All formulas for trigonometry. Equation sin x = a Formulas for converting products of functions

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

Trigonometric formulas are mathematical equalities for trigonometric functions that are satisfied for all valid values of the argument.

Formulas specify the relationships between the basic trigonometric functions - sine, cosine, tangent, cotangent.

The sine of an angle is the y coordinate of a point (ordinate) on the unit circle. The cosine of an angle is the x coordinate of a point (abscissa).

Tangent and cotangent are, respectively, the ratios 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 used less often - secant, cosecant. They represent the ratios of 1 to 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 definitions of trigonometric functions it is clear what signs they have in each quadrant. The sign of the function depends only on which quadrant the argument is located in.

When changing the sign of the argument from “+” to “-”, only the cosine function does not change its value. It's called even. Its graph is symmetrical about the y-axis.

The remaining functions (sine, tangent, cotangent) are odd. When changing the sign of the argument from “+” to “-”, their value also changes to negative. Their graphs are symmetrical about the origin.

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

Basic trigonometric identities

Basic trigonometric identities are formulas that establish a connection between 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 known other.
`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 angles of trigonometric functions

Formulas for adding and subtracting arguments express trigonometric functions of the sum or difference of two angles in terms of trigonometric functions of these angles.
`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)`

Double angle formulas

`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 angle formulas

`sin \ 3\alpha=3 \ sin \ \alpha-4sin^3 \alpha`
`cos \ 3\alpha=4cos^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 angle 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 these functions argument `\alpha`.

Their conclusion can be obtained from the previous group (addition and subtraction of arguments). For example, double angle identities are easily obtained by replacing `\beta` with `\alpha`.

Degree reduction formulas

Formulas of squares (cubes, etc.) of trigonometric functions allow you to move from 2,3,... degrees to trigonometric functions of the first degree, but multiple angles (`\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 the sum and difference of trigonometric functions

The formulas are transformations of the sum and difference of trigonometric functions of different arguments into a product.

`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 the transformation of addition and subtraction of functions of one argument into a product occurs.

`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 one and a trigonometric function into a product.

`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 converting products of functions

Formulas for converting the product of trigonometric functions with arguments `\alpha` and `\beta` into 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 in terms of the tangent of a half angle.
`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`

Reduction formulas

Reduction formulas can be obtained using such properties of trigonometric functions as periodicity, symmetry, and the property of shifting by a given angle. They allow functions of an arbitrary angle to be converted into functions whose angle is between 0 and 90 degrees.

For angle (`\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 angle (`\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 angle (`\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 angle (`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`

Expressing some trigonometric functions in terms of 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 literally translates to “measuring triangles.” It begins to be studied at school, and continues in more detail at universities. Therefore, basic formulas in trigonometry are needed starting from grade 10, as well as for passing the Unified State Exam. They denote connections between functions, and since there are many of these connections, there are also a lot of formulas themselves. It is not easy to remember them all, and it is not necessary - if necessary, they can all be displayed.

Trigonometric formulas are used in integral calculus, as well as in trigonometric simplifications, calculations, and transformations.

Exercise.
Find the value of x at .

Solution.
Finding the value of the function argument at which it is equal to any value means determining at which arguments the value of the sine will be exactly as indicated in the condition.
In this case, we need to find out at what values the sine value will be equal to 1/2. This can be done in several ways.
For example, use , by which to determine at what values of x the sine function will be equal to 1/2.
Another way is to use . Let me remind you that the values of the sines lie on the Oy axis.
The most common way is to refer to , especially if we are talking about values that are standard for this function, such as 1/2.
In all cases, one should not forget about one of the most important properties of the sine - its period.
Let's find the value 1/2 for sine in the table and see what arguments correspond to it. The arguments we are interested in are Pi / 6 and 5Pi / 6.
Let's write down all the roots that satisfy the given equation. To do this, we write down the unknown argument x that interests us and one of the values of the argument obtained from the table, that is, Pi / 6. We write down for it, taking into account the period of the sine, all the values of the argument:

Let's take the second value and follow the same steps as in the previous case:

The complete solution to the original equation will be:
And
q can take the value of any integer.

