Equation sin x \u003d a. Formulas trigonometry I group. Major identities

In trigonometry, many formulas are easier to withdraw than to drive. Cosuine dual corner - wonderful formula! It allows you to obtain the formulas for lowering the degree and formula of half angle.

So, we need a dual corner cosine and a trigonometric unit:

They are even similar: in the cosine formula of a double angle - the difference of squares of cosine and sinus, and in a trigonometric unit - their amount. If you express cosine from the trigonometric unit:

and to substitute it in the cosine of a double angle, we will get:

This is another dual corner cosine formula:

This formula is the key to obtaining a degree reduction formula:

So, the formula for lowering the degree of sine:

If an alpha angle is replaced in half an angle of alpha in half, and double the angle of two alpha - at the angle of alpha, then we get a formula of a half angle for sinus:

Now, from the trigonometric unit, we will express sinus:

We will substitute this expression in the dual corner cosine formula:

Received another cosine formula of a double angle:

This formula is the key to finding the formula for lowering the degree of cosine and half angle for cosine.

Thus, the formula for lowering the degree of cosine:

If it is replaced with α on α / 2, and 2α - on α, then we get a formula of a half argument for cosine:

Since Tangent is a sinus attitude to a cosine that formula for Tangent:

Kotangenes - the attitude of the cosine to sinus. Therefore, the formula for Kotangens:

Of course, in the process of simplifying trigonometric expressions of the formula of a half angle or a decrease of degree, it makes no sense every time to output. It is much easier to put a leaf with formulas. And simplification will move faster, and the visual memory will turn on for memorization.

But several times to remove these formulas still costs. Then you will be absolutely sure that on the exam, when there is no opportunity to use the crib, you can easily get them if necessary.

| BD | - The length of the arc of the circle with the center at the point a.
α - angle, expressed in radians.

Tangent ( tG α.) - This is a trigonometric function depending on the angle α between the hypothenooma and a cathe of the rectangular triangle, equal to the ratio of the length of the opposite category | BC | to the length of the adjacent category | AB | .
Kotnence ( cTG α.) is a trigonometric function, depending on the angle α between the hypothenooma and the ribal triangle cathet, equal to the ratio of the length of the adjacent category | AB | to the length of the opposite category | BC | .

Tangent

Where n. - whole.

In Western literature, Tangent is designated as:
.
;
;
.

Tangent function graph, Y \u003d TG X

Cotangent

Where n. - whole.

In Western literature, Kothanns is indicated as follows:
.
The following notation is also taken:
;
;
.

Cotanence function graph, Y \u003d CTG X

Properties of Tangent and Kotnence

Periodicity

Functions y \u003d. tG X. and y \u003d. cTG X. Periodic with a period π.

Parity

The functions of Tangent and Kotangenes are odd.

Fields of definition and values, increasing, decrease

The functions of Tangent and Cotangenes are continuous on their field of definition (see Proof of Continuity). The main properties of Tangent and Kotnence are presented in the table ( n. - whole).

	y \u003d. tG X.	y \u003d. cTG X.
Definition and continuity area
Region of values	-∞ < y < +∞	-∞ < y < +∞
Ascending		-
Disarmament	-
Extremes	-	-
Zeros, y \u003d 0
Point of intersection with the ordinate axis, x \u003d 0	y \u003d. 0	-

Formulas

Expressions through sinus and cosine

; ;
; ;
;

Tangent and Cotangent formulas from the amount and difference

The remaining formulas are easy to get, for example

Work tangent

The formula of the sum and the difference of tangents

This table presents the values \u200b\u200bof tangents and catangers at some values \u200b\u200bof the argument.

Integrated expressions

Expressions through hyperbolic functions

;
;

Derivatives

; .

.
N-th order derivative by variable x from function:
.
Output formulas for tangent \u003e\u003e\u003e; For Cotanza \u003e\u003e\u003e

Integrals

Decomposition in the ranks

To obtain a decomposition of tangent in degrees X, you need to take several decomposition members in a power row for functions sIN X. and cOS X. And divide these polynomials to each other ,. In this case, the following formulas are obtained.

At.

at.
Where B N. - Numbers Bernoulli. They are determined either from the recurrent ratio:
;
;
where.
Either by Laplace formula:

Reverse functions

Inverse functions to Tangent and Kotangent are Arctanens and Arkcotanence, respectively.

Arctgennes, Arctg.

where n. - whole.

Arkkothangenes, ArcCTG.

where n. - whole.

References:
I.N. Bronstein, K.A. Semendyaev, a reference book on mathematics for engineers and students of the attendants, "Lan", 2009.
Korn, Mathematics Directory for Scientists and Engineers, 2012.

I group. Major identities

sIN 2 α + COS 2 α \u003d 1;

tGα \u003d. ____ SINα COSα; Ctgα \u003d. ____ COSα SINα. ;

tGα · Ctgα \u003d 1;

1 + TG 2 α \u003d _____ 1 COS 2 α; 1 + CTG 2 α \u003d _____ 1 SIN 2 α.

This group contains the simplest and most popular formulas. Most students know them. But if still there is difficulties, then to remember the first three formulas, mentally imagine a rectangular triangle with a hypothenuclear equal one. Then its kartets will be equal, respectively, SINα to determine the sinus (the ratio of the opposite catech to hypotenuse) and COSα to determine the cosine (the ratio of the adjacent catech for hypotenuse).

The first formula is the Pythagoras theorem for such a triangle - the sum of the squares of the cathets is equal to the square of the hypotenuse (1 2 \u003d 1), the second and third is the definitions of the tangent (the ratio of the opposite category to the adjacent) and the catangen (the ratio of the adjacent category to the opposite).
The work of Tangent on Kotangenes is 1 because the catangent recorded in the form of a fraction (Formula Third) is an inverted tangent (second formula). The last consideration, by the way, makes it possible to exclude from among the formulas that it is necessary to memorize all subsequent long formulas with Kotangent. If you will meet CTGα in any difficult task, just replace it with a fraction ___ 1 TGα. And use the formulas for tangent.

