Trigonometric equations table. Methods for solving trigonometric equations

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Requires knowledge of the basic formulas of trigonometry - the sum of the squares of the sine and cosine, the expression of the tangent through the sine and cosine, and others. For those who have forgotten them or do not know, we recommend reading the article "".
So the main trigonometric formulas we know it's time to put them into practice. Solution trigonometric equations with the right approach, it is quite an exciting activity, like, for example, solving a Rubik's cube.

Based on the name itself, it is clear that a trigonometric equation is an equation in which the unknown is under the sign of the trigonometric function.
There are the so-called simplest trigonometric equations. This is how they look: sinx = a, cos x = a, tg x = a. Consider how to solve such trigonometric equations, for clarity, we will use the already familiar trigonometric circle.

sinx = a

cos x = a

tg x = a

cot x = a

Any trigonometric equation is solved in two stages: we bring the equation to the simplest form and then solve it as the simplest trigonometric equation.
There are 7 main methods by which trigonometric equations are solved.

1. Variable Substitution and Substitution Method

Solve the equation 2cos 2 (x + / 6) - 3sin (/ 3 - x) +1 = 0

Using the reduction formulas, we get:

2cos 2 (x + / 6) - 3cos (x + / 6) +1 = 0

Replace cos (x + / 6) with y for simplicity and get the usual quadratic equation:

2y 2 - 3y + 1 + 0

Whose roots y 1 = 1, y 2 = 1/2

Now let's go in reverse order

We substitute the found y values ​​and we get two answers:

2. Solving trigonometric equations through factorization

How to solve the equation sin x + cos x = 1?

Move everything to the left so that 0 remains on the right:

sin x + cos x - 1 = 0

Let's use the above identities to simplify the equation:

sin x - 2 sin 2 (x / 2) = 0

We do the factorization:

2sin (x / 2) * cos (x / 2) - 2 sin 2 (x / 2) = 0

2sin (x / 2) * = 0

We get two equations

3. Reduction to a homogeneous equation

An equation is homogeneous with respect to sine and cosine if all of its terms with respect to sine and cosine are the same power of the same angle. To solve a homogeneous equation, proceed as follows:

a) transfer all of its members to the left side;

b) take all common factors out of parentheses;

c) equate all factors and brackets to 0;

d) a homogeneous equation of a lesser degree is obtained in brackets, it in turn is divided into sine or cosine in the highest degree;

e) solve the resulting equation for tg.

Solve the equation 3sin 2 x + 4 sin x cos x + 5 cos 2 x = 2

Let's use the formula sin 2 x + cos 2 x = 1 and get rid of the open two on the right:

3sin 2 x + 4 sin x cos x + 5 cos x = 2sin 2 x + 2cos 2 x

sin 2 x + 4 sin x cos x + 3 cos 2 x = 0

Divide by cos x:

tg 2 x + 4 tg x + 3 = 0

Replace tg x with y and get a quadratic equation:

y 2 + 4y +3 = 0, whose roots y 1 = 1, y 2 = 3

From here we find two solutions to the original equation:

x 2 = arctan 3 + k

4. Solving equations by going to half angle

Solve the equation 3sin x - 5cos x = 7

Moving on to x / 2:

6sin (x / 2) * cos (x / 2) - 5cos 2 (x / 2) + 5sin 2 (x / 2) = 7sin 2 (x / 2) + 7cos 2 (x / 2)

Move everything to the left:

2sin 2 (x / 2) - 6sin (x / 2) * cos (x / 2) + 12cos 2 (x / 2) = 0

Divide by cos (x / 2):

tg 2 (x / 2) - 3tg (x / 2) + 6 = 0

5. Introduce an auxiliary angle

For consideration, take an equation of the form: a sin x + b cos x = c,

where a, b, c are some arbitrary coefficients, and x is unknown.

Divide both sides of the equation into:



Now the coefficients of the equation, according to the trigonometric formulas, have the properties sin and cos, namely: their modulus is not more than 1 and the sum of squares = 1. Let's denote them as cos and sin, respectively, where is the so-called auxiliary angle. Then the equation will take the form:

cos * sin x + sin * cos x = С

or sin (x +) = C

The solution to this simplest trigonometric equation is

x = (-1) k * arcsin С - + k, where



Note that cos and sin are used interchangeably.

Solve the equation sin 3x - cos 3x = 1

In this equation, the coefficients are:

a =, b = -1, so we divide both sides by = 2

Lesson in the complex application of knowledge.

