IN identical transformations trigonometric expressions The following algebraic techniques can be used: adding and subtracting the same terms; making a common factor for brackets; Multiplication and division on the same value; application of formulas of abbreviated multiplication; allocation of a full square; decomposition square three-shoe for multipliers; Introduction of new variables in order to simplify transformations.
In transformations of trigonometric expressions containing fractions, it is possible to use the properties of the proportion, a reduction of fractions or bringing fractions to a common denominator. In addition, you can use the selection of a piece of fraction, multiplying the numerator and denominator of the fraction on same value, as well as if possible, take into account the uniformity of the numerator or denominator. If necessary, you can represent a fraction in the form of a sum or a difference of several simpler fractions.
In addition, applying all the necessary methods for transforming trigonometric expressions, it is necessary to constantly take into account the obstacle values \u200b\u200bof the transformed expressions.
Consider several examples.
Example 1.
Calculate a \u003d (sin (2x - π) · cos (3π - x) + sin (2x - 9π / 2) · COS (X + π / 2)) 2 + (COS (X - π / 2) · COS ( 2x - 7π / 2) +
+
sin (3π / 2 - x) · sin (2x -5π / 2)) 2
Decision.
From the formula of bringing:
sin (2x - π) \u003d -sin 2x; COS (3π - X) \u003d -COS X;
sIN (2X - 9π / 2) \u003d -COS 2X; cos (x + π / 2) \u003d -sin x;
cos (x - π / 2) \u003d sin x; COS (2x - 7π / 2) \u003d -Sin 2x;
sIN (3π / 2 - X) \u003d -COS X; SIN (2X - 5π / 2) \u003d -COS 2X.
Where is the formula for the addition of arguments and the main trigonometric identity, we get
A \u003d (sin 2x · cos x + cos 2x · sin x) 2 + (-sin x · sin 2x + cos x · cos 2x) 2 \u003d sin 2 (2x + x) + cos 2 (x + 2x) \u003d
\u003d SIN 2 3X + COS 2 3X \u003d 1
Answer: 1.
Example 2.
Convert the expression M \u003d COS α + COS (α + β) · COS γ + COS β - sin (α + β) · sin γ + cos γ.
Decision.
From the formulas of the addition of arguments and the formulas of the transformation of the amount trigonometric functions in the work after the corresponding group we have
M \u003d (COS (α + β) · cos γ - sin (α + β) · sin γ) + cos α + (cos β + cos γ) \u003d
2cos ((β + γ) / 2) · COS ((β - γ) / 2) + (cos α + cos (α + β + γ)) \u003d
2cos ((β + γ) / 2) · COS ((β - γ) / 2) + 2cos (α + (β + γ) / 2) · cos ((β + γ) / 2)) \u003d
2COS ((β + γ) / 2) (COS ((β - γ) / 2) + COS (α + (β + γ) / 2)) \u003d
2cos ((β + γ) / 2) · 2cos ((β - γ) / 2 + α + (β + γ) / 2) / 2) · cos ((β - γ) / 2) - (α + ( β + γ) / 2) / 2) \u003d
4COS ((β + γ) / 2) · COS ((α + β) / 2) · COS ((α + γ) / 2).
Answer: M \u003d 4COS ((α + β) / 2) · COS ((α + γ) / 2) · COS ((β + γ) / 2).
Example 3..
Show that expression A \u003d COS 2 (X + π / 6) - COS (X + π / 6) · COS (X - π / 6) + COS 2 (X - π / 6) takes for all x from R one And the same meaning. Find this value.
Decision.
We give two ways to solve this problem. Applying the first method by selecting a full square and using the appropriate main trigonometric formulas, we obtain
A \u003d (COS (X + π / 6) - COS (X - π / 6)) 2 + COS (X - π / 6) · COS (X - π / 6) \u003d
4Sin 2 x · sin 2 π / 6 + 1/2 (COS 2X + COS π / 3) \u003d
SIN 2 X + 1/2 · COS 2X + 1/4 \u003d 1/2 · (1 - COS 2X) + 1/2 · COS 2X + 1/4 \u003d 3/4.
