Trigonometric shortening formulas. Trigonometric identities and transformation. Sum and difference of trigonometric functions

Basic formulas of trigonometry - these are formulas that establish connections between the basic trigonometric functions. Sine, cosine, tangent and cotangent are related to each other in an impersonal relationship. Below we outline the basic trigonometric formulas, and for clarity we group them according to their meanings. The following formulas can be used practically as a lesson from a standard trigonometry course. It is very important that the formulas themselves are listed below, and not their formulas, to which the statistics will be assigned.

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Basic aspects of trigonometry

Trigonometric equalities provide connections between the sine, cosine, tangent and cotangent of the same term, allowing one function to be determined through another.

Trigonometric Equalities

sin 2 a + cos 2 a = 1 t g α = sin α cos α , c t g α = cos α sin α t g α c t g α = 1 t g 2 α + 1 = 1 cos 2 α , c t g 2 α + 1 = 1 sin 2 α

These similarities directly arise from the meaning of a single number, sine (sin), cosine (cos), tangent (tg) and cotangent (ctg).

Guidance formulas

The given formulas allow you to go from working with enough and how many great cuts to working with cuts in the range from 0 to 90 degrees.

Guidance formulas

sin α + 2 π z = sin α , cos α + 2 π z = cos α t g α + 2 π z = t g α , c t g α + 2 π z = c t g α sin - α + 2 π z = - sin α, cos - α + 2 π z = cos α t g - α + 2 π z = - t g α , c t g - α + 2 π z = - c t g α sin π 2 + α + 2 π z = cos α, cos π 2 + α + 2 π z = - sin α t g π 2 + α + 2 π z = - c t g α , c t g π 2 + α + 2 π z = - t g α sin π 2 - α + 2 π z = cos α, cos π 2 - α + 2 π z = sin α t g π 2 - α + 2 π z = c t g α , c t g π 2 - α + 2 π z = t g α sin π + α + 2 π z = - sin α, cos π + α + 2 π z = - cos α t g π + α + 2 π z = t g α , c t g π + α + 2 π z = c t g α sin π - α + 2 π z = sin α, cos π - α + 2 π z = - cos α t g π - α + 2 π z = - t g α , c t g π - α + 2 π z = - c t g α sin 3 π 2 + α + 2 π z = - cos α, cos 3 π 2 + α + 2 π z = sin α t g 3 π 2 + α + 2 π z = - c t g α , c t g 3 π 2 + α + 2 π z = - t g α sin 3 π 2 - α + 2 π z = - cos α , cos 3 π 2 - α + 2 π z = - sin α t g 3 π 2 - α + 2 π z = c t g α , c t g 3 π 2 - α + 2 π z = t g α

The induction formulas are based on the periodicity of trigonometric functions.

Trigonometric folding formulas

Addition formulas in trigonometry allow you to express the trigonometric function of the sum or difference of parts through the trigonometric functions of these parts.

Trigonometric folding 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 t g α ± β = t g α ± t g β 1 ± t g α t g β c t g α ± β = - 1 ± c t g α c t g β c t g α ± c t g β

Based on the additional formulas, trigonometric formulas of multiple kut are derived.

Formulas for multiple kuta: podviynogo, triple kuta, etc.

Formulas for sub-double and triple kut

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 t g 2 α = 2 · t g α 1 - t g 2 α з t g 2 α = с t g 2 α - 1 2 · з t g α sin 3 α = 3 sin α · cos 2 α - sin 3 α , sin 3 α = 3 sin α - 4 sin 3 α cos 3 α = cos 3 α - 3 sin 2 α · cos α , cos 3 α = - 3 cos α + 4 cos 3 α t g 3 α = 3 t g α - t g 3 α 1 - 3 t g 2 α c t g 3 α = c t g 3 α - 3 c t g α 3 c t g 2 α - 1

Half kuta formulas

The formulas of the half-cut of trigonometry are inherited from the formulas of the subordinate cut and express the relationship between the basic functions of the half-cut and the cosine of the whole cut.

Half kuta 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

Lower level formulas

Lower level formulas

sin 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, when there is a breakdown, working with bulky steps is difficult. The formulas of the lower level allow you to reduce the level of the trigonometric function from as high as possible to the first. Let's take a look at his wicked look:

Back-of-the-envelope type of lower stage formulas

for guys n

sin n α = C n 2 n 2 n + 1 2 n - 1 ∑ k = 0 n 2 - 1 (- 1) n 2 - k · C k n · cos ((n - 2 k) α) cos n α = C n 2 n 2 n + 1 2 n - 1 ∑ k = 0 n 2 - 1 C k n cos ((n - 2 k) α)

for unpaired n

sin n α = 1 2 n - 1 ∑ k = 0 n - 1 2 (- 1) n - 1 2 - k C k n sin ((n - 2 k) α) cos n α = 1 2 n - 1 ∑ k = 0 n - 1 2 C k n cos ((n - 2 k) α)

