The formulas for the sum and difference of sines and cosines for two angles α and β allow you to go from the sum of the indicated angles to the product of the angles α + β 2 and α - β 2 . We note right away 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 derivation and show examples of application for specific problems.
Formulas for the sum and difference of sines and cosines
Let's write down how the sum and difference formulas for sines and cosines look like
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, respectively, the half-sum and half-difference of the angles alpha and beta. We give a formulation for each formula.
Definitions of sum and difference formulas for sines and cosines
The sum of the 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.
The sum of the 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 is 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.
Derivation of 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. We present 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
We also represent the angles themselves as the sum of half-sums and half-differences.
α \u003d α + β 2 + α - β 2 \u003d α 2 + β 2 + α 2 - β 2 β \u003d α + β 2 - α - β 2 \u003d 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. 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 the sine formula of the angle differences to the second one (see the 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
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 for deriving the rest of the 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 cosine difference formula
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
To begin with, we will check one of the formulas by substituting specific angle values into it. Let α = π 2 , β = π 6 . Let's calculate the value of the sum of the sines of these angles. First, we use the table of basic values of trigonometric functions, and then we apply the formula for the sum of sines.
Example 1. Checking the formula for the sum of the sines of two angles
α \u003d π 2, β \u003d π 6 sin π 2 + sin π 6 \u003d 1 + 1 2 \u003d 3 2 sin π 2 + sin π 6 \u003d 2 sin π 2 + π 6 2 cos π 2 - π 6 2 \u003d 2 sin π 3 cos π 6 \u003d 2 3 2 3 2 \u003d 3 2
Let us now consider the case when the values of the angles differ from the basic values presented in the table. Let α = 165°, β = 75°. Let us calculate the value of the difference between the sines of these angles.
Example 2. Applying the sine difference 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 go from the sum or difference to the product of trigonometric functions. Often these formulas are called formulas for the transition from sum to product. The formulas for the sum and difference of sines and cosines are widely used in solving trigonometric equations and when transforming trigonometric expressions.
If you notice a mistake in the text, please highlight it and press Ctrl+Enter
Double angle formulas are used to express sines, cosines, tangents, cotangents of an angle with a value of 2 α using trigonometric functions angle α . This article will introduce all double angle formulas with proofs. Examples of application of formulas will be considered. In the final part, the formulas for the triple, quadruple angles will be shown.
List of double angle formulas
To convert double angle formulas, remember that angles in trigonometry have the form n α notation, where n is natural number, the value of the expression is written without brackets. Thus, sin n α is considered to have the same meaning as sin (n α) . With the notation sin n α we have a similar notation (sin α) n . The usage of notation is applicable for all trigonometric functions with powers of n.
The following are the double angle formulas:
sin 2 α = 2 sin α cos α cos 2 α = cos 2 α - sin 2 α , cos 2 α = 1 - 2 sin 2 α , cos 2 α = 2 cos 2 α - 1 tg 2 α = 2 tg α 1 - tg 2 α ctg 2 α - ctg 2 α - 1 2 ctg α
Note that these sin and cos formulas are applicable with any value of the angle α. The formula for the tangent of a double angle is valid for any value of α, where t g 2 α makes sense, that is, α ≠ π 4 + π 2 · z, z is any integer. The cotangent of a double angle exists for any α , where c t g 2 α is defined on α ≠ π 2 · z .
The cosine of a double angle has a triple notation of a double angle. All of them are applicable.
Proof of double angle formulas
The proof of the formulas originates from the addition formulas. We apply the formulas for the sine of the sum:
sin (α + β) = sin α cos β + cos α sin β and the cosine of the sum cos (α + β) = cos α cos β - sin α sin β. Suppose that β = α , then we get that
sin (α + α) = sin α cos α + cos α sin α = 2 sin α cos α and cos (α + α) = cos α cos α - sin α sin α = cos 2 α - sin2α
Thus, the formulas for the sine and cosine of the double angle sin 2 α \u003d 2 sin α cos α and cos 2 α \u003d cos 2 α - sin 2 α are proved.