By solving the problem in the second way, we consider as a function from x from R and calculate its derivative. After transformation, we get
A'\u003d -2cos (x + π / 6) · sin (x + π / 6) + (sin (x + π / 6) · cos (x - π / 6) + cos (x + π / 6) · sin (x + π / 6)) - 2cos (x - π / 6) · sin (x - π / 6) \u003d
SIN 2 (X + π / 6) + SIN ((X + π / 6) + (X - π / 6)) - SIN 2 (X - π / 6) \u003d
SIN 2X - (SIN (2X + π / 3) + SIN (2X - π / 3)) \u003d
SIN 2X - 2SIN 2X · COS π / 3 \u003d SIN 2X - SIN 2X ≡ 0.
From here, due to the criterion for the constancy of the function differentiating on the interval, we conclude that
A (x) ≡ (0) \u003d COS 2 π / 6 - COS 2 π / 6 + COS 2 π / 6 \u003d (√3 / 2) 2 \u003d 3/4, x € R. 
Answer: A \u003d 3/4 for X € R.
The main techniques of evidence of trigonometric identities are:
but) the reduction of the left part of the identity to the right way to relevant transformations;
b) reducing the right side of the identity to the left;
in) The reduction of the right and left parts of the identity to the same form;
d) Minding to zero of the difference between the left and right parts of the proving identity.
Example 4.
Check that COS 3X \u003d -4COS X · COS (X + π / 3) · COS (X + 2π / 3).
Decision.
Converting the right side of this identity according to the appropriate trigonometric formulas, we have
4COS X · COS (X + π / 3) · COS (x + 2π / 3) \u003d
2COS X · (COS ((X + π / 3) + (X + 2π / 3)) + COS ((X + π / 3) - (x + 2π / 3))) \u003d
2COS X · (COS (2X + π) + COS π / 3) \u003d
2COS X · COS 2X - COS X \u003d (COS 3X + COS X) - COS X \u003d COS 3X.
The right side of the identity is reduced to the left.
Example 5.
Prove that SIN 2 α + SIN 2 β + SIN 2 γ - 2COS α · cos β · cos γ \u003d 2, if α, β, γ is the internal angles of some triangle.
Decision.
Considering that α, β, γ - the internal angles of some triangle, we get that
α + β + γ \u003d π and, therefore, γ \u003d π - α - β.
sIN 2 α + SIN 2 β + SIN 2 γ - 2COS α · COS β · COS γ \u003d
SIN 2 α + SIN 2 β + SIN 2 (π - α - β) - 2cos α · cos β · cos (π - α - β) \u003d
SIN 2 α + SIN 2 β + SIN 2 (α + β) + (COS (α + β) + COS (α - β) · (COS (α + β) \u003d
SIN 2 α + SIN 2 β + (SIN 2 (α + β) + COS 2 (α + β)) + COS (α - β) · (COS (α + β) \u003d
1/2 · (1 - Cos 2α) + ½ · (1 - COS 2β) + 1 + 1/2 · (COS 2α + COS 2β) \u003d 2.
The initial equality is proved.
Example 6.
To prove that in order for one of the angles α, β, the triangle of the triangle is 60 °, it is necessary and enough to sin 3α + sin 3β + sin 3γ \u003d 0.
Decision.
The condition of this task implies the proof of both the need and sufficiency.
First prove necessity.
You can show that
sIN 3α + SIN 3β + SIN 3γ \u003d -4COS (3α / 2) · COS (3β / 2) · COS (3γ / 2).
From here, considering that COS (3/2 · 60 °) \u003d COS 90 ° \u003d 0, we obtain that if one of the angles α, β or γ is 60 °, then
cOS (3α / 2) · COS (3β / 2) · COS (3γ / 2) \u003d 0 and, therefore, SIN 3α + SIN 3β + SIN 3γ \u003d 0.
We prove now adequacy specified condition.
If sin 3α + sin 3β + sin 3γ \u003d 0, then cos (3α / 2) · cos (3β / 2) · cos (3γ / 2) \u003d 0, and therefore
either COS (3α / 2) \u003d 0, or COS (3β / 2) \u003d 0, or COS (3γ / 2) \u003d 0.