Sum and difference of trigonometric functions

The value and sum of trigonometric functions can be given to the creator. The multiplication of the difference of sines and cosines can easily be combined with the highest trigonometric equations and simplified 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 = - 2 sin α + β 2 sin α - β 2 , cos α - cos β = 2 sin α + β 2 sin β - α 2

Additional trigonometric functions

Since formulas for the sum and difference functions allow one to go to the final solution, then the formulas for the creation of trigonometric functions make a return transition - leading to the solution to the sum. We look at the formulas for sines, cosines and sine by cosine.

Formulas for adding 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 through the tangent of the half-cut.

Universal trigonometric substitution

sin α = 2 t g α 2 1 + t g 2 α 2 cos α = 1 - t g 2 α 2 1 + t g 2 α 2 t g α = 2 t g α 2 1 - t g 2 α 2 c t g α = 1 - t g 2 α 2 t g α 2

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The equation that takes revenge on the unknown under the sign of the trigonometric function (`sin x, cos x, tan x` or `ctg x`) is called trigonometric equations, their formulas themselves will be discussed further.

The simplest ones are called the number `sin x=a, cos x=a, tg x=a, ctg x=a`, where `x` is the number that needs to be known, `a` is a number. Let's write down the root formula for the skin.

1. Rivnyanya `sin x=a`.

When `|a|>1` there is no solution.

When `|a| \leq 1` there is an infinite number of solutions.

Formula of roots: `x=(-1)^n arcsin a + \pi n, n \in Z`

2. Rivnyannya `cos x=a`

When `|a|>1` - as a result of the sine, there is no solution to the middle of the active numbers.

When `|a| \leq 1` there is no decision.

Formula of roots: x = p arccos a + 2 pi n, n in Z

Private variations for sine and cosine in graphs.

3. Rivnyannya `tg x=a`

There is no decision whatever the meaning of `a`.

Formula of roots: `x=arctg a + \pi n, n \in Z`

4. Rivnyannya `ctg x=a`

The same is true for any value of `a`.

Formula of roots: `x=arcctg a + \pi n, n \in Z`

Formulas for the roots of trigonometric equations in the table

For sine:
For cosine:
For tangent and cotangent:
Formulas for unraveling the equations to replace the gate trigonometric functions:

Methods for solving trigonometric equations

The connection of any trigonometric equation consists of two stages:

• for help, transform it to its simplest form;
• Find out the simplest formulas of the roots and tables.

Let's look at the butts at the main methods of tying.

Algebraic method.

In this whole method, it is necessary to replace the variable and substitute it for equalness.

butt. Divide the equation: `2cos^2(x+\frac \pi 6)-3sin(\frac \pi 3 - x)+1=0`

`2cos^2(x+frac \pi 6)-3cos(x+frac \pi 6)+1=0`,

Let's make a quick replacement: `cos(x+\frac \pi 6)=y`, then `2y^2-3y+1=0`,

we know the root: `y_1=1, y_2=1/2`, the stars show two forms:

1. ` cos (x + frac \ pi 6) = 1 `, ` x + \ frac \ pi 6 = 2 \ pi n `, ` x_1 = - \ frac \ pi 6 +2 \ pi n `.

2. `cos(x+\frac \pi 6)=1/2`, `x+\frac \pi 6=\pm arccos 1/2+2\pi n`, `x_2=\pm \frac \pi 3- \frac \pi 6+2\pi n`.

Version: `x_1=-\frac \pi 6+2\pi n`, `x_2=\pm \frac \pi 3-frac \pi 6+2\pi n`.

Unfolding into multiples.

butt. Extract the equation: `sin x+cos x=1`.

Decision. All terms of equality are moved to the left: `sin x+cos x-1=0`. Vikoristovuchi, reconcilable and decomposed into multipliers of the left part:

`sin x - 2sin^2 x/2=0`,

`2sin x/2 cos x/2-2sin^2 x/2=0`,

`2sin x/2 (cos x/2-sin x/2)=0`,

1. ` sin x/2 = 0 `, ` x/2 = \ pi n `, ` x_1 = 2 \ pi n `.
2. `cos x/2-sin x/2=0`, `tg x/2=1`, `x/2=arctg 1+ \pi n`, `x/2=\pi/4+ \pi n` , `x_2=pi/2+ 2pi n`.

Version: `x_1=2\pi n`, `x_2=\pi/2+ 2\pi n`.