Lesson objectives.

1. Consider different methods solutions of trigonometric equations.
2. Developing students' creativity by solving equations.
3. Encouraging students to self-control, mutual control, introspection of their educational activities.

Equipment: screen, projector, reference material.

During the classes

Introductory conversation.

The main method for solving trigonometric equations is to reduce them to the simplest. In this case, the usual methods are used, for example, factorization, as well as techniques used only for solving trigonometric equations. There are quite a few of these techniques, for example, various trigonometric substitutions, transformations of angles, transformations of trigonometric functions. The indiscriminate application of any trigonometric transformations usually does not simplify the equation, but catastrophically complicates it. To work out in general terms a plan for solving the equation, to outline the way to reduce the equation to the simplest one, you must first analyze the angles - the arguments of the trigonometric functions included in the equation.

Today we will talk about methods for solving trigonometric equations. A correctly chosen method often makes it possible to significantly simplify the solution, therefore, all the methods we have studied should always be kept in the area of ​​our attention in order to solve trigonometric equations by the most suitable method.

II. (Using the projector, we repeat the methods for solving equations.)

1. The method of reducing a trigonometric equation to an algebraic one.

It is necessary to express all trigonometric functions in terms of one, with the same argument. This can be done using the basic trigonometric identity and its consequences. Let's get an equation with one trigonometric function. Taking it as a new unknown, we get an algebraic equation. We find its roots and return to the old unknown, solving the simplest trigonometric equations.

2. Method of factorization.

To change the angles, conversion formulas, the sum and difference of arguments, as well as formulas for converting the sum (difference) of trigonometric functions into a product and vice versa, are often useful.

sin x + sin 3x = sin 2x + sin4x

3. Method of introducing an additional angle.

4. Method of using universal substitution.

Equations of the form F (sinx, cosx, tgx) = 0 are reduced to algebraic using the universal trigonometric substitution

By expressing sine, cosine and tangent in terms of the tangent of the half angle. This trick can lead to a higher order equation. The solution to which is difficult.

Many math 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. The principle of successful solution of each of the mentioned problems is as follows: it is necessary to establish what type of problem to be solved, to remember the necessary sequence of actions that will lead to the desired result, i.e. answer, and follow these steps.

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

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

By outward appearance the equation is sometimes difficult to determine its type. And without knowing the type of equation, it is almost impossible to choose the desired one from several tens of trigonometric formulas.

To solve the trigonometric equation, one should try:

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

Consider basic methods for solving trigonometric equations.

I. Reduction to the simplest trigonometric equations

Solution scheme

Step 1. Express trigonometric function through known components.

Step 2. Find the argument of a function by the formulas:

cos x = a; x = ± arccos a + 2πn, n ЄZ.

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

tg x = a; x = arctan a + πn, n Є Z.

ctg x = a; x = arcctg a + πn, n Є Z.

Step 3. Find 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 substitution

Solution scheme

Step 1. Bring the equation to an 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 scheme

Step 1. Replace the given equation with a linear one, using the degree reduction formulas for this:

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 scheme

Step 1. Bring this equation to the form

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

or to mind

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 tg x:

a) a tg x + b = 0;

b) a tg 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, so

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 for transforming an equation using trigonometric formulas

Solution scheme

Step 1. Using all kinds of trigonometric formulas, bring this equation to the equation solved by methods I, II, III, IV.

Step 2. Solve the resulting equation by 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 to solve trigonometric equations is very important, their development requires significant efforts, both on the part of the student and on the part of the teacher.

Many problems of stereometry, physics, etc. are connected with the solution of trigonometric equations. The process of solving such problems, as it were, contains many knowledge and skills that are acquired when studying the elements of trigonometry.

Trigonometric equations occupy an important place in the process of teaching mathematics and the development of personality in general.

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The concept of solving trigonometric equations.

• To solve a trigonometric equation, convert it to one or more basic trigonometric equations. Solving a trigonometric equation ultimately comes down to solving four basic trigonometric equations.

Solving basic trigonometric equations.

