Basic formulas of trigonometry. All trigonometry formulas Trigonometric formulas table 10
The relationships between the basic trigonometric functions - sine, cosine, tangent and cotangent - are given 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 basic 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.
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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 formulas addition 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 formulas double angle.
Their conclusion and examples of application can be found in the article.
Degree reduction formulas
Trigonometric formulas for reducing degrees are intended 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 you to reduce the powers of trigonometric functions to the first.
Formulas for the sum and difference of trigonometric functions
The 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, since they allow you to factorize 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 in terms of the tangent of a half angle rationally without roots.
Bibliography.
- 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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Trigonometry, trigonometric formulas
The relationships between the basic trigonometric functions - sine, cosine, tangent and cotangent - are given 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 basic 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.
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 basic trigonometric identities.
Top of page
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 reduction formulas.
Top of page
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.
For more information, see the article Addition formulas.
Top of page
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. corner.
Top of page
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 on half-angle formulas.
Top of page
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.
Top of page
Formulas for the sum and difference of trigonometric functions
The 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 when solving trigonometric equations, as they allow you to factor the sum and difference of sines and cosines.
For the derivation of formulas, as well as examples of their application, see the article formulas for the sum and difference of sine and cosine.
Top of page
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.
Top of page
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 in terms of the tangent of a half angle rationally without roots.
For more complete information see the article universal trigonometric substitution.
Top of page
- 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.
Trigonometric formulas- these are the most necessary formulas in trigonometry, necessary to express trigonometric functions that are performed for any value of the argument.
Addition formulas.
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
cos (α - β) = cos α cos β + sin α sin β
tg (α + β) = (tg α + tg β) ÷ (1 - tg α tg β)
tg (α - β) = (tg α - tg β) ÷ (1 + tg α · tg β)
ctg (α + β) = (ctg α · ctg β + 1) ÷ (ctg β - ctg α)
ctg (α - β) = (ctg α · ctg β - 1) ÷ (ctg β + ctg α)
Double angle formulas.
cos 2α = cos²α -sin²α
cos 2α = 2cos²α — 1
cos 2α = 1 - 2sin²α
sin 2α = 2sinα cosα
tg 2α = (2tg α) ÷ (1 - tg² α)
ctg 2α = (ctg²α — 1) ÷ (2ctgα )
Triple angle formulas.
sin 3α = 3sin α – 4sin³ α
cos 3α = 4cos³α - 3cosα
tg 3α = (3tgα — tg³α ) ÷ (1 — 3tg²α )
ctg 3α = (3ctg α - ctg³ α) ÷ (1 - 3ctg² α)
Half angle formulas.
Reduction formulas.
|
Function/angle in rad. |
π/2 - α |
π/2 + α |
3π/2 - α |
3π/2 + α |
2π - α |
2π + α |
||
|---|---|---|---|---|---|---|---|---|
|
Function/angle in ° |
90° - α |
90° + α |
180° - α |
180° + α |
270° - α |
270° + α |
360° - α |
360° + α |
Detailed description of reduction formulas.
Basic trigonometric formulas.
Basic trigonometric identity:
sin 2 α+cos 2 α=1
This identity is the result of applying the Pythagorean theorem to a triangle in the unit trigonometric circle.
The relationship between cosine and tangent is:
1/cos 2 α−tan 2 α=1 or sec 2 α−tan 2 α=1.
This formula is a consequence of the basic trigonometric identity and is obtained from it by dividing the left and right sides by cos2α. It is assumed that α≠π/2+πn,n∈Z.
Relation between sine and cotangent:
1/sin 2 α−cot 2 α=1 or csc 2 α−cot 2 α=1.
This formula also follows from the basic trigonometric identity (obtained from it by dividing the left and right sides by sin2α. Here it is assumed that α≠πn,n∈Z.
Tangent definition:
tanα=sinα/cosα,
Where α≠π/2+πn,n∈Z.
Definition of cotangent:
cotα=cosα/sinα,
Where α≠πn,n∈Z.
Corollary from the definitions of tangent and cotangent:
tanα⋅ cotα=1,
Where α≠πn/2,n∈Z.
Definition of secant:
secα=1/cosα,α≠π/2+πn,n∈ Z
Definition of cosecant:
cscα=1/sinα,α≠πn,n∈ Z
Trigonometric inequalities.
