topics necessary to successfully pass the Unified State Exam in mathematics with 60-65 points. Completely all tasks 1-13 of the Profile Unified State Exam in mathematics. Also suitable for passing the Basic Unified State Examination in mathematics. If you want to pass the Unified State Exam with 90-100 points, you need to solve part 1 in 30 minutes and without mistakes!

Preparation course for the Unified State Exam for grades 10-11, as well as for teachers. Everything you need to solve Part 1 of the Unified State Exam in mathematics (the first 12 problems) and Problem 13 (trigonometry). And this is more than 70 points on the Unified State Exam, and neither a 100-point student nor a humanities student can do without them.

All the necessary theory. Quick solutions, pitfalls and secrets of the Unified State Exam. All current tasks of part 1 from the FIPI Task Bank have been analyzed. The course fully complies with the requirements of the Unified State Exam 2018.

The course contains 5 large topics, 2.5 hours each. Each topic is given from scratch, simply and clearly.

Hundreds of Unified State Exam tasks. Word problems and probability theory. Simple and easy to remember algorithms for solving problems. Geometry. Theory, reference material, analysis of all types of Unified State Examination tasks. Stereometry. Tricky solutions, useful cheat sheets, development of spatial imagination. Trigonometry from scratch to problem 13. Understanding instead of cramming. Clear explanations of complex concepts. Algebra. Roots, powers and logarithms, function and derivative. A basis for solving complex problems of Part 2 of the Unified State Exam.

Lesson and presentation on the topic: "Solving simple trigonometric equations"

Additional materials
Dear users, do not forget to leave your comments, reviews, wishes! All materials have been checked by an anti-virus program.

Manuals and simulators in the Integral online store for grade 10 from 1C
We solve problems in geometry. Interactive tasks for building in space
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.

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.

Solution:

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:

Let us solve our equation in general form: 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.

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 of the 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.

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

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

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

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.

A) 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)

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.

Collection and use of personal information

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:

• When you submit an application on the site, we may collect various information, including your name, phone number, email address, etc.

How we use your personal information:

• The personal information we collect allows us to contact you with unique offers, promotions and other events and upcoming events.
• From time to time, we may use your personal information to send important notices and communications.
• We may also use personal information for internal purposes, such as conducting audits, data analysis and various research in order to improve the services we provide and provide you with recommendations regarding our services.
• If you participate in a prize draw, contest or similar promotion, we may use the information you provide to administer such programs.

Disclosure of information to third parties

We do not disclose the information received from you to third parties.

Exceptions:

• If necessary - in accordance with the law, judicial procedure, in legal proceedings, and/or on the basis of public requests or requests from government bodies in the Russian Federation - to disclose your personal information. We may also disclose information about you if we determine that such disclosure is necessary or appropriate for security, law enforcement, or other public importance purposes.
• In the event of a reorganization, merger, or sale, we may transfer the personal information we collect to the applicable successor third party.

Protection of personal information

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.

Respecting your privacy at the company level

To ensure that your personal information is secure, we communicate privacy and security standards to our employees and strictly enforce privacy practices.