Reduced to a uniform level

It is necessary to reduce the trigonometric equation to one of two types:

`a sin x+b cos x=0` (same level of the first step) or `a sin^2 x + b sin x cos x +c cos^2 x=0` (same level of the other step).

Then divide the offending parts into `cos x\ne 0` - for the first phase, and into `cos ^ 2 x\ne 0` - for the other. We exclude the calculation of `tg x`: `a tg x+b=0` and `a tg^2 x + b tg x +c =0`, as it is necessary to calculate in the following ways.

butt. Divide the equation: `2 sin ^ 2 x + sin x cos x - cos ^ 2 x = 1 `.

Decision. Let's write the right part as `1=sin^2 x+cos^2 x`:

`2 sin^2 x+sin x cos x - cos^2 x=`` sin^2 x+cos^2 x`,

`2 sin^2 x+sin x cos x - cos^2 x - `` sin^2 x - cos^2 x=0`

` sin ^ 2 x + sin x cos x - 2 cos ^ 2 x = 0 `.

This is the same trigonometric equal to the other stage, dividing its left and right parts by `cos^2 x \ne 0`, and is subtracted:

`\frac(sin^2 x)(cos^2 x)+\frac(sin x cos x)(cos^2 x) - \frac(2 cos^2 x)(cos^2 x)=0`

`tg^2 x + tg x - 2 = 0`. We introduce the replacement `tg x=t`, resulting in `t^2 + t - 2=0`. The root of this equation: `t_1=-2` and `t_2=1`. Todi:

1. `tg x=-2`, `x_1=arctg (-2)+\pi n`, `n \in Z`
2. `tg x=1`, `x=arctg 1+\pi n`, `x_2=\pi/4+\pi n`, `n \in Z`.

Confirmation. `x_1=arctg (-2)+\pi n`, `n \in Z`, `x_2=\pi/4+\pi n`, `n \in Z`.

Crossing to the halfway point

butt. Find the equation: `11 sin x - 2 cos x = 10`.

Decision. Let's sum up the formula of the undergrowth, as a result: `22 sin (x/2) cos (x/2) - ``2 cos^2 x/2 + 2 sin^2 x/2=``10 sin^2 x/2 +10 cos^2 x/2`

`4 tg^2 x/2 - 11 tg x/2 +6=0`

Having stagnated the descriptions of the superior method of algebra, we reject:

1. `tg x/2=2`, `x_1=2 arctg 2+2\pi n`, `n \in Z`,
2. `tg x/2=3/4`, `x_2=arctg 3/4+2\pi n`, `n \in Z`.

Confirmation. `x_1=2 arctg 2+2\pi n, n \in Z`, `x_2=arctg 3/4+2\pi n`, `n \in Z`.

Introduction of additional code

In trigonometric equation `a sin x + b cos x = c`, where a, b, c are coefficients, and x is variable, divisible into `sqrt (a^2+b^2)`:

`\frac a(sqrt (a^2+b^2)) sin x +` `\frac b(sqrt (a^2+b^2)) cos x =` `frac c(sqrt (a^2 + b^2))`.

The coefficients on the left side are based on the power of the sine and cosine, and the sum of their squares is equal to 1 and their modules are not more than 1. They are significant by the following order: `\frac a(sqrt(a^2+b^2))=cos\varphi` , ` \frac b(sqrt (a^2+b^2)) =sin \varphi`, `\frac c(sqrt (a^2+b^2))=C`, then:

` cos \ varphi sin x + sin \ varphi cos x = C `.

Let's look at the report on the side:

butt. Unravel the equation: `3 sin x+4 cos x=2`.

Decision. We divide the offending parts of jealousy into `sqrt (3^2+4^2)`, we exclude:

`\frac (3 sin x) (sqrt (3^2+4^2))+``\frac(4 cos x)(sqrt (3^2+4^2))=` `frac 2(sqrt ( 3^2+4^2))`

`3/5 sin x+4/5 cos x=2/5`.

Significantly `3/5 = cos\varphi`, `4/5 = sin\varphi`. So since ` sin \ varphi > 0 `, ` cos \ varphi > 0 `, then as an additional cut we take ` \ varphi = arcsin 4/5 `. Then let’s write down our jealousy in the form of:

`cos \varphi sin x+sin \varphi cos x=2/5`

Having established the sumi kuti formula for the sine, let’s write down our zeal in this form:

`sin (x+\varphi) = 2/5`,

`x+\varphi=(-1)^n arcsin 2/5+ \pi n`, `n \in Z`,

`x=(-1)^n arcsin 2/5-` `arcsin 4/5+ \pi n`, `n \in Z`.

Confirmation. `x=(-1)^n arcsin 2/5-` `arcsin 4/5+ \pi n`, `n \in Z`.