The remaining formulas cos 2 α \u003d 1 - 2 sin 2 α and cos 2 α \u003d 2 cos 2 α - 1 lead to the form cos 2 α \u003d cos 2 α \u003d cos 2 α - sin 2 α, when replacing 1 with the sum of squares by the basic identity sin 2 α + cos 2 α = 1 . We get that sin 2 α + cos 2 α = 1. So 1 - 2 sin 2 α \u003d sin 2 α + cos 2 α - 2 sin 2 α \u003d cos 2 α - sin 2 α and 2 cos 2 α - 1 \u003d 2 cos 2 α - (sin 2 α + cos 2 α) \u003d cos 2 α - sin 2 α.
To prove the formulas for the double angle of tangent and cotangent, we apply the equalities t g 2 α \u003d sin 2 α cos 2 α and c t g 2 α \u003d cos 2 α sin 2 α. After the transformation, we get that tan 2 α \u003d sin 2 α cos 2 α \u003d 2 sin α cos α cos 2 α - sin 2 α and ctg 2 α \u003d cos 2 α sin 2 α \u003d cos 2 α - sin 2 α 2 · sin α · cos α . Divide the expression by cos 2 α where cos 2 α ≠ 0 with any value of α when t g α is defined. Divide another expression by sin 2 α , where sin 2 α ≠ 0 with any values of α , when c t g 2 α makes sense. To prove the double angle formula for tangent and cotangent, we substitute and get:
The video course "Get an A" includes all the topics you need to successful delivery USE in mathematics for 60-65 points. Completely all tasks 1-13 of the Profile USE in mathematics. Also suitable for passing the Basic USE in mathematics. If you want to pass the exam with 90-100 points, you need to solve part 1 in 30 minutes and without mistakes!
Preparation course for the exam for grades 10-11, as well as for teachers. Everything you need to solve part 1 of the exam in mathematics (the first 12 problems) and problem 13 (trigonometry). And this is more than 70 points on the Unified State Examination, and neither a hundred-point student nor a humanist can do without them.
All the necessary theory. Quick Ways solutions, traps and secrets of the exam. All relevant tasks of part 1 from the Bank of FIPI tasks have been analyzed. The course fully complies with the requirements of the USE-2018.
The course contains 5 big topics, 2.5 hours each. Each topic is given from scratch, simply and clearly.
Hundreds of exam tasks. Text problems and probability theory. Simple and easy to remember problem solving algorithms. Geometry. Theory, reference material, analysis of all types of USE tasks. Stereometry. Cunning tricks for solving, useful cheat sheets, development of spatial imagination. Trigonometry from scratch - to task 13. Understanding instead of cramming. Visual explanation of complex concepts. Algebra. Roots, powers and logarithms, function and derivative. Base for solving complex problems of the 2nd part of the exam.
The basic formulas of trigonometry are formulas that establish relationships between basic trigonometric functions. Sine, cosine, tangent and cotangent are interconnected by many relationships. Below are the main trigonometric formulas, and for convenience, we group them according to their purpose. Using these formulas, you can solve almost any problem from the standard trigonometry course. We note right away that only the formulas themselves are given below, and not their derivation, to which separate articles will be devoted.
Basic identities of trigonometry
Trigonometric identities give a relationship between the sine, cosine, tangent, and cotangent of one angle, allowing one function to be expressed in terms of another.
Trigonometric identities
sin 2 a + cos 2 a = 1 tg α = sin α cos α , ctg α = cos α sin α tg α ctg α = 1 tg 2 α + 1 = 1 cos 2 α , ctg 2 α + 1 = 1 sin 2α
These identities follow directly from the definitions unit circle, sine (sin), cosine (cos), tangent (tg), and cotangent (ctg).
Cast formulas
Casting formulas allow you to move from working with arbitrary and arbitrarily large angles to working with angles ranging from 0 to 90 degrees.
