Tables of values ​​of sines (sin), cosines (cos), tangents (tg), cotangents (ctg) are a powerful and useful tool that helps to solve many problems, both theoretical and applied. In this article, we will provide a table of the main trigonometric functions(sines, cosines, tangents and cotangents) for angles 0, 30, 45, 60, 90, ..., 360 degrees (0, π 6, π 3, π 2, . . . , 2 π radians). Separate Bradis tables for sines and cosines, tangents and cotangents will also be shown, with an explanation of how to use them to find the values ​​of basic trigonometric functions.

Table of basic trigonometric functions for angles 0, 30, 45, 60, 90, ..., 360 degrees

Based on the definitions of sine, cosine, tangent and cotangent, you can find the values ​​of these functions for angles of 0 and 90 degrees

sin 0 = 0 , cos 0 = 1 , t g 0 = 0 , cotangent of zero - not defined,

sin 90 ° = 1 , cos 90 ° = 0 , with t g 90 ° = 0 , ninety degrees tangent not defined.

The values ​​of sines, cosines, tangents and cotangents in the course of geometry are defined as the ratio of the sides of a right triangle, the angles of which are 30, 60 and 90 degrees, and also 45, 45 and 90 degrees.

Definition of trigonometric functions for acute angle in a right triangle

Sinus is the ratio of the opposite leg to the hypotenuse.

Cosine is the ratio of the adjacent leg to the hypotenuse.

Tangent- the ratio of the opposite leg to the adjacent one.

Cotangent- the ratio of the adjacent leg to the opposite.

In accordance with the definitions, the values ​​of the functions are found:

sin 30 ° = 1 2 , cos 30 ° = 3 2 , t g 30 ° = 3 3 , c t g 30 ° = 3 , sin 45 ° = 2 2 , cos 45 ° = 2 2 , t g 45 ° = 1 , c t g 45 ° = 1 , sin 60 ° = 3 2 , cos 45 ° = 1 2 , t g 45 ° = 3 , c t g 45 ° = 3 3 .

Let's summarize these values ​​in a table and call it a table of basic values ​​of sine, cosine, tangent and cotangent.

Table of basic values ​​of sines, cosines, tangents and cotangents

One of the important properties of trigonometric functions is periodicity. Based on this property, this table can be expanded using cast formulas. Below we present an extended table of values ​​of the main trigonometric functions for angles 0, 30, 60, ..., 120, 135, 150, 180, ... , 360 degrees (0, π 6 , π 3 , π 2 , . . . , 2 pi radians).

Table of sines, cosines, tangents and cotangents

The periodicity of the sine, cosine, tangent and cotangent allows you to expand this table to arbitrarily large angles. The values ​​collected in the table are used most often in solving problems, so it is recommended to learn them by heart.

How to use the table of basic values ​​​​of trigonometric functions

The principle of using the table of values ​​of sines, cosines, tangents and cotangents is clear on an intuitive level. The intersection of a row and a column gives the function value for a particular angle.

Example. How to use the table of sines, cosines, tangents and cotangents

You need to find out what sin 7 π 6 is equal to

We find a column in the table, the value of the last cell of which is 7 π 6 radians - the same as 210 degrees. Then we select the term of the table in which the values ​​of the sines are presented. At the intersection of a row and a column, we find the desired value:

sin 7 π 6 \u003d - 1 2

Bradis tables

The Bradis table allows you to calculate the value of the sine, cosine, tangent or cotangent with an accuracy of up to 4 decimal places without the use of computer technology. This is a kind of replacement for an engineering calculator.

Reference

Vladimir Modestovich Bradis (1890 - 1975) - Soviet mathematician and teacher, since 1954 a corresponding member of the USSR APN. Tables of four-digit logarithms and natural trigonometric quantities, developed by Bradis, first appeared in 1921.

First, we give the Bradys table for sines and cosines. It allows one to accurately calculate the approximate values ​​of these functions for angles containing an integer number of degrees and minutes. The leftmost column of the table shows degrees, while the top row shows minutes. Note that all values ​​of the Bradys table angles are multiples of six minutes.