Fractional rational trigonometric equations

Concerns with fractions, in numbers and signs such as trigonometric functions.

butt. Virishity equal. frac (sin x) (1 + cos x) = 1-cos x`.

Decision. Let's multiply and divide the right part of the equality by `(1+cos x)`. As a result, we reject:

`\frac (sin x)(1+cos x)=``\frac ((1-cos x)(1+cos x))(1+cos x)`

`\frac (sin x)(1+cos x)=``\frac (1-cos^2 x)(1+cos x)`

`\frac (sin x)(1+cos x)=``\frac (sin^2 x)(1+cos x)`

`\frac (sin x)(1+cos x)-``\frac (sin^2 x)(1+cos x)=0`

`\frac (sin x-sin^2 x)(1+cos x)=0`

Vrahovuychi, because the sign of the faithful buti cannot be zero, we reject `1+cos x \ne 0`, `cos x \ne -1`, ` x \ne \pi+2\pi n, n \in Z`.

We equate the number of the fraction to zero: `sin x-sin^2 x=0`, `sin x(1-sin x)=0`. Either `sin x=0` or `1-sin x=0`.

1. `sin x=0`, `x=\pi n`, `n \in Z`
2. `1-sin x=0`, `sin x=-1`, `x=\pi /2+2\pi n, n \in Z`.

Doctors say that ` x \ne \pi+2\pi n, n \in Z`, the solutions will be `x=2\pi n, n \in Z` and `x=\pi /2+2\pi n` , `n\in Z`.

Confirmation. `x=2\pi n`, `n \in Z`, `x=\pi /2+2\pi n`, `n \in Z`.

Trigonometry and trigonometric equations are commonly used in all areas of geometry, physics, and engineering. Graduation begins in the 10th grade, and you will be required to attend EDI, so try to memorize all the formulas of trigonometric equations - you will need it!

However, there is no need to memorize them, but to understand the essence and take note. It's not as complicated as it sounds. Switch over and watch the video.

The relationship between the basic trigonometric functions - sine, cosine, tangent and cotangent - is specified trigonometric formulas. There are a lot of connections between trigonometric functions, which explain the layout of trigonometric formulas. Some formulas relate the trigonometric functions of a single cut, other functions of a multiple cut, others allow you to reduce the step, a fourth express all functions through the tangent of a half cut, etc.

This article covers in order all the basic trigonometric formulas that are sufficient for most trigonometry problems. To make it easier to remember, we group them together with their meanings and enter them in the table.

Navigation on the page.

Basic trigonometric equalities

Basic trigonometric equalities set the relationships between the sine, cosine, tangent and cotangent of one kut. The smell comes from the meaning of sine, cosine, tangent and cotangent, as well as the concept of a single stake. They allow you to express one trigonometric function through another.

A detailed description of these trigonometry formulas, their basics and applications can be found in the article.

Guidance formulas

Guidance formulas arise from the powers of sine, cosine, tangent and cotangent, so they represent the power of periodicity of trigonometric functions, the power of symmetry, as well as the power of zsuvo in this case. These trigonometric formulas allow you to move from work with sufficient cuts to work with cuts between zero and 90 degrees.

The basis of these formulas, the mnemonic rule for their memorization and the application of their application can be read from the statistics.

Addition formulas

Trigonometric folding formulas show how the trigonometric functions of the sum and differences of two parts are expressed through the trigonometric functions of these parts. These formulas are the basis for the derivation of lower trigonometric formulas.

Formulas for double, triple, etc. Kuta

Formulas for double, triple, etc. kut (they are also called multiple kut formulas) show how trigonometric functions are subordinate, triple, etc. kutiv () are expressed through the trigonometric functions of a single kut. Their symbols emerge from the folding formulas.

More detailed information is collected from the formula of the second, third and second one. Kuta.

Half kuta formulas

Half kuta formulas show how the trigonometric functions of a half kut are expressed through the cosine of the whole kut. These trigonometric formulas emerge from the formulas of the undergrowth.

Their designs and butts can be viewed from the statistics.

Lower level formulas

Trigonometric formulas of the lower level We welcome the transition from natural stages of trigonometric functions to sines and cosines at the first stage, or multiple stages. In other words, they allow the level of trigonometric functions to be reduced to the first level.

Formulas for sum and differences of trigonometric functions

Main purpose formulas sum and differences of trigonometric functions lies in the transition to the creation of functions, which is even worse when simplifying trigonometric expressions. The designated formulas are also widely used for the highest trigonometric equations, which allows one to multiply the sum and difference of sines and cosines.

Formulas for creating sines, cosines and sine by cosine

The transition to the creation of trigonometric functions up to the sum of the difference occurs using additional formulas for the creation of sines, cosines and sine by cosine.