The simplest trigonometric inequalities:
sinx > a, sinx ≥ a, sinx< a, sinx ≤ a,
cosx > a, cosx ≥ a, cosx< a, cosx ≤ a,
tanx > a, tanx ≥ a, tanx< a, tanx ≤ a,
cotx > a, cotx ≥ a, cotx< a, cotx ≤ a.
Squares of trigonometric functions.
Formulas for cubes of trigonometric functions.
TrigonometryMathematics. Trigonometry. Formulas. Geometry. Theory
We have looked at the most basic trigonometric functions (do not be fooled, in addition to sine, cosine, tangent and cotangent, there are many other functions, but more on them later), but for now let’s look at some basic properties of the functions already studied.
Trigonometric functions of numeric argument
Whichever real number t no matter what, it can be associated with a uniquely defined number sin(t).
True, the matching rule is quite complex and consists of the following.
To find the value of sin(t) from the number t, you need:
- place the number circle on coordinate plane so that the center of the circle coincides with the origin of coordinates, and the starting point A of the circle falls at point (1; 0);
- find a point on the circle corresponding to the number t;
- find the ordinate of this point.
- this ordinate is the desired sin(t).
Actually we're talking about about the function s = sin(t), where t is any real number. We know how to calculate some values of this function (for example, sin(0) = 0, \(sin \frac (\pi)(6) = \frac(1)(2) \), etc.), we know some of its properties.
Relationship between trigonometric functions
As you, I hope, can guess, all trigonometric functions are interconnected and even without knowing the meaning of one, it can be found through another.
For example, the most important formula in all trigonometry is basic trigonometric identity:
\[ sin^(2) t + cos^(2) t = 1 \]
As you can see, knowing the value of the sine, you can find the value of the cosine, and also vice versa.
Trigonometry formulas
Also very common formulas connecting sine and cosine with tangent and cotangent:
\[ \boxed (\tan\; t=\frac(\sin\; t)(\cos\; t), \qquad t \neq \frac(\pi)(2)+ \pi k) \]
\[ \boxed (\cot\; t=\frac(\cos\; )(\sin\; ), \qquad t \neq \pi k) \]
From the last two formulas one can derive another trigometric identity, this time connecting tangent and cotangent:
\[ \boxed (\tan \; t \cdot \cot \; t = 1, \qquad t \neq \frac(\pi k)(2)) \]
Now let's see how these formulas work in practice.
EXAMPLE 1. Simplify the expression: a) \(1+ \tan^2 \; t \), b) \(1+ \cot^2 \; t \)
a) First of all, let’s write the tangent, keeping the square:
\[ 1+ \tan^2 \; t = 1 + \frac(\sin^2 \; t)(\cos^2 \; t) \]
\[ 1 + \frac(\sin^2 \; t)(\cos^2 \; t)= \sin^2\; t + \cos^2\; t + \frac(\sin^2 \; t)(\cos^2 \; t) \]
Now let’s put everything under a common denominator, and we get:
\[ \sin^2\; t + \cos^2\; t + \frac(\sin^2 \; t)(\cos^2 \; t) = \frac(\cos^2 \; t + \sin^2 \; t)(\cos^2 \; t ) \]
And finally, as we see, the numerator can be reduced to one by the main trigonometric identity, as a result we get: \[ 1+ \tan^2 \; = \frac(1)(\cos^2 \; t) \]
b) With the cotangent we perform all the same actions, only the denominator will no longer be a cosine, but a sine, and the answer will be like this:
\[ 1+ \cot^2 \; = \frac(1)(\sin^2 \; t) \]
Having completed this task, we derived two more very important formulas that connect our functions, which we also need to know like the back of our hands:
\[ \boxed (1+ \tan^2 \; = \frac(1)(\cos^2 \; t), \qquad t \neq \frac(\pi)(2)+ \pi k) \]
\[ \boxed (1+ \cot^2 \; = \frac(1)(\sin^2 \; t), \qquad t \neq \pi k) \]
You must know all the formulas presented by heart, otherwise further study of trigonometry without them is simply impossible. In the future there will be more formulas and there will be a lot of them and I assure you that you will definitely remember all of them for a long time, or maybe you won’t remember them, but EVERYONE should know these six things!
A complete table of all basic and rare trigonometric reduction formulas.
