Centered 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 right triangle, equal to the ratio of the length of the opposite side |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

Where n- whole.

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

The tangent and cotangent functions are odd.

Areas of definition and values, increasing, decreasing

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 ( n- whole).

	y = tg x	y = ctg x
Scope and continuity
Range of values	-∞ < y < +∞	-∞ < y < +∞
Increasing		-
Descending	-
Extremes	-	-
Zeros, y = 0
Intercept points with the ordinate axis, x = 0	y = 0	-

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

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 And 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 .
Or according to Laplace's formula:

Inverse functions

The inverse functions of tangent and cotangent are arctangent and arccotangent, respectively.

Arctangent, arctg

, Where n- whole.

Arccotangent, arcctg

, Where n- whole.

Used literature:
I.N. Bronstein, K.A. Semendyaev, Handbook of mathematics for engineers and college students, “Lan”, 2009.
G. Korn, Handbook of Mathematics for Scientists and Engineers, 2012.

Allows you to establish a number of characteristic results - properties of sine, cosine, tangent and cotangent. In this article we will look at three main properties. The first of them indicates the signs of the sine, cosine, tangent and cotangent of the angle α depending on the angle of which coordinate quarter is α. Next, we will consider the property of periodicity, which establishes the invariance of the values ​​of sine, cosine, tangent and cotangent of the angle α when this angle changes by an integer number of revolutions. The third property expresses the relationship between the values ​​of sine, cosine, tangent and cotangent of opposite angles α and −α.

If you are interested in the properties of the functions sine, cosine, tangent and cotangent, then you can study them in the corresponding section of the article.

Page navigation.

Signs of sine, cosine, tangent and cotangent by quarters

Below in this paragraph the phrase “angle of I, II, III and IV coordinate quarter” will appear. Let's explain what these angles are.

Let's take a unit circle, mark the starting point A(1, 0) on it, and rotate it around the point O by an angle α, and we will assume that we will get to the point A 1 (x, y).

They say that angle α is the angle of the I, II, III, IV coordinate quadrant, if point A 1 lies in the I, II, III, IV quarters, respectively; if the angle α is such that point A 1 lies on any of the coordinate lines Ox or Oy, then this angle does not belong to any of the four quarters.

For clarity, here is a graphic illustration. The drawings below show rotation angles of 30, −210, 585, and −45 degrees, which are the angles of the I, II, III, and IV coordinate quarters, respectively.

Angles 0, ±90, ±180, ±270, ±360, … degrees do not belong to any of the coordinate quarters.

Now let's figure out what signs have the values ​​of sine, cosine, tangent and cotangent of the angle of rotation α, depending on which quadrant angle α is.

For sine and cosine this is easy to do.

By definition, the sine of angle α is the ordinate of point A 1. Obviously, in the I and II coordinate quarters it is positive, and in the III and IV quarters it is negative. Thus, the sine of angle α has a plus sign in the 1st and 2nd quarters, and a minus sign in the 3rd and 6th quarters.

In turn, the cosine of the angle α is the abscissa of point A 1. In the I and IV quarters it is positive, and in the II and III quarters it is negative. Consequently, the values ​​of the cosine of the angle α in the I and IV quarters are positive, and in the II and III quarters they are negative.

To determine the signs by quarters of tangent and cotangent, you need to remember their definitions: tangent is the ratio of the ordinate of point A 1 to the abscissa, and cotangent is the ratio of the abscissa of point A 1 to the ordinate. Then from rules for dividing numbers with the same and different signs it follows that tangent and cotangent have a plus sign when the abscissa and ordinate signs of point A 1 are the same, and have a minus sign when the abscissa and ordinate signs of point A 1 are different. Consequently, the tangent and cotangent of the angle have a + sign in the I and III coordinate quarters, and a minus sign in the II and IV quarters.

Indeed, for example, in the first quarter both the abscissa x and the ordinate y of point A 1 are positive, then both the quotient x/y and the quotient y/x are positive, therefore, tangent and cotangent have + signs. And in the second quarter, the abscissa x is negative, and the ordinate y is positive, therefore both x/y and y/x are negative, hence the tangent and cotangent have a minus sign.

Let's move on to the next property of sine, cosine, tangent and cotangent.