To solve some problems, a table of trigonometric identities will be useful, which will make it much easier to transform functions:

The simplest trigonometric identities

The quotient of dividing the sine of an angle alpha by the cosine of the same angle is equal to the tangent of this angle (Formula 1). See also the proof of the correctness of the transformation of the simplest trigonometric identities.
The quotient of dividing the cosine of an angle alpha by the sine of the same angle is equal to the cotangent of the same angle (Formula 2)
The secant of an angle is equal to one divided by the cosine of the same angle (Formula 3)
The sum of the squares of the sine and cosine of the same angle is equal to one (Formula 4). see also the proof of the sum of the squares of cosine and sine.
The sum of one and the tangent of an angle is equal to the ratio of one to the square of the cosine of this angle (Formula 5)
One plus the cotangent of an angle is equal to the quotient of one divided by the sine square of this angle (Formula 6)
The product of tangent and cotangent of the same angle is equal to one (Formula 7).

Converting negative angles of trigonometric functions (even and odd)

In order to get rid of the negative value of the degree measure of an angle when calculating the sine, cosine or tangent, you can use the following trigonometric transformations (identities) based on the principles of even or odd trigonometric functions.

As seen, cosine and the secant is even function, sine, tangent and cotangent are odd functions.

The sine of a negative angle is equal to the negative value of the sine of the same positive angle (minus sine alpha).
The cosine minus alpha will give the same value as the cosine of the alpha angle.
Tangent minus alpha is equal to minus tangent alpha.

Formulas for reducing double angles (sine, cosine, tangent and cotangent of double angles)

If you need to divide an angle in half, or vice versa, move from a double angle to a single angle, you can use the following trigonometric identities:

Double Angle Conversion (sine of a double angle, cosine of a double angle and tangent of a double angle) in single occurs according to the following rules:

Sine of double angle equal to twice the product of the sine and the cosine of a single angle

Cosine of double angle equal to the difference between the square of the cosine of a single angle and the square of the sine of this angle

Cosine of double angle equal to twice the square of the cosine of a single angle minus one

Cosine of double angle equal to one minus double sine squared single angle

Tangent of double angle is equal to a fraction whose numerator is twice the tangent of a single angle, and the denominator is equal to one minus the tangent squared of a single angle.

Cotangent of double angle is equal to a fraction whose numerator is the square of the cotangent of a single angle minus one, and the denominator is equal to twice the cotangent of a single angle

Formulas for universal trigonometric substitution

The conversion formulas below can be useful when you need to divide the argument of a trigonometric function (sin α, cos α, tan α) by two and reduce the expression to the value of half an angle. From the value of α we obtain α/2.

These formulas are called formulas of universal trigonometric substitution. Their value lies in the fact that with their help a trigonometric expression is reduced to expressing the tangent of half an angle, regardless of what trigonometric functions (sin cos tan ctg) were originally in the expression. After this, the equation with the tangent of half an angle is much easier to solve.

Trigonometric identities for half-angle transformations

The following are the formulas for trigonometric conversion of half an angle to its whole value.
The value of the argument of the trigonometric function α/2 is reduced to the value of the argument of the trigonometric function α.

Trigonometric formulas for adding angles

cos (α - β) = cos α cos β + sin α sin β

sin (α + β) = sin α cos β + sin β cos α

sin (α - β) = sin α cos β - sin β cos α
cos (α + β) = cos α cos β - sin α sin β

Tangent and cotangent of the sum of angles alpha and beta can be converted using the following rules for converting trigonometric functions:

Tangent of the sum of angles is equal to a fraction whose numerator is the sum of the tangent of the first and tangent of the second angle, and the denominator is one minus the product of the tangent of the first angle and the tangent of the second angle.

Tangent of angle difference is equal to a fraction whose numerator is equal to the difference between the tangent of the angle being reduced and the tangent of the angle being subtracted, and the denominator is one plus the product of the tangents of these angles.