Here you can find trigonometric formulas in a convenient form. And trigonometric reduction formulas can be found on another page.
Basic trigonometric identities
— mathematical expressions for trigonometric functions, executed for each value of the argument.
- sin² α + cos² α = 1
- tg α cot α = 1
- tg α = sin α ÷ cos α
- cot α = cos α ÷ sin α
- 1 + tg² α = 1 ÷ cos² α
- 1 + cotg² α = 1 ÷ sin² α
Addition formulas
- 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
- cos (α - β) = cos α cos β + sin α sin β
- tg (α + β) = (tg α + tg β) ÷ (1 - tg α tg β)
- tg (α - β) = (tg α - tg β) ÷ (1 + tg α · tg β)
- ctg (α + β) = (ctg α · ctg β + 1) ÷ (ctg β - ctg α)
- ctg (α - β) = (ctg α · ctg β - 1) ÷ (ctg β + ctg α)
https://uchim.org/matematika/trigonometricheskie-formuly - uchim.org
Double angle formulas
- cos 2α = cos² α - sin² α
- cos 2α = 2cos² α - 1
- cos 2α = 1 - 2sin² α
- sin 2α = 2sin α cos α
- tg 2α = (2tg α) ÷ (1 - tg² α)
- ctg 2α = (ctg² α - 1) ÷ (2ctg α)
Triple angle formulas
- sin 3α = 3sin α – 4sin³ α
- cos 3α = 4cos³ α – 3cos α
- tg 3α = (3tg α - tg³ α) ÷ (1 - 3tg² α)
- ctg 3α = (3ctg α - ctg³ α) ÷ (1 - 3ctg² α)
Degree reduction formulas
- sin² α = (1 - cos 2α) ÷ 2
- sin³ α = (3sin α – sin 3α) ÷ 4
- cos² α = (1 + cos 2α) ÷ 2
- cos³ α = (3cos α + cos 3α) ÷ 4
- sin² α · cos² α = (1 – cos 4α) ÷ 8
- sin³ α · cos³ α = (3sin 2α – sin 6α) ÷ 32
Transition from product to sum
- sin α cos β = ½ (sin (α + β) + sin (α - β))
- sin α sin β = ½ (cos (α - β) - cos (α + β))
- cos α cos β = ½ (cos (α - β) + cos (α + β))
We have listed quite a lot of trigonometric formulas, but if something is missing, please write.
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transformation of groups is considered in detail general solutions trigonometric equations. The third section examines non-standard trigonometric equations, the solutions of which are based on the functional approach.
All formulas (equations) of trigonometry: sin(x) cos(x) tan(x) ctg(x)
The fourth section discusses trigonometric inequalities. Methods for solving elementary trigonometric inequalities, both on the unit circle and...
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… Homework 19(3.6), 20(2.4) Goal setting Updating background knowledge Properties of trigonometric functions Reduction formulas New material Values of trigonometric functions Solving simple trigonometric equations Reinforcement Solving problems Lesson goal: today we will calculate the values of trigonometric functions and solve ...
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Trigonometric formulas
Trigonometric formulas
We present to your attention various formulas related to trigonometry.
| cotg(2α) = | ctg 2 (α) - 1 2ctg(α) |
General formulas
- print version
Definitions Sine of angle α (designation sin(α)) is the ratio of the leg opposite to angle α to the hypotenuse. Cosine of angle α (designation cos(α)) is the ratio of the leg adjacent to the angle α to the hypotenuse. Angle tangent α (designation tg(α)) is the ratio of the side opposite to angle α to the adjacent side. An equivalent definition is the ratio of the sine of an angle α to the cosine of the same angle - sin(α)/cos(α). Cotangent of angle α (designation cotg(α)) is the ratio of the leg adjacent to the angle α to the opposite one. An equivalent definition is the ratio of the cosine of an angle α to the sine of the same angle - cos(α)/sin(α). Other trigonometric functions: secant — sec(α) = 1/cos(α); cosecant - cosec(α) = 1/sin(α). Note We do not specifically write the sign * (multiply) - where two functions are written in a row, without a space, it is implied. Clue To derive formulas for cosine, sine, tangent or cotangent of multiple (4+) angles, it is enough to write them according to the formulas respectively. cosine, sine, tangent or cotangent of the sum, or reduce to the previous cases, reducing to the formulas of triple and double angles. Addition Derivatives table© Schoolboy. Mathematics (with the support of “Branched Tree”) 2009—2016
By doing trigonometric transformations follow these tips:
- Do not immediately try to come up with a scheme for solving the example from beginning to end.
