Formulas for the sum and difference of sines and cosines for two angles α and β allow us to move from the sum of these angles to the product of angles α + β 2 and α - β 2. Let us immediately note that you should not confuse the formulas for the sum and difference of sines and cosines with the formulas for sines and cosines of the sum and difference. Below we list these formulas, give their derivations and show examples of application for specific problems.
Formulas for the sum and difference of sines and cosines
Let's write down what the sum and difference formulas look like for sines and cosines
Sum and difference formulas for sines
sin α + sin β = 2 sin α + β 2 cos α - β 2 sin α - sin β = 2 sin α - β 2 cos α + β 2
Sum and difference formulas for cosines
cos α + cos β = 2 cos α + β 2 cos α - β 2 cos α - cos β = - 2 sin α + β 2 cos α - β 2 , cos α - cos β = 2 sin α + β 2 · β - α 2
These formulas are valid for any angles α and β. The angles α + β 2 and α - β 2 are called the half-sum and half-difference of the angles alpha and beta, respectively. Let us give the formulation for each formula.
Definitions of formulas for sums and differences of sines and cosines
Sum of sines of two angles is equal to twice the product of the sine of the half-sum of these angles and the cosine of the half-difference.
Difference of sines of two angles is equal to twice the product of the sine of the half-difference of these angles and the cosine of the half-sum.
Sum of cosines of two angles is equal to twice the product of the cosine of the half-sum and the cosine of the half-difference of these angles.
Difference of cosines of two angles equal to twice the product of the sine of the half-sum and the cosine of the half-difference of these angles, taken with a negative sign.
Deriving formulas for the sum and difference of sines and cosines
To derive formulas for the sum and difference of the sine and cosine of two angles, addition formulas are used. Let's list them below
sin (α + β) = sin α · cos β + cos α · sin β sin (α - β) = sin α · cos β - cos α · sin β cos (α + β) = cos α · cos β - sin α sin β cos (α - β) = cos α cos β + sin α sin β
Let’s also imagine the angles themselves as a sum of half-sums and half-differences.
α = α + β 2 + α - β 2 = α 2 + β 2 + α 2 - β 2 β = α + β 2 - α - β 2 = α 2 + β 2 - α 2 + β 2
We proceed directly to the derivation of the sum and difference formulas for sin and cos.
Derivation of the formula for the sum of sines
In the sum sin α + sin β, we replace α and β with the expressions for these angles given above. We get
sin α + sin β = sin α + β 2 + α - β 2 + sin α + β 2 - α - β 2
Now we apply the addition formula to the first expression, and to the second - the formula for the sine of angle differences (see formulas above)
sin α + β 2 + α - β 2 = sin α + β 2 cos α - β 2 + cos α + β 2 sin α - β 2 sin α + β 2 - α - β 2 = sin α + β 2 cos α - β 2 - cos α + β 2 sin α - β 2 sin α + β 2 + α - β 2 + sin α + β 2 - α - β 2 = sin α + β 2 cos α - β 2 + cos α + β 2 sin α - β 2 + sin α + β 2 cos α - β 2 - cos α + β 2 sin α - β 2 Open the brackets, add similar terms and get the required formula
sin α + β 2 cos α - β 2 + cos α + β 2 sin α - β 2 + sin α + β 2 cos α - β 2 - cos α + β 2 sin α - β 2 = = 2 sin α + β 2 cos α - β 2
The steps to derive the remaining formulas are similar.
Derivation of the formula for the difference of sines
sin α - sin β = sin α + β 2 + α - β 2 - sin α + β 2 - α - β 2 sin α + β 2 + α - β 2 - sin α + β 2 - α - β 2 = sin α + β 2 cos α - β 2 + cos α + β 2 sin α - β 2 - sin α + β 2 cos α - β 2 - cos α + β 2 sin α - β 2 = = 2 sin α - β 2 cos α + β 2
Derivation of the formula for the sum of cosines
cos α + cos β = cos α + β 2 + α - β 2 + cos α + β 2 - α - β 2 cos α + β 2 + α - β 2 + cos α + β 2 - α - β 2 = cos α + β 2 cos α - β 2 - sin α + β 2 sin α - β 2 + cos α + β 2 cos α - β 2 + sin α + β 2 sin α - β 2 = = 2 cos α + β 2 cos α - β 2
Derivation of the formula for the difference of cosines
cos α - cos β = cos α + β 2 + α - β 2 - cos α + β 2 - α - β 2 cos α + β 2 + α - β 2 - cos α + β 2 - α - β 2 = cos α + β 2 cos α - β 2 - sin α + β 2 sin α - β 2 - cos α + β 2 cos α - β 2 + sin α + β 2 sin α - β 2 = = - 2 sin α + β 2 sin α - β 2
Examples of solving practical problems
First, let's check one of the formulas by substituting specific angle values into it. Let α = π 2, β = π 6. Let us calculate the value of the sum of the sines of these angles. First, we will use the table of basic values of trigonometric functions, and then we will apply the formula for the sum of sines.