• There are 4 types of basic trigonometric equations:
• sin x = a; cos x = a
• tg x = a; ctg x = a
• Solving basic trigonometric equations involves looking at the various x positions on the unit circle and using a conversion table (or calculator).
• Example 1.sin x = 0.866. Using a conversion table (or calculator), you get the answer: x = π / 3. The unit circle gives another answer: 2π / 3. Remember: all trigonometric functions are periodic, that is, their values ​​are repeated. For example, the periodicity of sin x and cos x is 2πn, and the periodicity of tg x and ctg x is πn. Therefore, the answer is written as follows:
• x1 = π / 3 + 2πn; x2 = 2π / 3 + 2πn.
• Example 2.cos x = -1/2. Using a conversion table (or calculator), you get the answer: x = 2π / 3. The unit circle gives another answer: -2π / 3.
• x1 = 2π / 3 + 2π; x2 = -2π / 3 + 2π.
• Example 3.tg (x - π / 4) = 0.
• Answer: x = π / 4 + πn.
• Example 4. ctg 2x = 1.732.
• Answer: x = π / 12 + πn.

Transformations used to solve trigonometric equations.

• To transform trigonometric equations, algebraic transformations (factorization, reduction of homogeneous terms, etc.) and trigonometric identities are used.
• Example 5. Using trigonometric identities, the equation sin x + sin 2x + sin 3x = 0 is transformed into the equation 4cos x * sin (3x / 2) * cos (x / 2) = 0. Thus, you need to solve the following basic trigonometric equations: cos x = 0; sin (3x / 2) = 0; cos (x / 2) = 0.

• Finding angles from known values ​​of functions.

  • Before learning methods for solving trigonometric equations, you need to learn how to find angles from known values ​​of functions. This can be done using a conversion table or calculator.
  • Example: cos x = 0.732. The calculator will give the answer x = 42.95 degrees. The unit circle will give additional angles, the cosine of which is also 0.732.

• Set the solution aside on the unit circle.

  • You can defer the solutions to the trigonometric equation on the unit circle. The solutions of the trigonometric equation on the unit circle are the vertices of a regular polygon.
  • Example: The solutions x = π / 3 + πn / 2 on the unit circle are the vertices of a square.
  • Example: The solutions x = π / 4 + πn / 3 on the unit circle represent the vertices of a regular hexagon.

• Methods for solving trigonometric equations.

  • If a given trig equation contains only one trig function, solve that equation as the basic trig equation. If a given equation includes two or more trigonometric functions, then there are 2 methods for solving such an equation (depending on the possibility of its transformation).
    • Method 1.
  • Convert this equation to an equation of the form: f (x) * g (x) * h (x) = 0, where f (x), g (x), h (x) are the basic trigonometric equations.
  • Example 6.2cos x + sin 2x = 0. (0< x < 2π)
  • Solution. Using the double angle formula sin 2x = 2 * sin x * cos x, replace sin 2x.
  • 2cos x + 2 * sin x * cos x = 2cos x * (sin x + 1) = 0. Now solve the two basic trigonometric equations: cos x = 0 and (sin x + 1) = 0.
  • Example 7.cos x + cos 2x + cos 3x = 0. (0< x < 2π)
  • Solution: Using trigonometric identities, transform this equation into an equation of the form: cos 2x (2cos x + 1) = 0. Now solve the two basic trigonometric equations: cos 2x = 0 and (2cos x + 1) = 0.
  • Example 8.sin x - sin 3x = cos 2x. (0< x < 2π)
  • Solution: Using trigonometric identities, transform this equation into an equation of the form: -cos 2x * (2sin x + 1) = 0. Now solve the two basic trigonometric equations: cos 2x = 0 and (2sin x + 1) = 0.
    • Method 2.
  • Convert the given trigonometric equation to an equation containing only one trigonometric function. Then replace this trigonometric function with some unknown, for example, t (sin x = t; cos x = t; cos 2x = t, tg x = t; tg (x / 2) = t, etc.).
  • Example 9.3sin ^ 2 x - 2cos ^ 2 x = 4sin x + 7 (0< x < 2π).
  • Solution. In this equation, replace (cos ^ 2 x) with (1 - sin ^ 2 x) (by identity). The transformed equation is:
  • 3sin ^ 2 x - 2 + 2sin ^ 2 x - 4sin x - 7 = 0. Replace sin x with t. The equation now looks like this: 5t ^ 2 - 4t - 9 = 0. This is a quadratic equation with two roots: t1 = -1 and t2 = 9/5. The second root t2 does not satisfy the range of values ​​of the function (-1< sin x < 1). Теперь решите: t = sin х = -1; х = 3π/2.
  • Example 10.tg x + 2 tg ^ 2 x = ctg x + 2
  • Solution. Replace tg x with t. Rewrite the original equation as follows: (2t + 1) (t ^ 2 - 1) = 0. Now find t and then find x for t = tg x.