The last two formulas can not be memorized. They are less common. And if you need, you can always withdraw them on the draft anew. To do this, it is enough to substitute instead of a tangent or contact of their definition after a fraction (formula two and third, respectively) and lead the expression to the general denominator. But it is important to remember that such formulas that bind the squares of tangent and cosine, and the squares of Kotangens and Sinus exist. Otherwise, you can not guess which conversions are needed to solve a particular task.

II group. Formulas addition

sin (α + β) \u003d sinα · cosβ + cosα · sinβ;

sin (α - β) \u003d sinα · cosβ - cosα · sinβ;

cos (α + β) \u003d cosα · cosβ - sinα · sinβ;

cos (α - β) \u003d cosα · cosβ + sinα · sinβ;

tG (α + β) \u003d TGα + TGβ _________ 1 - TGα · TGβ;

tG (α - β) \u003d

Recall the accuracy of the parity / oddness of trigonometric functions:

sin (-α) \u003d - sin (α); cos (-α) \u003d cos (α); TG (-α) \u003d - TG (α).

Of all trigonometric functions, only cosine is an even function and does not change its sign when changing the argument sign (angle), the remaining functions are odd. The accuracy of the function, in fact, means that the minus sign can be made and put out the function sign. Therefore, if you encounter a trigonometric expression with a difference of two angles, you can always understand it as a sum of positive and negative angles.

For example, sin ( x. - 30º) \u003d sin ( x. + (-30º)).
Next, we use the formula sum of two angles and deal with signs:
sin ( x. + (-30º)) \u003d sin x.· COS (-30º) + COS x.· Sin (-30º) \u003d
\u003d SIN x.· COS30º - COS x.· Sin30º.

Thus, all formulas containing the difference of angles can be simply skipped at the first memorization. Then you should learn to restore them in general, first on the draft, and then mentally.

For example, TG (α - β) \u003d Tg (α + (-β)) \u003d TGα + TG (-β) ___________ 1 - TGα · TG (-β) = TGα - TGβ _________ 1 + TGα · TGβ.

This will help in further faster to guess which transformations need to be applied to solve a task of trigonometry.

Sh group. Formulas of multiple arguments

sin2α \u003d 2 · sinα · cosα;

cos2α \u003d cos 2 α - sin 2 α;

tG2α \u003d. 2tgα _______ 1 - TG 2 α;

sin3α \u003d 3sinα - 4sin 3 α;

cOS3α \u003d 4COS 3 α - 3COSα.

The need to use formulas for sine and cosine of a double angle occurs very often, for Tangent, too. These formulas should be known by heart. Moreover, there are no difficulties in their memorization. First, the formulas are short. Secondly, they are easily controlled by the formulas of the previous group, based on the fact that 2α \u003d α + α.
For example:
sin (α + β) \u003d sinα · cosβ + cosα · sinβ;
sin (α + α) \u003d sinα · cosα + cosα · sinα;
SIN2α \u003d 2SINα · COSα.

However, if you have learned these formulas faster, and not the previous ones, then you can act on the contrary: to remember the formula for the sum of two angles by the corresponding formula for a double angle.

For example, if you need a cosine formula of the sum of two angles:
1) Remember the dual corner cosine formula: cos2. x. \u003d COS 2. x. - SIN 2. x.;
2) We paint it long: cOS ( x. + x.) \u003d COS. x.· COS. x. - SIN x.· SIN x.;
3) replace one h. On α, the second on β: cOS (α + β) \u003d cosα · cosβ - sinα · sinβ.

Repeat similarly to restore formulas for sine sum and tangent amount. In responsible cases, such as the EGE, check the accuracy of the reduced formulas on the well-known first quarter: 0º, 30º, 45º, 60º, 90º.

Checking the previous formula (obtained by replacement in line 3):
let be α \u003d 60 °, β \u003d 30 °, α + β \u003d 90 °,
then cOS (α + β) \u003d cos90 ° \u003d 0, cosα \u003d cos60 ° \u003d 1/2, cosβ \u003d cos30 ° \u003d √3 _ / 2, sinα \u003d sin60 ° \u003d √3 _ / 2, sinβ \u003d sin30 ° \u003d 1/2;
We substitute the values \u200b\u200bin the formula: 0 \u003d (1/2) · ( √3_ /2) − (√3_ / 2) · (1/2);
0 ≡ 0, errors are not detected.

Formulas for a triple angle, in my opinion, not necessary to "tool". They are rarely found at the exams of the EGE. They are easily derived from the formulas that were higher, because sin3α \u003d sin (2α + α). And those students who for some reason still need to learn these formulas by heart, I advise you to pay attention to their some "symmetry" and remember not the formulas themselves, but mnemonic rules. For example, the order in which the numbers are located in two formulas "33433433", etc.

IV group. Amount / Difference -

sINα + SINβ \u003d 2 · SIN α + β ____ 2· COS. α - β ____ 2 ;

sINα - sinβ \u003d 2 · sin α - β ____ 2· COS. α + β ____ 2 ;

cOSα + COSβ \u003d 2 · COS α + β ____ 2· COS. α - β ____ 2 ;

cOSα - Cosβ \u003d -2 · Sin α - β ____ 2· SIN α + β ____ 2 ;

tGα + TGβ \u003d sIN (α + β) ________ COSα · COSβ ;

tGα - TGβ \u003d sIN (α - β) ________ COSα · COSβ .

Using the accuracy of the functions of Sinus and Tangent: sin (-α) \u003d - sin (α); TG (-α) \u003d - TG (α),
You can formulas for the differences of two functions to reduce formulas for their sums. For example,

sin90º - sin30º \u003d sin90º + sin (-30º) \u003d 2 · sin 90º + (-30º) __________ 2· COS. 90º - (-30º) __________ 2 =

2 · SIN30º · COS60º \u003d 2 · (1/2) · (1/2) \u003d 1/2.