- Don't try to convert the entire example at once. Take small steps forward.
- Remember that in addition to trigonometric formulas in trigonometry, you can still use all fair algebraic transformations (bracketing, abbreviating fractions, abbreviated multiplication formulas, and so on).
- Believe that everything will be fine.
Basic trigonometric formulas
Most formulas in trigonometry are often used both from right to left and from left to right, so you need to learn these formulas so well that you can easily apply some formula in both directions. Let us first write down the definitions of trigonometric functions. Let there be a right triangle:
Then, the definition of sine:
Definition of cosine:
Tangent definition:
Definition of cotangent:
Basic trigonometric identity:
The simplest corollaries from the basic trigonometric identity:
Double angle formulas. Sine of double angle:
Cosine of double angle:
Tangent of double angle:
Cotangent of double angle:
Additional trigonometric formulas
Trigonometric addition formulas. Sine of the sum:
Sine of the difference:
Cosine of the sum:
Cosine of the difference:
Tangent of the sum:
Tangent of difference:
Cotangent of the amount:
Cotangent of the difference:
Trigonometric formulas for converting a sum into a product. Sum of sines:
Sine difference:
Sum of cosines:
Difference of cosines:
Sum of tangents:
Tangent difference:
Sum of cotangents:
Cotangent difference:
Trigonometric formulas for converting a product into a sum. Product of sines:
Product of sine and cosine:
Product of cosines:
Degree reduction formulas.
Half angle formulas.
Trigonometric reduction formulas
The cosine function is called co-function sine functions and vice versa. Similarly, the tangent and cotangent functions are cofunctions. Reduction formulas can be formulated as the following rule:
- If in the reduction formula an angle is subtracted (added) from 90 degrees or 270 degrees, then the reduced function changes to a cofunction;
- If in the reduction formula the angle is subtracted (added) from 180 degrees or 360 degrees, then the name of the reduced function is retained;
- In this case, the sign that the reduced (i.e., original) function has in the corresponding quadrant is placed in front of the reduced function, if we consider the subtracted (added) angle to be acute.
Reduction formulas are given in table form:
By trigonometric circle easy to determine tabular values of trigonometric functions:
Trigonometric equations
To solve a certain trigonometric equation, it must be reduced to one of the simplest trigonometric equations, which will be discussed below. For this:
- You can use the trigonometric formulas given above. At the same time, you don’t need to try to transform the entire example at once, but you need to move forward in small steps.
- We must not forget about the possibility of transforming some expression using algebraic methods, i.e. for example, take something out of brackets or, conversely, open brackets, reduce a fraction, apply an abbreviated multiplication formula, bring fractions to a common denominator, and so on.
- When solving trigonometric equations you can use grouping method. It must be remembered that in order for the product of several factors to be equal to zero, it is sufficient that any of them be equal to zero, and the rest existed.
- Applying variable replacement method, as usual, the equation after introducing the replacement should become simpler and not contain the original variable. You also need to remember to perform a reverse replacement.
- Remember that homogeneous equations often appear in trigonometry.
- When opening modules or solving irrational equations with trigonometric functions, you need to remember and take into account all the subtleties of solving the corresponding equations with ordinary functions.
- Remember about ODZ (in trigonometric equations, restrictions on ODZ mainly come down to the fact that you cannot divide by zero, but do not forget about other restrictions, especially about the positivity of expressions in rational powers and under the roots of even powers). Also remember that the values of sine and cosine can only lie in the range from minus one to plus one, inclusive.
The main thing is, if you don’t know what to do, do at least something, and the main thing is to use trigonometric formulas correctly. If what you get gets better and better, then continue the solution, and if it gets worse, then go back to the beginning and try to apply other formulas, do this until you come across the correct solution.
Formulas for solutions of the simplest trigonometric equations. For sine there are two equivalent forms of writing the solution:
For other trigonometric functions, the notation is unambiguous. For cosine:
For tangent:
For cotangent:
Solving trigonometric equations in some special cases:
Successful, diligent and responsible implementation of these three points, as well as responsible study of the final training tests, will allow you to show an excellent result at the CT, the maximum of what you are capable of.
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