Universal trigonometric substitution

The review of the basic formulas of trigonometry ends with formulas that express trigonometric functions through the tangent of the half-cut. This replacement took away the name universal trigonometric substitution. The advantage is that these trigonometric functions are expressed through the tangent of the half-cut rationally without roots.

List of literature.

• Algebra: Navch. for 9th grade. middle school/Yu. N. Makarichev, N. G. Mindyuk, K. I. Neshkov, S. B. Suvorova; Per ed. S. A. Telyakovsky.- M.: Prosvitnitstvo, 1990.- 272 pp.: Ill.- ISBN 5-09-002727-7
• Bashmakov M. I. Algebra and analysis: Navch. for 10-11 grades. middle school - 3 types. - M: Prosvitnitstvo, 1993. - 351 p.: ill. - ISBN 5-09-004617-4.
• Algebra and start with analysis: Head. for 10-11 grades. zagalnosvit. installation / A. N. Kolmogorov, A. M. Abramov, Yu. P. Dudnitsin and in; Per ed. A. N. Kolmogorov. - 14 types. - M.: Prosvitnitstvo, 2004. - 384 pp.: Il. - ISBN 5-09-013651-3.
• Gusev V. A., Mordkovich A. G. Mathematics (a handbook for pre-technical school students): Navch. Pos_bnik.- M.; Visch. school, 1984.-351 p., ill.

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On this page you will find all the basic trigonometric formulas that will help you do a lot of right-handed work.

Trigonometric formulas are mathematical equations for trigonometric functions that are calculated for all valid values ​​of the argument.

Formulas define the relationship between the basic trigonometric functions - sine, cosine, tangent, cotangent.

Sine kut is the y coordinate of a point (ordinate) on a single number. Cosine cosine is the x coordinate of the point (abscis).

Tangent and cotangent are, obviously, the ratio of sine to cosine and vice versa.
`sin\alpha,\cos\alpha`
`tg \\alpha=\frac(sin\\alpha)(cos\\alpha), `` \alpha\ne\frac\pi2+\pi n, \n \in Z`
`ctg \\alpha=\frac(cos\\alpha)(sin\\alpha), `` \alpha\ne\pi+\pi n, \n \in Z`

And two that are discussed earlier – secant, cosecant. They indicate the relationship between 1 and cosine and sine.

`sec \\alpha=\frac(1)(cos\\alpha),`` \alpha\ne\frac\pi2+\pi n,\n \in Z`
` cosec \ \ alpha = \ frac (1) (sin \ \ alpha), `` \ alpha \ ne \ pi + \ pi n, \ n \ in Z `

From the value of trigonometric functions, you can see what signs of stench appear on the skin. The sign of the function depends only on which of the quarters the argument is expanded.

When changing the argument symbol from “+” to “-”, only the cosine function does not change its value. That one is called a steam room. This graph is symmetrical along the ordinate axis.

Other functions (sine, tangent, cotangent) are unpaired. When you change the argument symbol from “+” to “-”, their values ​​also change negatively. Their graphs are symmetrical to the core of coordinates.

`sin(-\alpha)=-sin \\alpha`
`cos(-\alpha)=cos \\alpha`
`tg(-\alpha)=-tg \\alpha`
`ctg(-\alpha)=-ctg \\alpha`

Basic trigonometric equalities

Basic trigonometric equalities - these are formulas that establish links between trigonometric functions of one unit (`sin\alpha,\cos\\alpha,\tg\alpha,\ctg\\alpha`) and which allow you to find the values ​​of the skin x function through anyone else's knowledge.
`sin^2 \alpha+cos^2 \alpha=1`
`tg \ \alpha \cdot ctg \ \alpha=1, \ \alpha\ne\frac(\pi n) 2, \n \in Z`
`1+tg^2 \alpha=\frac 1(cos^2 \alpha)=sec^2 \alpha, `` \alpha\ne\frac\pi2+\pi n, \n \in Z`
`1+ctg^2 \alpha=\frac 1(sin^2 \alpha)=cosec^2 \alpha, `` \alpha\ne\pi n, \n \in Z`

Formulas for the sum and difference of trigonometric functions

The formulas of the folded and obvious arguments express the trigonometric functions of the sum or the difference of two parts through the trigonometric functions of these parts.
`sin(\alpha+\beta)=` `sin \ \alpha\ cos \ \beta+cos \ \alpha\ sin \ beta`
` sin ( \ alpha - \ beta ) = `` sin \ \ alpha \ cos \ \ beta - cos \ \ alpha \ sin \ \ beta `
`cos(\alpha+\beta)=` `cos \ \alpha\ cos \ \beta-sin \ \alpha\ sin \ \beta`
`cos(\alpha-\beta)=` `cos \ \alpha\ cos \ \beta+sin \ alpha\ sin \ beta`
`tg(\alpha+\beta)=\frac(tg \ \alpha+tg \ \beta)(1-tg \ \alpha\ tg \ \beta)`
`tg(\alpha-\beta)=\frac(tg \ \alpha-tg \ \beta)(1+tg \ \alpha \ tg \ \beta)`
`ctg(\alpha+\beta)=\frac(ctg \ \alpha \ ctg \ \beta-1)(ctg \ \beta+ctg \ \alpha)`
`ctg(\alpha-\beta)=\frac(ctg \ \alpha\ ctg \ \beta+1)(ctg \ \beta-ctg \ \alpha)`