Bradis table for sines and cosines

To find the values ​​of the sines and cosines of angles that are not presented in the table, it is necessary to use corrections.

Now we give the Bradys table for tangents and cotangents. It contains the values ​​of the tangents of angles from 0 to 76 degrees, and the cotangents of angles from 14 to 90 degrees.

Bradis table for tangent and cotangent

How to use Bradys tables

Consider the Bradys table for sines and cosines. Everything related to the sinuses is at the top and left. If we need cosines, we look at the right side at the bottom of the table.

To find the values ​​of the sine of the angle, you need to find the intersection of the row containing the required number of degrees in the leftmost cell and the column containing the required number of minutes in the upper cell.

If the exact value of the angle is not in the Bradis table, we resort to the help of corrections. Corrections for one, two and three minutes are given in the rightmost columns of the table. To find the value of the sine of an angle that is not in the table, we find the closest value to it. After that, we add or subtract the correction corresponding to the difference between the angles.

If we are looking for the sine of an angle that is greater than 90 degrees, we first need to use the reduction formulas, and only then - the Bradis table.

Example. How to use the Bradis table

Let it be necessary to find the sine of the angle 17 ° 44 ". According to the table, we find what the sine is 17 ° 42" and add an amendment to its value for two minutes:

17° 44" - 17° 42" = 2" (correct ion needed) sin 17° 44" = 0. 3040 + 0 . 0006 = 0 . 3046

The principle of working with cosines, tangents and cotangents is similar. However, it is important to remember the sign of the corrections.

Important!

When calculating the values ​​of sines, the correction has a positive sign, and when calculating cosines, the correction must be taken with a negative sign.

If you notice a mistake in the text, please highlight it and press Ctrl+Enter

Table of basic trigonometric functions for angles 0, 30, 45, 60, 90, ... degrees

From the trigonometric definitions of the functions $\sin$, $\cos$, $\tan$, and $\cot$, one can find their values ​​for angles $0$ and $90$ degrees:

$\sin⁡0°=0$, $\cos0°=1$, $\tan 0°=0$, $\cot 0°$ not defined;

$\sin90°=1$, $\cos90°=0$, $\cot90°=0$, $\tan 90°$ is not defined.

AT school course geometries in the study of right triangles find the trigonometric functions of the angles $0°$, $30°$, $45°$, $60°$ and $90°$.

The found values ​​of trigonometric functions for the specified angles in degrees and radians respectively ($0$, $\frac(\pi)(6)$, $\frac(\pi)(4)$, $\frac(\pi)(3) $, $\frac(\pi)(2)$) for ease of memorization and use are entered in a table called trigonometric table, table of basic values ​​of trigonometric functions etc.

When using reduction formulas, the trigonometric table can be expanded to an angle of $360°$ and $2\pi$ radians respectively:

Applying the periodicity properties of trigonometric functions, each angle that differs from the already known by $360°$ can be calculated and recorded in a table. For example, the trigonometric function for the angle $0°$ will have the same value for the angle $0°+360°$, and for the angle $0°+2 \cdot 360°$, and for the angle $0°+3 \cdot 360°$ and etc.

Using a trigonometric table, you can determine the values ​​​​of all angles of a unit circle.

In the school geometry course, it is supposed to memorize the basic values ​​​​of trigonometric functions collected in a trigonometric table for the convenience of solving trigonometric problems.

Using a table

In the table, it is enough to find the necessary trigonometric function and the value of the angle or radian for which this function needs to be calculated. At the intersection of the row with the function and the column with the value, we get the desired value of the trigonometric function of the given argument.

In the figure you can see how to find the value $\cos⁡60°$ which is equal to $\frac(1)(2)$.

The extended trigonometric table is used similarly. The advantage of using it is, as already mentioned, the calculation of the trigonometric function of almost any angle. For example, you can easily find the value $\tan 1 380°=\tan (1 380°-360°)=\tan(1 020°-360°)=\tan(660°-360°)=\tan300°$:

Bradis tables of basic trigonometric functions

The ability to calculate the trigonometric function of absolutely any angle value for an integer value of degrees and an integer value of minutes gives the use of Bradis tables. For example, find the value $\cos⁡34°7"$. The tables are divided into 2 parts: the table of $\sin$ and $\cos$ values ​​and the table of $\tan$ and $\cot$ values.