Thus, the formulas of the difference of sinuses and tangents do not necessarily immediately memorize.
With the sum and the difference of cosine, the situation is more complicated. These formulas are not interchangeable. But again, using the parity of the cosine, you can remember the following rules.

The amount of COSα + COSβ cannot change its sign for any changes in the signs of the angles, so the product should also consist of even functions, i.e. Two cosines.

The COSα - COSβ difference sign depends on the values \u200b\u200bof the functions themselves, which means the work mark should depend on the correlation of the angles, so the product should consist of odd functions, i.e. two sines.

Nevertheless, this group of formulas is not the easiest to memorize. This is the case when it is better to sharpen, but more check. To prevent errors in the formula in a given exam, be sure to first record it on the draft and check in two ways. First substitutions β \u003d α and β \u003d -α, then by known values \u200b\u200bof functions for simple angles. To do this, it is best to take 90º and 30º, as it was done in the example above, because the half-diet and the sedimentality of these values, again give simple angles, and you can easily see how equality becomes the identity for the correct option. Or, on the contrary, not executed if you are mistaken.

Examplechecks of the formula COSα - Cosβ \u003d 2 · sin α - β ____ 2· SIN α + β ____ 2 For the difference of cosinees with a mistake !

1) let β \u003d α, then cosα - cosα \u003d 2 · sin α - α _____ 2· SIN α + α _____ 2 \u003d 2sin0 · sinα \u003d 0 · sinα \u003d 0. cosα - cosα ≡ 0.

2) Let β \u003d - α, then cosα - cos (- α) \u003d 2 · sin α - (-α) _______ 2· SIN α + (-α) _______ 2 \u003d 2sinα · sin0 \u003d 0 · sinα \u003d 0. cosα - cos (- α) \u003d cosα - cosα ≡ 0.

These checks showed that the functions in the formula are used correctly, but due to the fact that the identity obtained the type 0 ≡ 0, an error with a sign or a coefficient could be missed. We make a third check.

3) Let α \u003d 90º, β \u003d 30º, then COS90º - COS30º \u003d 2 · SIN 90º - 30º ________ 2· SIN 90º + 30º ________ 2 \u003d 2sin30º · sin60º \u003d 2 · (1/2) · (√3 _ /2) = √3_ /2.

cOS90 - COS30 \u003d 0 - √3 _ /2 = −√3_ /2 ≠ √3_ /2.

The error was really in the sign and only in the sign before the work.

V band. Work - in the amount / difference

sinα · sinβ \u003d 1 _ 2 · (COS (α - β) - COS (α + β));

cosα · cosβ \u003d 1 _ 2 · (COS (α - β) + COS (α + β));

sinα · cosβ \u003d 1 _ 2 · (Sin (α - β) + sin (α + β)).

The name of the fifth group of formulas itself suggests that these formulas are reverse with respect to the previous group. It is clear that in this case it is easier to restore the formula on the draft, than to learn it again, increasing the risk of creating "porridge in the head". The only thing that makes sense to focus for faster recovery of the formula, these are the following equalities (check them):

α = α + β ____ 2 + α - β ____ 2; β = α + β ____ 2 − α - β ____ 2.

Consider example: need to convert sin5 x.· COS3. x. in the sum of two trigonometric functions.
Since the work includes sinus, and cosine, then we take from the previous group the formula for the amount of sinuses, which was already learned, and write it on the draft.

sINα + SINβ \u003d 2 · SIN α + β ____ 2· COS. α - β ____ 2

Let 5. x. = α + β ____ 2 and 3. x. = α - β ____ 2 , then α \u003d α + β ____ 2 + α - β ____ 2 = 5x. + 3x. = 8x., β = α + β ____ 2 − α - β ____ 2 = 5x. − 3x. = 2x..

We replace in the formula on the draft the values \u200b\u200bof the angles, expressed through the variables α and β, on the values \u200b\u200bof the angles, expressed through the variable x..
Receive sin8. x. + SIN2. x. \u003d 2 · sin5 x.· COS3. x.

We divide both part of the Justice for 2 and write it to the final to the right left sin5 x.· COS3. x. = 1 _ 2 (SIN8. x. + SIN2. x.). The answer is ready.

As an exercise: Explain why in the textbook formula for transforming the amount / difference in the work of 6, and inverse (for converting a product in sum or difference) - only 3?

VI group. Degree reduction formulas

cOS 2 α \u003d 1 + COS2α _________ 2;

sIN 2 α \u003d 1 - COS2α _________ 2;

cOS 3 α \u003d 3COSα + COS3α ____________ 4;

sIN 3 α \u003d 3sinα - sin3α ____________ 4.

The first two formulas of this group are very necessary. It is often used in solving trigonometric equations, including the level of a single exam, as well as when calculating integrals containing the elemental functions of a trigonometric type.

It may be easier to remember them in the following "one-story" form
2cos 2 α \u003d 1 + cos2α;
2 SIN 2 α \u003d 1 - COS2α,
And you can always divide into 2 or in the draft.

The need to use the following two formulas (with cubes of functions) on the exams is much less common. In another setting, you will always have time to use the draft. The following options are possible:
1) If you remember the last two formulas of the group III, then use them to express SIN 3 α and COS 3 α by simple transformations.
2) If in the last two formulas of this group you noticed the elements of symmetry, which contribute to their memorization, then write down the sketches of formulas on the draft and check them by the values \u200b\u200bof the main corners.
3) If, besides that such degree reduction formulas exist, you do not know anything about them, then solve the problem in stages, based on the fact that Sin 3 α \u003d SIN 2 α · SINα and other learned formulas. Degree reduction formulas for the square and the formula for the transformation of the work in the amount.

VII group. Half argument

sin. α _ 2. = ± √ 1 - COSα ________ 2;_____

cos. α _ 2. = ± √ 1 + cosα ________ 2;_____

tG. α _ 2. = ± √ 1 - COSα ________ 1 + COSα._____

I do not see the point in memorizing by heart of this group of formulas in the form in which they are presented in textbooks and reference books. If you understand that α is half of 2α, That this is enough to quickly derive the desired formula of half argument, based on the first two formulas to lower the degree.