Hence,
either 3α / 2 \u003d π / 2 + πk, i.e. α \u003d π / 3 + 2πk / 3, 
either 3β / 2 \u003d π / 2 + πk, i.e. β \u003d π / 3 + 2πk / 3,
either 3γ / 2 \u003d π / 2 + πk,
those. γ \u003d π / 3 + 2πk / 3, where k ε z.
From the fact that α, β, γ are the corners of the triangle, we have
0 < α < π, 0 < β < π, 0 < γ < π.
Therefore, for α \u003d π / 3 + 2πk / 3 or β \u003d π / 3 + 2πk / 3 or
γ \u003d π / 3 + 2πk / 3 of all kεz is suitable only k \u003d 0.
From where it follows that either α \u003d π / 3 \u003d 60 °, or β \u003d π / 3 \u003d 60 °, or γ \u003d π / 3 \u003d 60 °.
The statement is proved.
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To solve some tasks, there will be a useful table of trigonometric identities, which will make it much easier to perform conversion functions:Simplest trigonometric identities
The private from the division of the sinus an angle of alpha on the cosine of the same angle is equal to the tangent of this angle (Formula 1). See also proof of the correctness of the transformation of the simplest trigonometric identities.
The private from the division of the cosine of the angle of alpha on the sinus of the same angle is equal to Kotannce of the same angle (Formula 2)
The angle session is equal to a unit divided by the cosine of the same corner (Formula 3)
The sum of the squares of sine and cosine of the same angle is equal to one (formula 4). See also proof of the sum of cosine and sine squares.
The sum of the unit and tangent of the angle is equal to the attitude of the unit to the square of the cosine of this angle (Formula 5)
Unit plus a Cotangent angle is equal to the private from dividing the unit on the sinus square of this angle (Formula 6)
The work of Tangent on the catangent of the same angle is equal to one (formula 7).
Transformation of negative angles of trigonometric functions (parity and oddness)
In order to get rid of negative value degree measures The angle when calculating sinus, cosine or tangent, you can use the following trigonometric transformations (identities) based on the principles of parity or the oddness of trigonometric functions.
As seen, cosine And the session is even function, sinus, Tangent and Kotangent - odd functions.
The sinus of the negative angle is equal to the negative value of the sinus of the same positive angle (minus sinus alpha).
Cosine "Minus Alpha" will give the same value as the causin of the angle of alpha.
Tangent minus Alpha is equal to minus Tangent Alpha.
Double angle reduction formulas (sinus, cosine, tangent and catangent double angle)
If it is necessary to divide the angle in half, or vice versa, move from the double angle to the single, you can use the following trigonometric identities:
Transformation of double corner (double angle sinus, dual corner cosine and double corner tangent) in single occurs the following rules:
Double corner sinus equal to double product of sinus on a single corner cosine
Kosinus double corner equal to the difference in the square of the cosine of the single angle and the square of the sinus of this angle
Kosinus double corner equal to double square cosine single angle minus one
Kosinus double corner equal to one minus double sine square single angle
Tangent Double Angle It is equal to the fraction, the numerator of which is a double tangent of a single angle, and the denominator is equal to one minus tangent square of a single angle.
Cotanence Double Angle It is equal to the fraction, the numerator of which is the square of the single angle of the coal coal minus, and the denominator is equal to the double corner catangen.
Universal trigonometric substitution formulas
The following formulas for the conversion can be useful when the argument of the trigonometric function (SIN α, COS α, TG α) is divided into two and lead the expression to the value of half angle. From the value α we obtain α / 2.These formulas are called Universal trigonometric substitution formulas. Their value is that the trigonometric expression with their help is reduced to the expression of half angle tangent, regardless of which trigonometric functions ( sin Cos. TG CTG) were in terms of expression initially. After that, the equation with the tangent of half an angle to solve is much easier.
Trigonometric identities of half angle conversion
The following formulas trigonometric transformation Half the corner to its whole value.The value of the argument of the trigonometric function α / 2 is given to the value of the argument of the trigonometric function α.