Bradys tables make it possible to obtain an approximate value of trigonometric functions with an accuracy of up to 4 decimal places.

Using Bradis Tables

Using the Bradis tables for sines, we find $\sin⁡17°42"$. To do this, in the column on the left of the table of sines and cosines we find the value of degrees - $17°$, and in the top line we find the value of minutes - $42"$. At their intersection, we get the desired value:

$\sin17°42"=0.304$.

To find the value of $\sin17°44"$, you need to use the correction on the right side of the table. In this case, to the value of $42"$, which is in the table, you need to add a correction for $2"$, which is equal to $0.0006$. We get:

$\sin17°44"=0.304+0.0006=0.3046$.

To find the value of $\sin17°47"$, we also use the correction on the right side of the table, only in this case we take the value of $\sin17°48"$ as a basis and subtract the correction for $1"$:

$\sin17°47"=0.3057-0.0003=0.3054$.

When calculating the cosines, we perform similar actions, but we look at the degrees in the right column, and the minutes in the bottom column of the table. For example, $\cos20°=0.9397$.

There are no corrections for tangent values ​​up to $90°$ and small angle cotangent. For example, let's find $\tan 78°37"$, which according to the table is $4,967$.

Table of values ​​of trigonometric functions

Note. This table of values ​​of trigonometric functions uses the sign √ to denote square root. To denote a fraction - 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, a sine of 30 degrees - we are looking for a column with the heading sin (sine) and find the intersection of this column of the table with the line "30 degrees", at their intersection we read the result - one second. Similarly, we find cosine 60 degrees, sine 60 degrees (once again, at the intersection of the sin (sine) column and the 60 degree row, we find the value sin 60 = √3/2), etc. In the same way, the values ​​of sines, cosines and tangents of other "popular" angles are found.

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

The table of cosines, sines and tangents below 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 60 degree angle in the first line and read its value in radians under it. 60 degrees is equal to π/3 radians.

The number pi uniquely expresses the dependence of the circumference of a circle on the degree measure of the angle. So pi radians equals 180 degrees.

Any number expressed in terms of pi (radian) can be easily converted to degrees by replacing the number 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 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 is equal to minus one.

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

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

If in the table of values ​​of trigonometric functions, instead of the value of the function, a dash is indicated (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 definite value. If there is no dash, the cell is empty, so we have not yet entered the desired value. We are interested in what requests users come to us for and supplement the table with new values, despite the fact that the current data on the values ​​of cosines, sines and tangents of the most common angle values ​​is enough 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
(numerical values ​​"as per Bradis tables")

In the article, we will fully understand what it looks like table of trigonometric values, sine, cosine, tangent and cotangent. Consider the basic value of trigonometric functions, from an angle of 0,30,45,60,90,...,360 degrees. And let's see how to use these tables in calculating the value of trigonometric functions.
First consider table of cosine, sine, tangent and cotangent from an angle of 0, 30, 45, 60, 90,.. degrees. The definition of these quantities makes it possible to determine the value of the functions of angles of 0 and 90 degrees:

sin 0 0 \u003d 0, cos 0 0 \u003d 1. tg 0 0 \u003d 0, the cotangent of 0 0 will be indefinite
sin 90 0 = 1, cos 90 0 =0, ctg90 0 = 0, the tangent of 90 0 will be undefined

If we take right-angled triangles whose angles are from 30 to 90 degrees. We get:

sin 30 0 = 1/2, cos 30 0 = √3/2, tg 30 0 = √3/3, ctg 30 0 = √3
sin 45 0 = √2/2, cos 45 0 = √2/2, tg 45 0 = 1, ctg 45 0 = 1
sin 60 0 = √3/2, cos 60 0 = 1/2, tg 60 0 =√3, ctg 60 0 = √3/3

We represent all the obtained values ​​in the form trigonometric table:

Table of sines, cosines, tangents and cotangents!