This also applies to a half angle tangent, the formula for which is obtained by dividing the expression for sinus to the corresponding expression for cosine.

Do not forget only when removing the square root to put a sign ± .

VIII group. Universal substitution

sinα \u003d 2tg (α / 2) _________ 1 + TG 2 (α / 2);

cosα \u003d. 1 - TG 2 (α / 2) __________ 1 + TG 2 (α / 2);

tGα \u003d. 2TG (α / 2) _________ 1 - TG 2 (α / 2).

These formulas may be extremely useful for solving trigonometric tasks of all types. They allow you to realize the principle of "one argument is one function", which allows you to replace variables that reduce complex trigonometric expressions to algebraic. No wonder this substitution is called universal.
The first two formulas learn must. The third one can be obtained by dividing the first two on each other by definition of TGα tangent \u003d sINα ___ COSα.

IX group. Claim formulas.

To deal with this group of trigonometric formulas, fie

X group. Values \u200b\u200bfor main corners.

The values \u200b\u200bof trigonometric functions for the main corners of the first quarter are given.

So do it output: Formulas trigonometry need to know. The bigger, the better. But what to spend your time and effort - to memorize the formulas or on their recovery in the process of solving tasks, everyone should solve independently.

Example of the task of using trigonometry formulas

Solve equation sin5 x.· COS3. x. - SIN8. x.· COS6. x. = 0.

We have two different functions sin () and cos () and four! Different arguments 5. x., 3x., 8x. and 6. x.. Without preliminary transformations, it will not be possible to reduce the simplest types of trigonometric equations. Therefore, we first try to replace the works on the amounts or difference of functions.
We do it the same way as in the example above (see section).

sIN (5. x. + 3x.) + sin (5 x. − 3x.) \u003d 2 · sin5 x.· COS3. x.
sin8. x. + SIN2. x. \u003d 2 · sin5 x.· COS3. x.

sIN (8. x. + 6x.) + sin (8 x. − 6x.) \u003d 2 · sin8 x.· COS6. x.
SIN14. x. + SIN2. x. \u003d 2 · sin8 x.· COS6. x.

Expressing the work from these equalities, we substitute them to the equation. We get:

(SIN8. x. + SIN2. x.) / 2 - (sin14 x. + SIN2. x.)/2 = 0.

We multiply on 2 of both parts of the equation, reveal brackets and give such members

Sin8. x. + SIN2. x. - SIN14. x. - SIN2. x. = 0;
sin8. x. - SIN14. x. = 0.

The equation has simplified significantly, but to solve it so sin8 x. \u003d SIN14. x., therefore, 8. x. = 14x. + T, where t - the period is incorrect, since we do not know the value of this period. Therefore, we use that in the right part of equality it is worth 0, with which it is easy to compare multipliers in any expression.
To decompose sin8 x. - SIN14. x. For multipliers, you need to go from the difference to the work. To do this, you can use the sinus difference formula, or again the formula sum of the sinuses and the oddity of the sinus function (see example in the section).

sin8. x. - SIN14. x. \u003d sin8. x. + sin (-14 x.) \u003d 2 · sin 8x. + (−14x.) __________ 2 · COS. 8x. − (−14x.) __________ 2 \u003d sin (-3 x.) · COS11 x. \u003d -Sin3 x.· COS11 x..

So, equation sin8 x. - SIN14. x. \u003d 0 is equivalent to sin3 equation x.· COS11 x. \u003d 0, which, in turn, is equivalent to the combination of two simple SIN3 equations x. \u003d 0 and COS11 x. \u003d 0. Solving the latter, we get two series of responses
x. 1 \u003d π. n./3, n.εz.
x. 2 \u003d π / 22 + π k./11, k.εz.

If you have detected an error or typical in the text, please inform it to the email address [Email Protected] . I will be very grateful.

The task.
Find the value of x at.

Decision.
Find the value of the function of the function at which it is equal to any value means to determine at what arguments the size of the sine will be exactly as indicated in the condition.
In this case, we need to find out at what values \u200b\u200bthe sinus value will be 1/2. This can be done in several ways.
For example, to use by which to determine at what values \u200b\u200bx the sinus function will be 1/2.
Another way is to use. Let me remind you that the sinus values \u200b\u200blie on the OU axis.
The most common way is to appeal to, especially if we are talking about the values \u200b\u200bof such standard functions as 1/2.
In all cases, you should not forget about one of the most important properties of Sinus - about his period.
Find in the table value 1/2 for sinus and let's see what arguments it corresponds to it. The arguments you are interested in are pi / 6 and 5p / 6.
We write all the roots that satisfy the specified equation. To do this, write to us the unknown argument and one of the values \u200b\u200bof the argument obtained from the table, that is, PI / 6. We write to it, given the period of sine, all the values \u200b\u200bof the argument:

Take the second value, and we do the same steps as in the previous case:

A complete solution of the source equation will be:
and
q. Can take the value of any integer.

On this page you will find all the main trigonometric formulas that will help you solve many exercises, significantly simplifying the expression itself.

Trigonometric formulas - mathematical equality for trigonometric functions that are performed with all valid values \u200b\u200bof the argument.

The formulas are given by relations between the main trigonometric functions - sine, cosine, tangent, Kotangent.

The sine of the angle is the coordinate Y of the point (ordinate) on a single circle. Cosine angle is the coordinate X point (abscissa).

Tangent and Kotangenes are, accordingly, the ratio of sine to cosine and vice versa.
`sin \\ \\ \\ alpha, \\ cos \\ \\ alpha`
`TG \\ \\ ALPHA \u003d \\ FRAC (sin \\ \\ alpha) (COS \\ \\ Alpha),` `\\ alpha \\ ne \\ frac \\ pi2 + \\ pi n, \\ n \\ in z`
`Ctg \\ \\ Alpha \u003d \\ FRAC (cos \\ \\ alpha) (sin \\ \\ alpha),` `\\ alpha \\ ne \\ pi + \\ pi n, \\ n \\ in z`

And the two, which are used less often - sessions, sosekans. They denote the ratios 1 to cosine and sinus.