Example 1. Checking the formula for the sum of sines of two angles
α = π 2, β = π 6 sin π 2 + sin π 6 = 1 + 1 2 = 3 2 sin π 2 + sin π 6 = 2 sin π 2 + π 6 2 cos π 2 - π 6 2 = 2 sin π 3 cos π 6 = 2 3 2 3 2 = 3 2
Let us now consider the case when the angle values differ from the basic values presented in the table. Let α = 165°, β = 75°. Let's calculate the difference between the sines of these angles.
Example 2. Application of the difference of sines formula
α = 165 °, β = 75 ° sin α - sin β = sin 165 ° - sin 75 ° sin 165 - sin 75 = 2 sin 165 ° - 75 ° 2 cos 165 ° + 75 ° 2 = = 2 sin 45 ° cos 120 ° = 2 2 2 - 1 2 = 2 2
Using the formulas for the sum and difference of sines and cosines, you can move from the sum or difference to the product of trigonometric functions. Often these formulas are called formulas for moving from a sum to a product. The formulas for the sum and difference of sines and cosines are widely used in solving trigonometric equations and in converting trigonometric expressions.
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Trigonometric formulas have a number of properties, one of which is the use of formulas for reducing the degree. They help simplify expressions by reducing the degree.
Definition 1
Reduction formulas work on the principle of expressing the degree of sine and cosine through the sine and cosine of the first degree, but a multiple of the angle. When simplified, the formula becomes convenient for calculations, and the multiplicity of the angle increases from α to n α.
Formulas for reducing degrees, their proof
Below is a table of formulas for reducing degrees from 2 to 4 for sin and cos angles. After reading them we will ask general formula for all degrees.
sin 2 α = 1 - cos 2 α 2 cos 2 α = 1 + cos 2 α 2 sin 3 = 3 sin α - sin 3 α 4 sin 4 = 3 - 4 cos 2 α + cos 4 α 8 cos 4 α = 3 + 4 cos 2 α + cos 4 α 8
These formulas are intended to reduce the degree.
There are formulas for the double angle of cosine and sine, from which the formulas for reducing the degree cos 2 α = 1 - 2 · sin 2 α and cos 2 α = 2 · cos 2 α - 1 follow. Equalities are resolved with respect to the square of sine and cosine, which are given as sin 2 α = 1 - cos 2 α 2 and cos 2 α = 1 + cos 2 α 2 .
Formulas for reducing powers of trigonometric functions have something in common with the formulas for sine and cosine half angle.
The triple angle formula sin 3 α = 3 · sin α - 4 · sin 3 α and cos 3 α = - 3 · cos α + 4 · cos 3 α takes place.
If we solve the equality with respect to sine and cosine cubed, we obtain formulas for reducing powers for sine and cosine:
sin 3 α = 3 - 4 cos 2 α + cos 4 α 8 and cos 3 α = 3 cos α + cos 3 α 4.
The formulas for the fourth degree of trigonometric functions look like this: sin 4 α = 3 - 4 · cos 2 α + cos 4 α 8 and cos 4 α = 3 + 4 · cos 2 α + cos 4 α 8.
To lower the degrees of these expressions, you can act in 2 stages, that is, lower them twice, then it looks like this:
sin 4 α = (sin 2 α) 2 = (1 - cos 2 α 2) 2 = 1 - 2 cos 2 α + cos 2 2 α 4 = = 1 - 2 cos 2 α + 1 + cos 4 α 2 4 = 3 - 4 · cos 2 α + cos 4 α 8 ; cos 4 α = (cos 2 α) 2 = (1 + cos 2 α 2) 2 = 1 + 2 cos 2 α + cos 2 2 α 4 = = = 1 + 2 cos 2 α + 1 + cos 4 α 2 4 = 3 + 4 cos 2 α + cos 4 α 8
To solve some problems, a table of trigonometric identities will be useful, which will make it much easier to transform functions:The simplest trigonometric identities
The quotient of dividing the sine of an angle alpha by the cosine of the same angle is equal to the tangent of this angle (Formula 1). See also the proof of the correctness of the transformation of the simplest trigonometric identities.