Formulas for rootstock

`sin\2\alpha = 2\sin\\alpha\cos\\alpha = ``frac (2\tg\\alpha) (1 + tg^2\alpha) = \frac (2\ctg\\alpha) (1+ctg^2 \alpha)=` `\frac 2(tg \ \alpha+ctg \ \alpha)`
`cos \ 2\alpha=cos^2 \alpha-sin^2 \alpha=``1-2 \ sin^2 \alpha=2 \cos^2 \alpha-1=` `frac(1-tg^ 2 \alpha)(1+tg^2\alpha)=\frac(ctg^2\alpha-1)(ctg^2\alpha+1)=` `frac(ctg \ alpha-tg \ alpha) (ctg \ alpha + tg\alpha)`
`tg \ 2\alpha=\frac(2 \ tg \ \alpha)(1-tg^2 \alpha)=``\frac(2 \ ctg \ \alpha)(ctg^2 \alpha-1)=` `\frac 2(\ctg \ \alpha-tg \ \alpha)`
`ctg \ 2\alpha=\frac(ctg^2 \alpha-1)(2 \ ctg \ \alpha)=``\frac ( \ctg \ \alpha-tg \ \alpha)2`

Triple kuta formulas

`sin \ 3\alpha=3 \ sin \ \alpha-4sin^3 \alpha`
` cos \ 3 \ alpha = 4 cos ^ 3 \ alpha-3 \ cos \ \ alpha `
`tg \ 3\alpha=\frac(3 \ tg \ \alpha-tg^3 \alpha)(1-3 \ tg^2 \alpha)`
`ctg \ 3\alpha=\frac(ctg^3 \alpha-3 \ ctg \ \alpha)(3 \ ctg^2 \alpha-1)`

Half kuta formulas

`sin \ \frac \alpha 2=\pm \sqrt(\frac (1-cos \ \alpha)2)`
`cos \ \frac \alpha 2=\pm \sqrt(\frac (1+cos \ \alpha)2)`
`tg \frac \alpha 2=\pm \sqrt(\frac (1-cos \alpha)(1+cos \alpha))=` `frac (sin \ alpha)(1+cos \ \alpha)=\frac (1-cos\alpha)(sin\alpha)`
`ctg \ \frac \alpha 2=\pm \sqrt(\frac (1+cos \ \alpha)(1-cos \ \alpha))=``\frac (sin \ \alpha)(1-cos \ \ alpha) = frac (1 + cos \ alpha) (sin \ alpha)`

Formulas for half, double and triple arguments express the functions `sin, \cos, \tg, \ctg` of these arguments (`\frac(\alpha)2, \2\alpha, \3\alpha, ... ') through ci functions to the argument `\alpha`.

They can be removed from the front group (folded and visible arguments). For example, the similarity of the undergrowth can be easily removed by replacing `beta` with `alpha`.

Lower level formulas

Formulas of squares (cubes, etc.) of trigonometric functions allow you to go from 2,3, ... stage to trigonometric functions of the first stage, or multiples (`\alpha, \3\alpha, \...' or `2 \alpha, \4 \alpha, \...`).
`sin^2 \alpha=\frac(1-cos \ 2\alpha)2,`` (sin^2 \frac \alpha 2=\frac(1-cos \ \alpha)2)`
`cos^2 \alpha=\frac(1+cos \ 2\alpha)2,`` (cos^2 \frac \alpha 2=\frac(1+cos \alpha)2)`
`sin^3 \alpha=\frac(3sin \ \alpha-sin \ 3\alpha)4`
`cos^3 \alpha=\frac(3cos \ \alpha+cos \ 3\alpha)4`
`sin^4 \alpha=\frac(3-4cos \ 2\alpha+cos \ 4\alpha)8`
`cos^4 \alpha=\frac(3+4cos \ 2\alpha+cos \ 4\alpha)8`

Formulas for sum and differences of trigonometric functions

The formulas are a transformation of the sum and difference of trigonometric functions of different arguments on the surface.