`sec \\ \\ \\ alpha \u003d \\ frac (1) (cos \\ \\ alpha),` `\\ alpha \\ ne \\ frac \\ pi2 + \\ pi n, \\ n \\ in z`
`COSEC \\ \\ Alpha \u003d \\ FRAC (1) (SIN \\ \\ ALPHA),` `'\\ alpha \\ ne \\ pi + \\ pi n, \\ n \\ in in z`

Of the definitions of trigonometric functions, you can see which signs they have in every quarter. The function of the function depends only on which of the quarters is the argument.

When the argument sign changes with "+" to "-", only the cosine function does not change its value. It is called even. Its graph is symmetrical about the axis of the ordinate.

The remaining functions (sinus, tangent, catangent) are odd. When changing the sign of the argument with "+" to "-" their meaning is also changed to the negative. Their graphs are symmetrical on the start of the coordinates.

`sin (- \\ alpha) \u003d - sin \\ \\ alpha`
`COS (- \\ Alpha) \u003d COS \\ \\ Alpha`
`TG (- \\ Alpha) \u003d - TG \\ \\ Alpha`
`CTG (- \\ Alpha) \u003d - CTG \\ \\ Alpha`

Basic trigonometric identities

Basic trigonometric identities are formulas that establish communication between the trigonometric functions of one angle (`sin \\ \\ alpha, \\ cos \\ \\ alpha, \\ tg \\ \\ alpha, \\ ctg \\ \\ alpha`) and which allow you to find the value of each of these functions through Any famous other.
`sin ^ 2 \\ alpha + cos ^ 2 \\ alpha \u003d 1`
`TG \\ \\ \\ Alpha \\ Cdot CTG \\ \\ Alpha \u003d 1, \\ \\ Alpha \\ NE \\ FRAC (\\ pi n) 2, \\ N \\ in z`
`1 + TG ^ 2 \\ alpha \u003d \\ FRAC 1 (COS ^ 2 \\ Alpha) \u003d sec ^ 2 \\ alpha,` `\\ alpha \\ ne \\ frac \\ pi2 + \\ pi n, \\ n \\ in z`
`1 + CTG ^ 2 \\ alpha \u003d \\ FRAC 1 (sin ^ 2 \\ alpha) \u003d COSEC ^ 2 \\ alpha,` `\\ alpha \\ ne \\ pi n, \\ n \\ in z`

Formulas of the sum and difference of corners of trigonometric functions

The formulas of addition and subtraction of arguments express the trigonometric functions of the sum or the difference of two angles through the trigonometric functions of these angles.
`sin (\\ alpha + \\ Beta) \u003d` `sin \\ \\ alpha \\ cos \\ \\ beta + cos \\ \\ alpha \\ sin \\ \\ Beta`
`sin (\\ Alpha- \\ Beta) \u003d` `sin \\ \\ alpha \\ cos \\ \\ beta-cos \\ \\ alpha \\ sin \\ \\ beta`
`COS (\\ Alpha + \\ Beta) \u003d` `cos \\ \\ alpha \\ cos \\ \\ beta-sin \\ \\ alpha \\ sin \\ \\ Beta`
`COS (\\ Alpha- \\ Beta) \u003d` `cos \\ \\ alpha \\ cos \\ \\ beta + sin \\ \\ alpha \\ sin \\ \\ \\ beta`
`TG (\\ Alpha + \\ Beta) \u003d \\ FRAC (TG \\ \\ ALPHA + TG \\ \\ BETA) (1-TG \\ \\ Alpha \\ TG \\ \\ Beta)`
`TG (\\ Alpha- \\ Beta) \u003d \\ FRAC (TG \\ \\ Alpha-TG \\ \\ Beta) (1 + TG \\ \\ Alpha \\ TG \\ \\ Beta)`
`CTG (\\ Alpha + \\ Beta) \u003d \\ FRAC (CTG \\ \\ Alpha \\ Ctg \\ \\ Beta-1) (CTG \\\\ Beta + CTG \\ \\ Alpha)`
`CTG (\\ Alpha- \\ Beta) \u003d \\ FRAC (CTG \\ \\ Alpha \\ Ctg \\ \\ Beta + 1) (CTG \\ \\ Beta-Ctg \\ \\ Alpha)`

Double corner formulas

`sin \\ 2 \\ alpha \u003d 2 \\ sin \\ \\ alpha \\ cos \\ \\ alpha \u003d` `\\ frac (2 \\ tg \\ \\ alpha) (1 + TG ^ 2 \\ alpha) \u003d \\ FRAC (2 \\ CTG \\ \\ Alpha ) (1 + CTG ^ 2 \\ alpha) \u003d `` \\ FRAC 2 (TG \\ \\ ALPHA + CTG \\ \\ Alpha) `
`cos \\ 2 \\ alpha \u003d cos ^ 2 \\ alpha-sin ^ 2 \\ alpha \u003d` `1-2 \\ sin ^ 2 \\ alpha \u003d 2 \\ cos ^ 2 \\ alpha-1 \u003d` `\\ FRAC (1-TG ^ 2 \\ alpha) (1 + TG ^ 2 \\ Alpha) \u003d \\ FRAC (CTG ^ 2 \\ alpha-1) (CTG ^ 2 \\ Alpha + 1) \u003d `` \\ FRAC (CTG \\ \\ Alpha-TG \\ \\ Alpha) (CTG \\ \\ ALPHA + TG \\ \\ Alpha) `
`TG \\ 2 \\ ALPHA \u003d \\ FRAC (2 \\ TG \\ \\ Alpha) (1-TG ^ 2 \\ alpha) \u003d` `\\ FRAC (2 \\ CTG \\ \\ Alpha) (CTG ^ 2 \\ alpha-1) \u003d` `\\ FRAC 2 (\\ CTG \\ \\ Alpha-TG \\ \\ Alpha)`
`CTG \\ 2 \\ alpha \u003d \\ FRAC (CTG ^ 2 \\ alpha-1) (2 \\ CTG \\ \\ Alpha) \u003d` `\\ FRAC (\\ CTG \\ \\ Alpha-TG \\ \\ Alpha) 2`