Cast formulas
sin α + 2 π z = sin α , cos α + 2 π z = cos α tg α + 2 π z = tg α , ctg α + 2 π z = ctg α sin - α + 2 π z = - sin α , cos - α + 2 π z = cos α tg - α + 2 π z = - tg α , ctg - α + 2 π z = - ctg α sin π 2 + α + 2 π z = cos α , cos π 2 + α + 2 π z = - sin α tg π 2 + α + 2 π z = - ctg α , ctg π 2 + α + 2 π z = - tg α sin π 2 - α + 2 π z = cos α , cos π 2 - α + 2 π z = sin α tg π 2 - α + 2 π z = ctg α , ctg π 2 - α + 2 π z = tg α sin π + α + 2 π z = - sin α , cos π + α + 2 π z = - cos α tg π + α + 2 π z = tg α , ctg π + α + 2 π z = ctg α sin π - α + 2 π z = sin α , cos π - α + 2 π z = - cos α tg π - α + 2 π z = - tg α , ctg π - α + 2 π z = - ctg α sin 3 π 2 + α + 2 π z = - cos α , cos 3 π 2 + α + 2 π z = sin α tg 3 π 2 + α + 2 π z = - ctg α , ctg 3 π 2 + α + 2 π z = - tg α sin 3 π 2 - α + 2 π z = - cos α , cos 3 π 2 - α + 2 π z = - sin α tg 3 π 2 - α + 2 π z = ctg α , ctg 3 π 2 - α + 2 π z = tg α
The reduction formulas are a consequence of the periodicity of trigonometric functions.
Trigonometric addition formulas
The addition formulas in trigonometry allow you to express the trigonometric function of the sum or difference of angles in terms of the trigonometric functions of these angles.
Trigonometric addition formulas
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 tg α ± β = tg α ± tg β 1 ± tg α tg β ctg α ± β = - 1 ± ctg α ctg β ctg α ± ctg β
Based on the addition formulas, trigonometric formulas for a multiple angle are derived.
Multiple angle formulas: double, triple, etc.
Double and triple angle formulassin 2 α \u003d 2 sin α cos α cos 2 α \u003d cos 2 α - sin 2 α, cos 2 α \u003d 1 - 2 sin 2 α, cos 2 α \u003d 2 cos 2 α - 1 tg 2 α \u003d 2 tg α 1 - tg 2 α with tg 2 α \u003d with tg 2 α - 1 2 with tg α sin 3 α \u003d 3 sin α cos 2 α - sin 3 α, sin 3 α \u003d 3 sin α - 4 sin 3 α cos 3 α \u003d cos 3 α - 3 sin 2 α cos α, cos 3 α \u003d - 3 cos α + 4 cos 3 α tg 3 α \u003d 3 tg α - tg 3 α 1 - 3 tg 2 α ctg 3 α = ctg 3 α - 3 ctg α 3 ctg 2 α - 1
Half Angle Formulas
The half angle formulas in trigonometry are a consequence of the double angle formulas and express the relationship between the basic functions of the half angle and the cosine of the whole angle.
Half Angle Formulas
sin 2 α 2 = 1 - cos α 2 cos 2 α 2 = 1 + cos α 2 t g 2 α 2 = 1 - cos α 1 + cos α c t g 2 α 2 = 1 + cos α 1 - cos α
Reduction formulas
Reduction formulassin 2 α = 1 - cos 2 α 2 cos 2 α = 1 + cos 2 α 2 sin 3 α = 3 sin α - sin 3 α 4 cos 3 α = 3 cos α + cos 3 α 4 sin 4 α = 3 - 4 cos 2 α + cos 4 α 8 cos 4 α = 3 + 4 cos 2 α + cos 4 α 8
Often, in calculations, it is inconvenient to operate with cumbersome powers. Degree reduction formulas allow you to reduce the degree of a trigonometric function from arbitrarily large to the first. Here is their general view:
General form of reduction formulas
for even n
sin n α = C n 2 n 2 n + 1 2 n - 1 ∑ k = 0 n 2 - 1 (- 1) n 2 - k C kn cos ((n - 2 k) α) cos n α = C n 2 n 2 n + 1 2 n - 1 ∑ k = 0 n 2 - 1 C kn cos ((n - 2 k) α)
for odd n
sin n α = 1 2 n - 1 ∑ k = 0 n - 1 2 (- 1) n - 1 2 - k C kn sin ((n - 2 k) α) cos n α = 1 2 n - 1 ∑ k = 0 n - 1 2 C kn cos ((n - 2 k) α)
Sum and difference of trigonometric functions
The difference and sum of trigonometric functions can be represented as a product. Factoring the differences of sines and cosines is very convenient to use when solving trigonometric equations and simplifying expressions.
Sum and difference of trigonometric functions
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 cos α + cos β = 2 cos α + β 2 cos α - β 2 cos α - cos β \u003d - 2 sin α + β 2 sin α - β 2, cos α - cos β \u003d 2 sin α + β 2 sin β - α 2
Product of trigonometric functions
If the formulas for the sum and difference of functions allow you to go to their product, then the formulas for the product of trigonometric functions carry out the reverse transition - from the product to the sum. Formulas for the product of sines, cosines and sine by cosine are considered.