Trigonometric Corner Addition Formulas
cOS (α - β) \u003d cos α · cos β + sin α · sin β
sin (α + β) \u003d sin α · cos β + sin β · cos α
sin (α - β) \u003d sin α · cos β - sin β · cos α
COS (α + β) \u003d cos α · cos β - sin α · sin β
Tangent and Cotangence Cornersalpha and beta can be converted according to the following rules for converting trigonometric functions:
Tangent Angle Amounts It is equal to the fraction, the numerator of which is the sum of the first and tangent of the second angle, and the denominator - the unit minus the product of the Tangent of the first angle on the tangent of the second angle.
Tangent Angle difference It is equal to the fraction, the numerator of which is equal to the difference in the tangent of the reduced angle and the tangent of the subtracted angle, and the denominator is a unit plus the product of the tangents of these angles.
Cotanence amounts of corners It is equal to the fraction, the numerator of which is equal to the product of the catangents of these angles plus a unit, and the denominator is equal to the difference in the cotannce of the second angle and the cotangence of the first angle.
Cotangence angles difference It is equal to the fraction, the numerator of which is the product of the catangents of these angles minus one, and the denominator is equal to the amount of catangents of these angles.
These trigonometric identities are conveniently used when it is necessary to calculate, for example, tangent 105 degrees (TG 105). If it is to imagine both TG (45 + 60), you can use the above identical transformations Tangent The sum of the corners, after which it is simply to substitute the tables of Tangent 45 and Tangent 60 degrees.
Formulas for the transformation of the amount or difference of trigonometric functions
Expressions that represent the sum of the type SIN α + sin β can be converted using the following formulas:
Triple Angle Formulas - SIN3α COS3α TG3α transformation in SINα COSα TGα
Sometimes it is necessary to convert a triple angle so that the argument of the trigonometric function instead of 3α becomes an angle α.In this case, you can use the formulas (identities) of the triple angle conversion:
Transformation formulas for trigonometric functions
If there is a need to transform the product of the sinuses of different angles of cosine of different angles or even the works of sinus on the cosine, then you can use the following trigonometric identities:
In this case, the product of the functions of sinus, cosine or tangent of different angles will be transformed in an amount or difference.
Trigonometric Formulas
You need to use the trigger table as follows. In the line, choose the function that interests us. In the column - the angle. For example, an angle sinus (α + 90) at the intersection of the first row and the first column we find out that Sin (α + 90) \u003d COS α.
Running with all values \u200b\u200bof the argument (from common area definitions).
Universal substitution formulas.
With these formulas, an easy expression that contains various trigonometric functions of one argument turns into a rational expression of one function tG. (α / 2):
Formulas transformation amounts in the work and works in the amount.
Previously, the above formulas were used to simplify the calculations. They were calculated using logarithmic tables, and later - the logarithmic ruler, since the logarithms are best suited for multiplication of numbers. That is why each initial expression was kept to the form that would be convenient for logarithming, that is, to works, eg:
2 sin. α sin. b. = cos. (α - b.) - cos. (α + b.);
2 cos. α cos. b. = cos. (α - b.) + cos. (α + b.);
2 sin. α cos. b. = sin. (α - b.) + sin. (α + b.).
where is the angle for which in particular
Formulas for the functions of Tangent and Kotangent are easily obtained from the above mentioned.
Degree reduction formulas.
|
sIN 2 α \u003d (1 - COS 2α) / 2; |
cOS 2 α \u003d (1 + COS 2α) / 2; |
|
sIN 3.α \u003d (3 sinα - SIN 3.α )/4; |
cOS 3 A \u003d (3 COSα + COS 3.α )/4. |
With the help of these formulas, trigonometric equations are easily given to equations with lower degrees. Similarly, the decrease formulas are derived for higher degrees. sin. and cos..
The expression of trigonometric functions through one of them of the same argument.
Sign before the root depend on the quarter of the arrangement of coal α .