Formulas of the triple corner

`sin \\ 3 \\ alpha \u003d 3 \\ sin \\ \\ alpha-4sin ^ 3 \\ alpha`
`cos \\ 3 \\ alpha \u003d 4cos ^ 3 \\ alpha-3 \\ cos \\ \\ alpha`
`TG \\ 3 \\ alpha \u003d \\ FRAC (3 \\ TG \\ \\ Alpha-Tg ^ 3 \\ Alpha) (1-3 \\ TG ^ 2 \\ Alpha)`
`CTG \\ 3 \\ Alpha \u003d \\ FRAC (CTG ^ 3 \\ Alpha-3 \\ Ctg \\ \\ Alpha) (3 \\ CTG ^ 2 \\ alpha-1)`

Formulas of half angle

`sin \\ \\ frac \\ alpha 2 \u003d \\ pm \\ sqrt (\\ FRAC (1-COS \\ \\ ALPHA) 2)`
`COS \\ \\ FRAC \\ ALPHA 2 \u003d \\ PM \\ SQRT (\\ FRAC (1 + COS \\ \\ ALPHA) 2)`
`TG \\ \\ FRAC \\ ALPHA 2 \u003d \\ PM \\ SQRT (\\ FRAC (1-COS \\ \\ ALPHA) (1 + COS \\ \\ ALPHA)) \u003d` `\\ FRAC (SIN \\ \\ ALPHA) (1 + COS \\ \\ `CTG \\ \\ FRAC \\ ALPHA 2 \u003d \\ PM \\ SQRT (\\ FRAC (1 + COS \\ \\ ALPHA) (1-COS \\ \\ Alpha)) \u003d` `\\ FRAC (SIN \\ \\ ALPHA) (1-COS \\ \\ Formulas of half, double and triple arguments express the functions `sin, \\ cos, \\ tg, \\ ctg` of these arguments (` \\ FRAC (\\ Alpha) 2, \\ 2 \\ alpha, \\ 3 \\ alpha, ... `) through these functions Argument `\\ Alpha`.
The conclusion can be obtained from the previous group (addition and subtraction of arguments). For example, a double angle identity is easy to get, replacing `\\ beta` on` \\ alpha`.

Degree reduction formulas

Square formulas (cubes, etc.) of trigonometric functions allow you to move from 2.3, ... degree to trigonometric functions of the first degree, but multiple angles (`\\ alpha, \\ 3 \\ alpha, \\ ...` or `2 \\ alpha, \\ ... `sin ^ 2 \\ alpha \u003d \\ FRAC (1-COS \\ 2 \\ alpha) 2,` `(sin ^ 2 \\ frac \\ alpha 2 \u003d \\ FRAC (1-COS \\ \\ ALPHA) 2)`

`cos ^ 2 \\ alpha \u003d \\ FRAC (1 + COS \\ 2 \\ alpha) 2,` `(COS ^ 2 \\ FRAC \\ Alpha 2 \u003d \\ FRAC (1 + COS \\ \\ ALPHA) 2)`

`Sin ^ 3 \\ alpha \u003d \\ FRAC (3SIN \\ \\ Alpha-Sin \\ 3 \\ Alpha) 4`
`cos ^ 3 \\ alpha \u003d \\ FRAC (3cos \\ \\ alpha + cos \\ 3 \\ alpha) 4`
`sin ^ 4 \\ alpha \u003d \\ FRAC (3-4cos \\ 2 \\ alpha + cos \\ 4 \\ alpha) 8`
`cos ^ 4 \\ alpha \u003d \\ FRAC (3 + 4COS \\ 2 \\ alpha + cos \\ 4 \\ alpha) 8`
Formulas of the sum and difference of trigonometric functions
Formulas are transformations of the amount and difference of trigonometric functions of different arguments into the work.
{!LANG-589fc8f3702621443953dfc504870ff3!}

{!LANG-453760d4629f235c598388d5f6dfd13c!}

{!LANG-e5c21cea6640a9d19a54a10dfbb1bc9d!}

`sin \\ \\ \\ alpha + sin \\ \\ \\ bata \u003d` `2 \\ sin \\ frac (\\ alpha + \\ beta) 2 \\ COS \\ FRAC (\\ Alpha- \\ Beta) 2`
`sin \\ \\ \\ \\ alpha-sin \\ \\ bata \u003d` `2 \\ cos \\ frac (\\ alpha + \\ \\ beta) 2 \\ sin \\ frac (\\ alpha- \\ beta) 2`
`cos \\ \\ \\ \\ alpha + cos \\ \\ \\ bata \u003d` `2 \\ cos \\ frac (\\ alpha + \\ beta) 2 \\ COS \\ FRAC (\\ Alpha- \\ Beta) 2`
`cos \\ \\ \\ alpha-cos \\ \\ beta \u003d` `-2 \\ sin \\ frac (\\ alpha + \\ beta) 2 \\ sin \\ frac (\\ alpha- \\ Beta) 2 \u003d` `2 \\ sin \\ frac (\\ alpha + \\ `TG \\ \\ ALPHA \\ PM TG \\ \\ BETA \u003d \\ FRAC (sin (\\ alpha \\ pm \\ beta)) (cos \\ \\ alpha \\ cos \\ \\ beta)`
`CTG \\ \\ Alpha \\ Pm CTG \\ \\ Beta \u003d \\ FRAC (sin (\\ beta \\ pm \\ alpha)) (sin \\ \\ alpha \\ sin \\ \\ beta)`
`TG \\ \\ ALPHA \\ PM CTG \\ \\ Beta \u003d` `\\ PM \\ FRAC (COS (\\ Alpha \\ Mp \\ Beta)) (COS \\ \\ Alpha \\ sin \\ \\ Beta)`
Here is the conversion of addition and subtracts of the functions of one argument into the work.