Periodicity property

Now we will look at perhaps the most obvious property of sine, cosine, tangent and cotangent of an angle. It is as follows: when the angle changes by an integer number of full revolutions, the values ​​of the sine, cosine, tangent and cotangent of this angle do not change.

This is understandable: when the angle changes by an integer number of revolutions, we will always get from the starting point A to point A 1 on the unit circle, therefore, the values ​​of sine, cosine, tangent and cotangent remain unchanged, since the coordinates of point A 1 are unchanged.

Using formulas, the considered property of sine, cosine, tangent and cotangent can be written as follows: sin(α+2·π·z)=sinα, cos(α+2·π·z)=cosα, tan(α+2·π· z)=tgα , ctg(α+2·π·z)=ctgα , where α is the angle of rotation in radians, z is any , absolute value which indicates the number of full revolutions by which the angle α changes, and the sign of the number z indicates the direction of rotation.

If the rotation angle α is specified in degrees, then the indicated formulas will be rewritten as sin(α+360° z)=sinα , cos(α+360° z)=cosα , tg(α+360° z)=tgα , ctg(α+360°·z)=ctgα .

Let's give examples of using this property. For example, , because , A . Here's another example: or .

This property, together with reduction formulas, is very often used when calculating the values ​​of sine, cosine, tangent and cotangent of “large” angles.

The considered property of sine, cosine, tangent and cotangent is sometimes called the property of periodicity.

Properties of sines, cosines, tangents and cotangents of opposite angles

Let A 1 be the point obtained by rotating the initial point A(1, 0) around point O by an angle α, and point A 2 be the result of rotating point A by an angle −α, opposite to angle α.

The property of sines, cosines, tangents and cotangents of opposite angles is based on a fairly obvious fact: the points A 1 and A 2 mentioned above either coincide (at) or are located symmetrically relative to the Ox axis. That is, if point A 1 has coordinates (x, y), then point A 2 will have coordinates (x, −y). From here, using the definitions of sine, cosine, tangent and cotangent, we write the equalities and .
Comparing them, we come to relationships between sines, cosines, tangents and cotangents of opposite angles α and −α of the form.
This is the property under consideration in the form of formulas.

Let's give examples of using this property. For example, the equalities and .

It remains only to note that the property of sines, cosines, tangents and cotangents of opposite angles, like the previous property, is often used when calculating the values ​​of sine, cosine, tangent and cotangent, and allows you to completely avoid negative angles.

References.

• 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
• 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.
• 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.
• Gusev V. A., Mordkovich A. G. Mathematics (a manual for those entering technical schools): Proc. allowance.- M.; Higher school, 1984.-351 p., ill.

Table of values trigonometric functions

Note. This table of trigonometric function values ​​uses the √ sign to indicate square root. To indicate a fraction, use the symbol "/".

See also useful materials:

For determining the value of a trigonometric function, find it at the intersection of the line indicating the trigonometric function. For example, sine 30 degrees - we look for the column with the heading sin (sine) and find the intersection of this table column with the row “30 degrees”, at their intersection we read the result - one half. Similarly we find cosine 60 degrees, sine 60 degrees (once again, at the intersection of the sin column and the 60 degree line we find the value sin 60 = √3/2), etc. The values ​​of sines, cosines and tangents of other “popular” angles are found in the same way.

Sine pi, cosine pi, tangent pi and other angles in radians

The table below of cosines, sines and tangents is also suitable for finding the value of trigonometric functions whose argument is given in radians. To do this, use the second column of angle values. Thanks to this, you can convert the value of popular angles from degrees to radians. For example, let's find the angle of 60 degrees in the first line and read its value in radians under it. 60 degrees is equal to π/3 radians.

The number pi unambiguously expresses the dependence of the circumference on degree measure corner. Thus, pi radians are equal to 180 degrees.

Any number expressed in terms of pi (radians) can be easily converted to degrees by replacing pi (π) with 180.

Examples:
1. Sine pi.
sin π = sin 180 = 0
thus, the sine of pi is the same as the sine of 180 degrees and it is equal to zero.

2. Cosine pi.
cos π = cos 180 = -1
thus, the cosine of pi is the same as the cosine of 180 degrees and it is equal to minus one.