The quotient of dividing the cosine of an angle alpha by the sine of the same angle is equal to the cotangent of the same angle (Formula 2)
The secant of an angle is equal to one divided by the cosine of the same angle (Formula 3)
The sum of the squares of the sine and cosine of the same angle is equal to one (Formula 4). see also the proof of the sum of the squares of cosine and sine.
The sum of one and the tangent of an angle is equal to the ratio of one to the square of the cosine of this angle (Formula 5)
One plus the cotangent of an angle is equal to the quotient of one divided by the sine square of this angle (Formula 6)
The product of tangent and cotangent of the same angle is equal to one (Formula 7).
Converting negative angles of trigonometric functions (even and odd)
To get rid of the negative value degree measure angle when calculating sine, cosine or tangent, you can use the following trigonometric transformations (identities) based on the principles of even or odd trigonometric functions.
As you can see, cosine and the secant is even function
, sine, tangent and cotangent are odd functions.
The sine of a negative angle is equal to negative value sine of the same positive angle (minus sine alpha).
The cosine minus alpha will give the same value as the cosine of the alpha angle.
Tangent minus alpha is equal to minus tangent alpha.
Formulas for reducing double angles (sine, cosine, tangent and cotangent of double angles)
If you need to divide an angle in half, or vice versa, move from a double angle to a single angle, you can use the following trigonometric identities:
Double Angle Conversion (sine of a double angle, cosine of a double angle and tangent of a double angle) in single occurs according to the following rules:
Sine of double angle equal to twice the product of the sine and the cosine of a single angle
Cosine of double angle equal to the difference between the square of the cosine of a single angle and the square of the sine of this angle
Cosine of double angle equal to twice the square of the cosine of a single angle minus one
Cosine of double angle equal to one minus double sine squared single angle
Tangent of double angle is equal to a fraction whose numerator is twice the tangent of a single angle, and the denominator is equal to one minus the tangent squared of a single angle.
Cotangent of double angle is equal to a fraction whose numerator is the square of the cotangent of a single angle minus one, and the denominator is equal to twice the cotangent of a single angle
Formulas for universal trigonometric substitution
The conversion formulas below can be useful when you need to divide the argument of a trigonometric function (sin α, cos α, tan α) by two and reduce the expression to the value of half an angle. From the value of α we obtain α/2.These formulas are called formulas of universal trigonometric substitution. Their value lies in the fact that a trigonometric expression with their help is reduced to expressing the tangent of half an angle, regardless of what trigonometric functions (sincos tg ctg) were in the expression initially. After this, the equation with the tangent of half an angle is much easier to solve.
Trigonometric identities for half-angle transformations
The following are the formulas for trigonometric conversion of half an angle to its whole value.The value of the argument of the trigonometric function α/2 is reduced to the value of the argument of the trigonometric function α.
Trigonometric formulas for adding angles
cos (α - β) = cos α cos β + sin α sin β
sin (α + β) = sin α cos β + sin β cos α
sin (α - β) = sin α cos β - sin β cos α
cos (α + β) = cos α cos β - sin α sin β
Tangent and cotangent of the sum of angles alpha and beta can be converted using the following rules for converting trigonometric functions:
Tangent of the sum of angles is equal to a fraction whose numerator is the sum of the tangent of the first and tangent of the second angle, and the denominator is one minus the product of the tangent of the first angle and the tangent of the second angle.
Tangent of angle difference is equal to a fraction whose numerator is equal to the difference between the tangent of the angle being reduced and the tangent of the angle being subtracted, and the denominator is one plus the product of the tangents of these angles.
Cotangent of the sum of angles is equal to a fraction whose numerator is equal to the product of the cotangents of these angles plus one, and the denominator is equal to the difference between the cotangent of the second angle and the cotangent of the first angle.
Cotangent of angle difference is equal to a fraction whose numerator is the product of the cotangents of these angles minus one, and the denominator is equal to the sum of the cotangents of these angles.
These trigonometric identities are convenient to use when you need to calculate, for example, the tangent of 105 degrees (tg 105). If you imagine it as tg (45 + 60), then you can use the given identical transformations of the tangent of the sum of angles, and then simply substitute the tabulated values of tangent 45 and tangent 60 degrees.
Formulas for converting the sum or difference of trigonometric functions
Expressions representing a sum of the form sin α + sin β can be transformed using the following formulas:
Triple angle formulas - converting sin3α cos3α tan3α to sinα cosα tanα
Sometimes it is necessary to transform the triple value of an angle so that the argument of the trigonometric function becomes the angle α instead of 3α.In this case, you can use the triple angle transformation formulas (identities):
Formulas for converting products of trigonometric functions
If there is a need to transform the product of sines of different angles, cosines of different angles, or even the product of sine and cosine, then you can use the following trigonometric identities:
In this case, the product of the sine, cosine or tangent functions of different angles will be converted into a sum or difference.