Formulas for the product of trigonometric functions
sin α sin β = 1 2 (cos (α - β) - cos (α + β)) cos α cos β = 1 2 (cos (α - β) + cos (α + β)) sin α cos β = 1 2 (sin (α - β) + sin (α + β))
Universal trigonometric substitution
All basic trigonometric functions - sine, cosine, tangent and cotangent - can be expressed in terms of the tangent of a half angle.
Universal trigonometric substitution
sin α = 2 tg α 2 1 + tg 2 α 2 cos α = 1 - tg 2 α 2 1 + tg 2 α 2 tg α = 2 tg α 2 1 - tg 2 α 2 ctg α = 1 - tg 2 α 2 2tgα 2
If you notice a mistake in the text, please highlight it and press Ctrl+Enter
The ratios between the main trigonometric functions - sine, cosine, tangent and cotangent - are given trigonometric formulas. And since there are quite a lot of connections between trigonometric functions, this also explains the abundance of trigonometric formulas. Some formulas connect the trigonometric functions of the same angle, others - the functions of a multiple angle, others - allow you to lower the degree, the fourth - to express all functions through the tangent of a half angle, etc.
In this article, we list in order all the basic trigonometric formulas, which are enough to solve the vast majority of trigonometry problems. For ease of memorization and use, we will group them according to their purpose, and enter them into tables.
Page navigation.
Basic trigonometric identities
Basic trigonometric identities set the relationship between the 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 through any other.
For a detailed description of these trigonometry formulas, their derivation and application examples, see the article.
Cast formulas
Cast 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, and also 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 the trigonometric functions of the sum or difference of two angles are expressed in terms of the trigonometric functions of these angles. These formulas serve as the basis for the derivation of the following trigonometric formulas.
Formulas for double, triple, etc. corner
Formulas for double, triple, etc. angle (they are also called multiple angle formulas) show how the 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 the trigonometric functions of a half angle are expressed in terms of the cosine of an integer angle. These trigonometric formulas follow from the double angle formulas.
Their conclusion and examples of application can be found in the article.
Reduction formulas
Trigonometric formulas for decreasing degrees designed to facilitate the transition from natural degrees trigonometric functions to sines and cosines to the first degree, but multiple angles. In other words, they allow one to reduce the powers of trigonometric functions to the first.
Formulas for the sum and difference of trigonometric functions
main destination sum and difference formulas for trigonometric functions consists in the transition 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 factoring 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 the sum or difference is carried out through the formulas for the product of sines, cosines and sine by cosine.
Universal trigonometric substitution
We complete the review of the basic formulas of trigonometry with formulas expressing trigonometric functions in terms of the tangent of a half angle. This replacement is called universal trigonometric substitution. Its convenience lies in the fact that all trigonometric functions are expressed in terms of the tangent of a half angle rationally without roots.
Bibliography.
- Algebra: Proc. for 9 cells. avg. school / Yu. N. Makarychev, N. G. Mindyuk, K. I. Neshkov, S. B. Suvorova; Ed. S. A. Telyakovsky.- M.: Enlightenment, 1990.- 272 p.: Ill.- ISBN 5-09-002727-7
- Bashmakov M.I. Algebra and the beginning of analysis: Proc. for 10-11 cells. avg. school - 3rd ed. - M.: Enlightenment, 1993. - 351 p.: ill. - ISBN 5-09-004617-4.
- Algebra and the beginning of the analysis: Proc. for 10-11 cells. general education institutions / A. N. Kolmogorov, A. M. Abramov, Yu. P. Dudnitsyn and others; Ed. A. N. Kolmogorova.- 14th ed.- M.: Enlightenment, 2004.- 384 p.: ill.- ISBN 5-09-013651-3.
- Gusev V. A., Mordkovich A. G. Mathematics (a manual for applicants to technical schools): Proc. allowance.- M.; Higher school, 1984.-351 p., ill.
Copyright by clever students
All rights reserved.
Protected by copyright law. No part of the site, including internal materials and external design, may be reproduced in any form or used without the prior written permission of the copyright holder.