`sin \ \alpha+sin \ \beta=` `2 \ sin \frac(\alpha+\beta)2 \ cos \frac(\alpha-\beta)2`
`sin \ \alpha-sin \ \beta=` `2 \cos \frac(\alpha+\beta)2 \sin \frac(\alpha-\beta)2`
` cos \ \ alpha + cos \ \ beta = `` 2 \ cos \ frac ( \ alpha + \ beta ) 2 \ cos \ frac ( \ alpha - beta )2 `
`cos \ \alpha-cos \ \beta=` `-2 \ sin \frac(\alpha+\beta)2 \ sin \frac(\alpha-\beta)2=` `2 \ sin \frac(\alpha+\ ) beta)2 \ sin \frac(\beta-\alpha)2`
`tg \ \alpha \pm tg \ \beta=\frac(sin(\alpha \pm \beta))(cos \ \alpha \ cos \ \beta)`
`ctg \ \alpha \pm ctg \ \beta=\frac(sin(\beta \pm \alpha))(sin \ \alpha \ sin \ \beta)`
`tg \ \alpha \pm ctg \ \beta=``\pm \frac(cos(\alpha \mp \beta))(cos \ \alpha \ sin \ beta)`

Here you need to change the addition and function of one argument per type.

` cos \ \ alpha + sin \ \ alpha = \ sqrt (2) \ cos ( \ frac ( \ pi) 4- \ alpha)`
`cos \ \alpha-sin \ \alpha=\sqrt(2) \ sin (\frac(\pi)4-\alpha)`
`tg \ alpha+ctg \ \ alpha = 2 \ cosec \ 2 \ alpha; ``tg\\alpha-ctg\\alpha = -2\ctg\2\alpha

The following formulas convert the sum and difference of a unit and a trigonometric function in additions.

`1 + cos\\alpha = 2\cos^2\frac (\alpha)2`
`1-cos \\alpha=2\sin^2\frac(\alpha)2`
`1 + sin\\alpha = 2\cos^2 (\frac (\pi)4-\frac (\alpha)2)`
`1-sin \alpha=2 \sin^2 (\frac (\pi) 4-\frac(\alpha)2)`
`1 \pm tg \ \alpha=\frac(sin(\frac(\pi)4 \pm \alpha))(cos \frac(\pi)4 \cos \ \alpha)=` `\frac(\sqrt (2) sin(\frac(\pi)4 \pm \alpha))(cos \ \alpha)`
`1 \pm tg\alpha\tg\\beta =\frac (cos (\alpha\mp\beta)) (cos\\alpha\cos\\beta); `` \ ctg \ \ alpha \ ctg \ \ beta \pm 1=\frac(cos(\alpha \mp \beta))(sin \ \alpha \ sin \ \beta)`

Formulas for the transformation of creative functions

Formulas for converting trigonometric functions with arguments '\alpha' and '\beta' to the sum (difference) of these arguments.
`sin \ \alpha \sin \ \beta = `` \frac(cos(\alpha - \beta)-cos(\alpha + \beta))(2)`
`sin\alpha \cos\beta =` `\frac(sin(\alpha - \beta)+sin(\alpha + \beta))(2)`
` cos \ \ alpha \ cos \ \ beta =` ` \frac(cos(\alpha - \beta)+cos(\alpha + \beta))(2)`
`tg\\alpha\tg\\beta = ``frac (cos (\alpha - \beta) - cos (\alpha + \beta)) ( cos (\alpha - \beta) + cos (\alpha + \beta )) = ``\frac(tg\alpha + tg\beta)(ctg\alpha + ctg\beta)`
`ctg \ \alpha \ ctg \ \beta =` `frac(cos(\alpha - \beta) + cos(\alpha + \beta)) (cos(\alpha - \beta)-cos(\alpha + \ beta )) = ``frac(ctg\alpha + ctg\beta)(tg\alpha+tg\beta)`
`tg \ \alpha \ ctg \ \beta =` `frac(sin(\alpha - \beta) + sin(\alpha + \beta)) (sin(\alpha + \beta)-sin(\alpha - \beta ))`

Universal trigonometric substitution

These formulas express trigonometric functions through the tangent of the half-cut.
`sin \ \alpha= \frac(2tg\frac(\alpha)(2))(1 + tg^(2)\frac(\alpha)(2)), `` \alpha\ne \pi +2\ pi n, n \in Z`
`cos\\alpha =\frac (1 - tg^(2)\frac (\alpha) (2)) (1 + tg^(2)\frac (\alpha) (2)), ``\alpha\ ne \pi +2\pi n, n \in Z`
`tg\\alpha =\frac (2tg\frac (\alpha) (2)) (1 - tg^(2)\frac (\alpha) (2)), ``\alpha\ne\pi +2\ pi n, n \in Z, `` \alpha \ne \frac(\pi)(2)+ \pi n, n \in Z`
`ctg \ \alpha = \frac(1 - tg^(2)\frac(\alpha)(2))(2tg\frac(\alpha)(2)), `` \alpha \ne \pi n, n \in Z, ``\alpha \ne \pi + 2\pi n, n \in Z`

Guidance formulas

Reducing formulas can be obtained, vikorystvo and such power of trigonometric functions, such as periodicity, symmetry, power of the sum of data. They allow the functions of a certain amount of heat to change to functions that are between 0 and 90 degrees.