Formulas for reducing trigonometric functions
You need to use the reduction table as follows. In the line we select the function that interests us. In the column there is an angle. For example, the sine of the angle (α+90) at the intersection of the first row and the first column, we find out that sin (α+90) = cos α.
The relationships between the basic trigonometric functions - sine, cosine, tangent and cotangent - are specified trigonometric formulas. And since there are quite a lot of connections between trigonometric functions, this explains the abundance of trigonometric formulas. Some formulas connect trigonometric functions of the same angle, others - functions of a multiple angle, others - allow you to reduce the degree, fourth - express all functions through the tangent of a half angle, etc.
In this article we will list in order all the main trigonometric formulas, which are sufficient to solve the vast majority of trigonometry problems. For ease of memorization and use, we will group them by purpose and enter them into tables.
Page navigation.
Basic trigonometric identities
Basic trigonometric identities define the relationship between sine, cosine, tangent and cotangent of one angle. They follow from the definition of sine, cosine, tangent and cotangent, as well as the concept of the unit circle. They allow you to express one trigonometric function in terms of any other.
For a detailed description of these trigonometry formulas, their derivation and examples of application, see the article.
Reduction formulas
Reduction formulas follow from the properties of sine, cosine, tangent and cotangent, that is, they reflect the property of periodicity of trigonometric functions, the property of symmetry, as well as the property of shift by a given angle. These trigonometric formulas allow you to move from working with arbitrary angles to working with angles ranging from zero to 90 degrees.
The rationale for these formulas, a mnemonic rule for memorizing them and examples of their application can be studied in the article.
Addition formulas
Trigonometric addition formulas show how trigonometric functions of the sum or difference of two angles are expressed in terms of trigonometric functions of those angles. These formulas serve as the basis for deriving the following trigonometric formulas.
Formulas for double, triple, etc. angle
Formulas for double, triple, etc. angle (they are also called multiple angle formulas) show how trigonometric functions of double, triple, etc. angles () are expressed in terms of trigonometric functions of a single angle. Their derivation is based on addition formulas.
More detailed information is collected in the article formulas for double, triple, etc. angle
Half angle formulas
Half angle formulas show how trigonometric functions of a half angle are expressed in terms of the cosine of a whole angle. These trigonometric formulas follow from the double angle formulas.
Their conclusion and examples of application can be found in the article.
Degree reduction formulas
Trigonometric formulas for reducing degrees are designed to facilitate the transition from natural powers of trigonometric functions to sines and cosines in the first degree, but multiple angles. In other words, they allow you to reduce the powers of trigonometric functions to the first.
Formulas for the sum and difference of trigonometric functions
Main purpose formulas for the sum and difference of trigonometric functions is to go to the product of functions, which is very useful when simplifying trigonometric expressions. These formulas are also widely used in solving trigonometric equations, as they allow you to factor the sum and difference of sines and cosines.
Formulas for the product of sines, cosines and sine by cosine
The transition from the product of trigonometric functions to a sum or difference is carried out using the formulas for the product of sines, cosines and sine by cosine.
Universal trigonometric substitution
We complete our review of the basic formulas of trigonometry with formulas expressing trigonometric functions in terms of the tangent of a half angle. This replacement was called universal trigonometric substitution. Its convenience lies in the fact that all trigonometric functions are expressed rationally in terms of the tangent of a half angle without roots.
References.
- Algebra: Textbook for 9th grade. avg. school/Yu. N. Makarychev, N. G. Mindyuk, K. I. Neshkov, S. B. Suvorova; Ed. S. A. Telyakovsky. - M.: Education, 1990. - 272 pp.: ill. - ISBN 5-09-002727-7
- Bashmakov M. I. Algebra and the beginnings of analysis: Textbook. for 10-11 grades. avg. school - 3rd ed. - M.: Education, 1993. - 351 p.: ill. - ISBN 5-09-004617-4.
- Algebra and the beginning of analysis: Proc. for 10-11 grades. general education institutions / A. N. Kolmogorov, A. M. Abramov, Yu. P. Dudnitsyn and others; Ed. A. N. Kolmogorov. - 14th ed. - M.: Education, 2004. - 384 pp.: ill. - ISBN 5-09-013651-3.
- Gusev V. A., Mordkovich A. G. Mathematics (a manual for those entering technical schools): Proc. allowance.- M.; Higher school, 1984.-351 p., ill.
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