3. Tangent pi
tg π = tg 180 = 0
thus, tangent pi is the same as tangent 180 degrees and it is equal to zero.

Table of sine, cosine, tangent values ​​for angles 0 - 360 degrees (common values)

angle α value (degrees)	angle α value in radians (via pi)	sin (sinus)	cos (cosine)	tg (tangent)	ctg (cotangent)	sec (secant)	cosec (cosecant)
0	0	0	1	0	-	1	-
15	π/12			2 - √3	2 + √3
30	π/6	1/2	√3/2	1/√3	√3	2/√3	2
45	π/4	√2/2	√2/2	1	1	√2	√2
60	π/3	√3/2	1/2	√3	1/√3	2	2/√3
75	5π/12			2 + √3	2 - √3
90	π/2	1	0	-	0	-	1
105	7π/12		-	- 2 - √3	√3 - 2
120	2π/3	√3/2	-1/2	-√3	-√3/3
135	3π/4	√2/2	-√2/2	-1	-1	-√2	√2
150	5π/6	1/2	-√3/2	-√3/3	-√3
180	π	0	-1	0	-	-1	-
210	7π/6	-1/2	-√3/2	√3/3	√3
240	4π/3	-√3/2	-1/2	√3	√3/3
270	3π/2	-1	0	-	0	-	-1
360	2π	0	1	0	-	1	-

If in the table of values ​​of trigonometric functions a dash is indicated instead of the function value (tangent (tg) 90 degrees, cotangent (ctg) 180 degrees), then for a given value of the degree measure of the angle the function does not have a specific value. If there is no dash, the cell is empty, which means we have not yet entered the required value. We are interested in what queries users come to us for and supplement the table with new values, despite the fact that current data on the values ​​of cosines, sines and tangents of the most common angle values ​​is quite sufficient to solve most problems.

Table of values ​​of trigonometric functions sin, cos, tg for the most popular angles
0, 15, 30, 45, 60, 90 ... 360 degrees
(numeric values ​​“as per Bradis tables”)

angle α value (degrees)	angle α value in radians	sin (sine)	cos (cosine)	tg (tangent)	ctg (cotangent)
0	0
15		0,2588	0,9659	0,2679
30		0,5000		0,5774
45		0,7071
		0,7660
60		0,8660	0,5000	1,7321
	7π/18

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Problem 6.12. Same question as in the previous problem, but for a regular pentagon (hint: see problem 3.5).

Problem 6.13. In Problem 4.8 it was said that as an approximate value of the cosine of a small angle α, we can take the number 1, that is, the value of the cosine function at zero. What if, without further ado, we take 0 = sin 0 as an approximate value for the sine of a small angle α? Why is this bad?

Rice. 6.4. Point M moves along a cycloid.

Problem 6.14. Consider a wheel of radius 1 touching the x-axis at the origin (Fig. 6.4). Let's assume that the wheel rolls along the x-axis in the positive direction with a speed of 1 (that is, during time t its center shifts t to the right).

a) Draw (approximately) a curve that will be described by point M, touching the abscissa axis at the first moment.

b) Find what the abscissa and ordinate of point M will be after time t after the start of the movement.

6.1. Tangent axis

In this section we defined sine and cosine geometrically, as the ordinate and abscissa of a point, and tangent - algebraically, as sin t/ cos t. It is possible, however, to give the tangent a geometric meaning.

To do this, draw through the point with coordinates (1; 0) (the origin on the trigonometric circle) a tangent to the trigonometric circle - a straight line parallel to the axis

Rice. 6.5. Tangent axis.

ordinate Let's call this straight line the tangent axis (Fig. 6.5). This name is justified this way: let M be a point on the trigonometric circle corresponding to the number t. Let us continue the radius SM until it intersects with the tangent axis. Then it turns out that the ordinate of the intersection point is equal to tg t.

In fact, triangles NOS and MP S in Fig. 6.5, obviously

	but similar. From here
	which is what was stated.									or (0; −1), then directly
	If point M has coordinates (0; 1)

May SM is parallel to the tangent axis, and the tangent cannot be determined using our method. This is not surprising: the abscissa of these points is 0, so cos t = 0 for the corresponding values ​​of t, and tg t = sin t/ cos t is not defined.

6.2. Signs of trigonometric functions

Let's figure out at what values ​​of t the sine, cosine and tangent are positive, and at what values ​​they are negative. According to the definition, sin t is the ordinate of a point on the trigonometric circle corresponding to the number t. Therefore sin t > 0 if point t is on