Ratios between the main trigonometric functions - sinus, cosine, tangens and catangent - set trigonometric formulas. And since there are many connections between trigonometric functions, then the abundance of trigonometric formulas is also explained by this. Some formulas bind trigonometric functions of the same angle, others - the functions of a multiple angle, the third - allow to reduce the degree, the fourth - to express all functions through a half angle tangent, etc.
In this article, we list all major trigonometric formulas that are sufficient to solve the overwhelming majority of trigonometry problems. For ease of memorization and use, we will group them on purpose, and enter the table.
Navigating page.
Basic trigonometric identities
Basic trigonometric identities Set the relationship between sinus, cosine, tangent and catangent of one corner. They arise from the definition of sinus, cosine, tangent and catangent, as well as concepts of a single circle . They allow you to express one trigonometric function through any other.
A detailed description of these trigonometry formulas, their conclusion and examples of application see the article.
Formulas of the cast
Formulas of the cast follow out properties of sinus, cosine, tangent and catangens That is, they reflect the properties of the frequency of trigonometric functions, the symmetrical property, as well as the shift property for this angle. These trigonometric formulas allow you to work with arbitrary angles to switch to operation with angles ranging from zero to 90 degrees.
The rationale for these formulas, the mnemonic rule for their memorization and examples of their application can be explored in the article.
Formulas addition
Trigonometric formulas Additions Show, as trigonometric functions of the sum or difference of two angles, are expressed through the trigonometric functions of these angles. These formulas serve as a base for the conclusion following trigonometric formulas.
Formulas double, triple, etc. Angle
Formulas double, triple, etc. The angle (they are also called multiple corner formulas) show how trigonometric functions of double, triple, etc. The angles () are expressed through the trigonometric functions of the single angle. Their conclusion is based on formulas of addition.
More detailed information is collected in the article. formulas double, triple, etc. Angle.
Formulas of half angle
Formulas of half angle Show, as trigonometric functions of a half angle are expressed through a kosineus of a whole angle. These trigonometric formulas follow from the formulas of the double angle.
Their conclusion and examples of application can be viewed in the article.
Degree reduction formulas
Trigonometric degree reduction formulas It is called for promoting the transition from natural degrees of trigonometric functions to sinus and cosine in the first degree, but multiple corners. In other words, they allow to reduce the degrees of trigonometric functions to the first.
Formulas of the sum and difference of trigonometric functions
main destination formulas of the sum and difference of trigonometric functions It is to switch to the product of functions, which is very useful when simplifying trigonometric expressions. These formulas are also widely used in solving trigonometric equationsSince they allow us to lay out the amount and difference in sinuses and cosine.
Formulas works of sinuses, cosine and sine on cosine
Transition from the product of trigonometric functions to sum or difference is carried out by formulas for the works of sinuses, cosine and sinus on cosine.
Universal trigonometric substitution
Overview of the basic formulas of trigonometry Complete by formulas expressing trigonometric functions through a half angle tangent. Such a replacement was named universal trigonometric substitution. Its convenience is that all trigonometric functions are expressed through a half angle tangent rationally without roots.
Bibliography.
- Algebra: Studies. For 9 cl. environments Shk. /u. N. Makarychev, N. G. Mindyuk, K. I. Neshkov, S. B. Suvorov; Ed. S. A. Telikovsky. - M.: Education, 1990.- 272 C.: Il.- ISBN 5-09-002727-7
- Bashmakov M. I. Algebra and start analysis: studies. for 10-11 cl. environments shk. - 3rd ed. - M.: Enlightenment, 1993. - 351 C.: Il. - ISBN 5-09-004617-4.
- Algebra and starting analysis: studies. for 10-11 cl. general education. institutions / A. N. Kolmogorov, A. M. Abramov, Yu. P. Dudnitsyn, etc.; Ed. A. N. Kolmogorova.- 14th ed. - M.: Enlightenment, 2004.- 384 C.: Il.- ISBN 5-09-013651-3.
- Gusev V. A., Mordkovich A. G. Mathematics (benefit for applicants in technical schools): studies. benefit. - m.; Higher. Shk., 1984.-351 p., Il.
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