`cos \\ \\ \\ alpha + sin \\ \\ alpha \u003d \\ sqrt (2) \\ COS (\\ FRAC (\\ PI) 4- \\ Alpha)`

`cos \\ \\ \\ alpha-sin \\ \\ alpha \u003d \\ sqrt (2) \\ sin (\\ FRAC (\\ PI) 4- \\ Alpha)`
`TG \\ \\ Alpha + CTG \\ \\ Alpha \u003d 2 \\ COSEC \\ 2 \\ Alpha;` `TG \\ \\ Alpha-CTG \\ \\ Alpha \u003d -2 \\ CTG \\ 2 \\ alpha`
The following formulas convert the amount and difference of units and trigonometric function into the work.

`1 + cos \\ \\ alpha \u003d 2 \\ cos ^ 2 \\ FRAC (\\ Alpha) 2`

`1-cos \\ \\ alpha \u003d 2 \\ sin ^ 2 \\ FRAC (\\ Alpha) 2`
`1 + sin \\ \\ alpha \u003d 2 \\ cos ^ 2 (\\ FRAC (\\ PI) 4- \\ FRAC (\\ ALPHA) 2)`
`1-sin \\ \\ alpha \u003d 2 \\ sin ^ 2 (\\ FRAC (\\ PI) 4- \\ FRAC (\\ Alpha) 2)`
`1 \\ pm TG \\ \\ Alpha \u003d \\ FRAC (SIN (\\ FRAC (\\ PI) 4 \\ PM \\ Alpha) (COS \\ FRAC (\\ PI) 4 \\ COS \\ \\ ALPHA) \u003d` `\\ FRAC (\\ SQRT (2) SIN (\\ FRAC (\\ PI) 4 \\ PM \\ Alpha)) (COS \\ \\ Alpha) `
`1 \\ pm tg \\ \\ alpha \\ tg \\ \\ beta \u003d \\ frac (cos (\\ alpha \\ mp \\ beta)) (cos \\ \\ alpha \\ cos \\ \\ Beta);` `\\ ctg \\ \\ alpha \\ ctg \\ \\ Formulas for converting works of functions
Formulas for converting the product of trigonometric functions with the arguments `\\ alpha` and` \\ beta` in the amount (difference) of these arguments.

`sin \\ \\ \\ alpha \\ sin \\ \\ \\ beta \u003d` `\\ FRAC (COS (\\ Alpha - \\ Beta) -COS (\\ Alpha + \\ Beta)) (2)`

`sin \\ alpha \\ cos \\ beta \u003d` `\\ FRAC (sin (\\ alpha - \\ beta) + sin (\\ alpha + \\ beta)) (2)`
`cos \\ \\ \\ alpha \\ cos \\ \\ \\ beta \u003d` `\\ FRAC (COS (\\ Alpha - \\ Beta) + COS (\\ Alpha + \\ Beta)) (2)`
`TG \\ \\ Alpha \\ Tg \\ \\ Beta \u003d` `\\ FRAC (COS (\\ Alpha - \\ Beta) -COS (\\ Alpha + \\ Beta)) (COS (\\ Alpha - \\ Beta) + COS (\\ Alpha + \\ `CTG \\ \\ Alpha \\ Ctg \\ \\ Beta \u003d` `\\ FRAC (COS (\\ Alpha - \\ Beta) + COS (\\ Alpha + \\ Beta)) (COS (\\ Alpha - \\ Beta) -COS (\\ Alpha + \\
{!LANG-17bf6206fa66a1decfbf97e9950fee73!}
{!LANG-fd027f9ee4ac9d7235964700517add4a!}
{!LANG-8ffc9a4aac746f7913fae1e0f1cf7b2a!}
`TG \\ \\ ALPHA \\ CTG \\ \\ Beta \u003d` `\\ FRAC (sin (\\ alpha - \\ beta) + sin (\\ alpha + \\ beta)) (sin (\\ alpha + \\ beta) -sin (\\ alpha - \\ Universal trigonometric substitution

These formulas express trigonometric functions through a half angle tangent.

`sin \\ \\ \\ alpha \u003d \\ FRAC (2TG \\ FRAC (\\ alpha) (2)) (1 + TG ^ (2) \\ FRAC (\\ Alpha) (2)),` `\\ alpha \\ ne \\ pi +2 \\ `cos \\ \\ \\ alpha \u003d \\ FRAC (1 - TG ^ (2) \\ FRAC (\\ Alpha) (2)) (1 + TG ^ (2) \\ FRAC (\\ Alpha) (2)),` `\\ alpha \\ `TG \\ \\ ALPHA \u003d \\ FRAC (2TG \\ FRAC (\\ alpha) (2)) (1 - TG ^ (2) \\ FRAC (\\ alpha) (2)),` `\\ alpha \\ ne \\ pi +2 \\ `Ctg \\ \\ Alpha \u003d \\ 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`
Formulas of the cast
The resulting formulas can be obtained using such properties of trigonometric functions, as frequency, symmetry, shift property for the angle. They allow the functions of an arbitrary angle to convert to the function, the angle of which is in the limit between 0 and 90 degrees.
For angle (`\\ frac (\\ pi) 2 \\ pm \\ alpha`) or (` 90 ^ \\ CIRC \\ PM \\ Alpha`):
`sin (\\ FRAC (\\ PI) 2 - \\ Alpha) \u003d cos \\ \\ alpha;` `sin (\\ frac (\\ pi) 2 + \\ alpha) \u003d cos \\ \\ alpha`

`COS (\\ FRAC (\\ pi) 2 - \\ alpha) \u003d sin \\ \\ alpha;` `COS (\\ FRAC (\\ PI) 2 + \\ Alpha) \u003d - sin \\ \\ alpha`