Cotangent of the sum of angles is equal to a fraction whose numerator is equal to the product of the cotangents of these angles plus one, and the denominator is equal to the difference between the cotangent of the second angle and the cotangent of the first angle.

Cotangent of angle difference is equal to a fraction whose numerator is the product of the cotangents of these angles minus one, and the denominator is equal to the sum of the cotangents of these angles.

These trigonometric identities are convenient to use when you need to calculate, for example, the tangent of 105 degrees (tg 105). If you imagine it as tg (45 + 60), then you can use the given identical transformations of the tangent of the sum of angles, and then simply substitute the tabulated values of tangent 45 and tangent 60 degrees.

Formulas for converting the sum or difference of trigonometric functions

Expressions representing a sum of the form sin α + sin β can be transformed using the following formulas:

Triple angle formulas - converting sin3α cos3α tan3α to sinα cosα tanα

Sometimes it is necessary to transform the triple value of an angle so that the argument of the trigonometric function becomes the angle α instead of 3α.
In this case, you can use the triple angle transformation formulas (identities):

Formulas for converting products of trigonometric functions

If there is a need to transform the product of sines of different angles, cosines of different angles, or even the product of sine and cosine, then you can use the following trigonometric identities:

In this case, the product of the sine, cosine or tangent functions of different angles will be converted into a sum or difference.

Formulas for reducing trigonometric functions

You need to use the reduction table as follows. In the line we select the function that interests us. In the column there is an angle. For example, the sine of the angle (α+90) at the intersection of the first row and the first column, we find out that sin (α+90) = cos α.

|BD|
- length of the arc of a circle with center at point A.

α is the angle expressed in radians. Tangent ( tan α
) is a trigonometric function depending on the angle α between the hypotenuse and the leg of a right triangle, equal to the ratio of the length of the opposite leg |BC| to the length of the adjacent leg |AB| . Cotangent (

ctg α

) is a trigonometric function depending on the angle α between the hypotenuse and the leg of a right triangle, equal to the ratio of the length of the adjacent leg |AB| to the length of the opposite leg |BC| . Tangent

Where
.
;
;
.

n

- whole.

In Western literature, tangent is denoted as follows:
.
Graph of the tangent function, y = tan x
;
;
.

Cotangent

In Western literature, cotangent is denoted as follows:

The following notations are also accepted:

Graph of the cotangent function, y = ctg x Properties of tangent and cotangent Periodicity Functions y = tg x

and y =

ctg x

are periodic with period π.

Parity to the length of the opposite leg |BC| . The tangent and cotangent functions are odd.

	Areas of definition and values, increasing, decreasing Properties of tangent and cotangent	Areas of definition and values, increasing, decreasing Functions y =
The tangent and cotangent functions are continuous in their domain of definition (see proof of continuity). The main properties of tangent and cotangent are presented in the table (
- whole).	-∞ < y < +∞	-∞ < y < +∞
y =		-
Scope and continuity	-
Range of values	-	-
Increasing 0
Descending 0	Areas of definition and values, increasing, decreasing 0	-

Extremes

Zeros, y =

; ;
; ;
;

Intercept points with the ordinate axis, x =

Formulas

Expressions using sine and cosine

Formulas for tangent and cotangent from sum and difference

The remaining formulas are easy to obtain, for example

Product of tangents

Formula for the sum and difference of tangents

;
;

This table presents the values of tangents and cotangents for certain values of the argument.

; .

.
Expressions using complex numbers
.
Expressions through hyperbolic functions

Derivatives

Derivative of the nth order with respect to the variable x of the function:

Deriving formulas for tangent > > > ; for cotangent > > > Integrals And Series expansions To obtain the expansion of the tangent in powers of x, you need to take several terms of the expansion in a power series for the functions

sin x

cos x
and divide these polynomials by each other, . This produces the following formulas. At .
;
;
at .
Where

Bn

- Bernoulli numbers. They are determined either from the recurrence relation:

Where .

, Where to the length of the opposite leg |BC| . Tangent

Arccotangent, arcctg