For kut (`\frac (\pi)2 \pm \alpha`) or (`90^\circ \pm \alpha`):
`sin(\frac (\pi)2 - \alpha) = cos \ \alpha; `` sin (\frac (\pi)2 + \alpha) = cos \ \alpha`
`cos(\frac(\pi)2 - \alpha)=sin \\alpha;``cos(\frac(\pi)2 + \alpha)=-sin \\alpha`
`tg(\frac(\pi)2 - \alpha) = ctg \\alpha;``tg(\frac(\pi)2 + \alpha)=-ctg \\alpha`
`ctg(\frac(\pi)2 - \alpha)=tg \\alpha;``ctg(\frac(\pi)2 + \alpha)=-tg \\alpha`
For kut (`\pi \pm \alpha`) or (`180^\circ \pm \alpha`):
` sin (\pi - \alpha) = sin\\alpha; ``sin(\pi +\alpha) = - sin\\alpha`
`cos(\pi - \alpha)=-cos \\alpha;``cos(\pi + \alpha)=-cos \\alpha`
`tg(\pi - \alpha)=-tg \ \alpha;`` tg(\pi + \alpha)=tg \ \alpha`
`ctg(\pi - \alpha) = -ctg \ \alpha;`` ctg(\pi + \alpha) = ctg \ \alpha`
For kut (`\frac (3\pi)2 \pm \alpha`) or (`270^\circ \pm \alpha`):
`sin(\frac(3\pi)2 - \alpha)=-cos \\alpha;``sin(\frac(3\pi)2 + \alpha)=-cos\alpha`
`cos(\frac(3\pi)2 - \alpha)=-sin \\alpha;``cos(\frac(3\pi)2 + \alpha)=sin \\alpha`
`tg(\frac(3\pi)2 - \alpha)=ctg \\alpha;``tg(\frac(3\pi)2 + \alpha)=-ctg \\alpha`
`ctg(\frac(3\pi)2 - \alpha) = tg \\alpha;`` ctg(\frac(3\pi)2 + \alpha)=-tg \\alpha`
For kut (`2\pi \pm \alpha`) or (`360^\circ \pm \alpha`):
` sin(2\pi -\alpha) = - sin\\alpha; ``sin(2\pi+\alpha) = sin\\alpha`
` cos(2\pi -\alpha) = cos\\alpha; ``cos(2\pi+\alpha) = cos\\alpha`
`tg(2\pi - \alpha)=-tg \ \alpha;`` tg(2\pi + \alpha)=tg \ \alpha`
`ctg(2\pi - \alpha) = -ctg \ \alpha; ``ctg(2\pi+\alpha) = ctg\\alpha`

Expression of some trigonometric functions through others

`sin \ \alpha=\pm \sqrt(1-cos^2 \alpha)=` `\frac(tg \ \alpha)(\pm \sqrt(1+tg^2 \alpha))=\frac 1( \pm \sqrt(1+ctg^2 \alpha))`
`cos \ \alpha=\pm \sqrt(1-sin^2 \alpha)=` `\frac 1(\pm \sqrt(1+tg^2 \alpha))=\frac (ctg \ \alpha)( \pm \sqrt(1+ctg^2 \alpha))`
`tg \ \alpha=\frac (sin \ \alpha)(\pm \sqrt(1-sin^2 \alpha))=``\frac (\pm \sqrt(1-cos^2 \alpha))( cos\\alpha) = \frac 1 (ctg\\alpha)`
`ctg \ \alpha=\frac (\pm \sqrt(1-sin^2 \alpha))(sin \ \alpha)=` `frac (cos \ \alpha)(\pm \sqrt(1-cos^ 2 \alpha))=\frac 1(tg \ \alpha)`

Trigonometry is literally translated as “the world of tricutaneous people.” Vaughn begins to study at school, and will continue in more detail at VNZ. Therefore, the basic formulas of trigonometry are needed starting from the 10th grade, as well as for completing the EDI. The stench signifies connections between functions, and while the fragments of these connections are plentiful, the formulas themselves are few. It’s not easy to remember them all, but it’s not necessary - they can be taken out of necessity.

Trigonometric formulas are used in integral calculation, as well as in trigonometric reductions, calculations, and conversions.