`TG (\\ FRAC (\\ pi) 2 - \\ alpha) \u003d CTG \\ \\ Alpha;` `TG (\\ FRAC (\\ PI) 2 + \\ Alpha) \u003d - CTG \\ \\ Alpha`

`CTG (\\ FRAC (\\ PI) 2 - \\ Alpha) \u003d TG \\ \\ Alpha;` `CTG (\\ FRAC (\\ PI) 2 + \\ Alpha) \u003d - TG \\ \\ Alpha`
For an angle (`\\ pi \\ pm \\ alpha`) or (` 180 ^ \\ Circ \\ PM \\ Alpha`):
`sin (\\ pi - \\ alpha) \u003d sin \\ \\ alpha;` `sin (\\ pi + \\ alpha) \u003d - sin \\ \\ alpha`
`COS (\\ pi - \\ alpha) \u003d - cos \\ \\ alpha;` `cos (\\ pi + \\ alpha) \u003d - COS \\ \\ ALPHA`
`TG (\\ pi - \\ alpha) \u003d - TG \\ \\ Alpha;` `TG (\\ pi + \\ alpha) \u003d TG \\ \\ Alpha`
`CTG (\\ pi - \\ alpha) \u003d - CTG \\ \\ alpha;` `CTG (\\ pi + \\ alpha) \u003d CTG \\ \\ Alpha`
For angle (`\\ FRAC (3 \\ PI) 2 \\ pm \\ alpha`) or (` 270 ^ \\ CIRC \\ PM \\ Alpha`):
`Sin (\\ FRAC (3 \\ PI) 2 - \\ Alpha) \u003d - cos \\ \\ \\ alpha;` `sin (\\ FRAC (3 \\ PI) 2 + \\ Alpha) \u003d - COS \\ \\ ALPHA`
`COS (\\ FRAC (3 \\ PI) 2 - \\ Alpha) \u003d - sin \\ \\ \\ alpha;` `COS (\\ FRAC (3 \\ PI) 2 + \\ Alpha) \u003d sin \\ \\ alpha`
`TG (\\ FRAC (3 \\ PI) 2 - \\ Alpha) \u003d CTG \\ \\ Alpha;` `TG (\\ FRAC (3 \\ PI) 2 + \\ Alpha) \u003d - CTG \\ \\ Alpha`
`CTG (\\ FRAC (3 \\ PI) 2 - \\ alpha) \u003d TG \\ \\ Alpha;` `` CTG (\\ FRAC (3 \\ PI) 2 + \\ Alpha) \u003d - TG \\ \\ Alpha`
For angle (`2 \\ pi \\ pm \\ alpha`) or (` 360 ^ \\ Circ \\ PM \\ Alpha`):
`sin (2 \\ pi - \\ alpha) \u003d - sin \\ \\ \\ alpha;` `sin (2 \\ pi + \\ alpha) \u003d sin \\ \\ alpha`
`COS (2 \\ pi - \\ alpha) \u003d cos \\ \\ alpha;` `cos (2 \\ pi + \\ alpha) \u003d cos \\ \\ alpha`
{!LANG-865bec17cccb210506f4219cf5f59471!}
{!LANG-933005be39c9c9a504cace44e2e2ef14!}
{!LANG-512a92bffe418564e180a66982465b74!}
{!LANG-3463fc8cb381283b2167dc4fa236ca5c!}
`TG (2 \\ pi - \\ alpha) \u003d - TG \\ \\ Alpha;` `TG (2 \\ Pi + \\ Alpha) \u003d TG \\ \\ Alpha`
`CTG (2 \\ pi - \\ alpha) \u003d - CTG \\ \\ Alpha;` `` CTG (2 \\ pi + \\ alpha) \u003d CTG \\ \\ Alpha`

Expression of one trigonometric functions through other

`sin \\ \\ \\ alpha \u003d \\ pm \\ sqrt (1-cos ^ 2 \\ alpha) \u003d` `\\ FRAC (TG \\ \\ Alpha) (\\ pm \\ sqrt (1 + TG ^ 2 \\ Alpha)) \u003d \\ FRAC 1 ( \\ PM \\ SQRT (1 + CTG ^ 2 \\ Alpha)) `
`cos \\ \\ \\ alpha \u003d \\ pm \\ sqrt (1-sin ^ 2 \\ alpha) \u003d` `\\ FRAC 1 (\\ pm \\ sqrt (1 + TG ^ 2 \\ alpha)) \u003d \\ FRAC (CTG \\ \\ Alpha) ( \\ PM \\ SQRT (1 + CTG ^ 2 \\ Alpha)) `
`TG \\ \\ \\ ALPHA \u003d \\ FRAC (sin \\ \\ alpha) (\\ pm \\ sqrt (1-sin ^ 2 \\ alpha)) \u003d` `\\ FRAC (\\ pm \\ sqrt (1-COS ^ 2 \\ Alpha)) ( COS \\ \\ ALPHA) \u003d \\ FRAC 1 (CTG \\ \\ Alpha) `
`CTG \\ \\ Alpha \u003d \\ FRAC (\\ pm \\ sqrt (1-sin ^ 2 \\ alpha)) (sin \\ \\ \\ alpha) \u003d` `\\ FRAC (COS \\ \\ ALPHA) (\\ PM \\ SQRT (1-COS ^ 2 \\ Alpha)) \u003d \\ FRAC 1 (TG \\ \\ Alpha) `

Trigonometry literally translates as "the measurement of triangles". She begins to study in school, and continues in more detail in universities. Therefore, the basic formulas on trigonometry are needed, starting from the 10th grade, as well as for passing the USE. They denote links between functions, and since these connections are many, then the most formulas are a lot. It is not easy to remember, and it is not necessary - if necessary, everything can be outlined.

Trigonometric formulas are applied in integral terms, as well as trigonometric simplifications, calculations, transformations.