If we use the cast formula, our table will increase, values ​​for angles up to 360 degrees will be added. It will look like:

Also, based on the properties of periodicity, the table can be increased if we replace the angles by 0 0 +360 0 *z .... 330 0 +360 0 *z, in which z is an integer. In this table, it is possible to calculate the value of all angles corresponding to points in a single circle.

Let's see clearly how to use the table in the solution.
Everything is very simple. Since the value we need lies at the intersection point of the cells we need. For example, let's take cos of an angle of 60 degrees, in the table it will look like this:

In the final table of the main values ​​of trigonometric functions, we act in the same way. But in this table it is possible to find out how much the tangent from an angle of 1020 degrees will be, it = -√3 Let's check 1020 0 = 300 0 +360 0 *2. Let's find the table.

For a more search for trigonometric angle values ​​accurate to minutes, are used. Detailed instructions on how to use them on the page

Bradis table. For sine, cosine, tangent and cotangent.

The tables of Bradys are divided into several parts, they consist of tables of cosine and sine, tangent and cotangent - which is divided into two parts (tg of an angle up to 90 degrees and ctg of small angles).

Sine and cosine

tg angle starting from 0 0 ending 76 0 , ctg angle starting from 14 0 ending 90 0 .

tg up to 90 0 and ctg small angles.

Let's figure out how to use Bradis tables in solving problems.

Let's find the designation sin (the designation in the column from the left edge) 42 minutes (the designation is on the top line). By crossing we are looking for a designation, it is = 0.3040.

The values ​​of the minutes are indicated with an interval of six minutes, what if the value we need falls within this interval. Let's take 44 minutes, and there are only 42 in the table. We take 42 as a basis and use the additional columns on the right side, take the 2nd correction and add to 0.3040 + 0.0006 we get 0.3046.

With sin 47 min, we take 48 min as a basis and subtract 1 correction from it, i.e. 0.3057 - 0.0003 = 0.3054

When calculating cos, we work similarly to sin, only we take the bottom row of the table as a basis. For example cos 20 0 = 0.9397

Values ​​tg of an angle up to 90 0 and cot of a small angle are correct and there are no corrections in them. For example, find tg 78 0 37min = 4.967


and ctg 20 0 13 min = 25.83

Well, here we have considered the main trigonometric tables. We hope this information was extremely useful for you. Your questions on the tables, if any, be sure to write in the comments!

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TABLE OF VALUES OF TRIGONOMETRIC FUNCTIONS

The table of values ​​of trigonometric functions is compiled for angles of 0, 30, 45, 60, 90, 180, 270 and 360 degrees and their corresponding angles in radians. Of the trigonometric functions, the table shows the sine, cosine, tangent, cotangent, secant and cosecant. For the convenience of solving school examples, the values ​​\u200b\u200bof trigonometric functions in the table are written as a fraction with the preservation of the signs of extracting the square root from numbers, which very often helps to reduce complex mathematical expressions. For tangent and cotangent, the values ​​of some angles cannot be determined. For the values ​​of the tangent and cotangent of such angles, there is a dash in the table of values ​​of trigonometric functions. It is generally accepted that the tangent and cotangent of such angles is equal to infinity. On a separate page are formulas for reducing trigonometric functions.

The table of values ​​for the trigonometric function sine shows the values ​​\u200b\u200bfor the following angles: sin 0, sin 30, sin 45, sin 60, sin 90, sin 180, sin 270, sin 360 in degree measure, which corresponds to sin 0 pi, sin pi / 6 , sin pi / 4, sin pi / 3, sin pi / 2, sin pi, sin 3 pi / 2, sin 2 pi in radian measure of angles. School table of sines.

For the trigonometric cosine function, the table shows the values ​​​​for the following angles: cos 0, cos 30, cos 45, cos 60, cos 90, cos 180, cos 270, cos 360 in degree measure, which corresponds to cos 0 pi, cos pi to 6, cos pi by 4, cos pi by 3, cos pi by 2, cos pi, cos 3 pi by 2, cos 2 pi in radian measure of angles. School table of cosines.

The trigonometric table for the trigonometric function tangent gives values ​​​​for the following angles: tg 0, tg 30, tg 45, tg 60, tg 180, tg 360 in degree measure, which corresponds to tg 0 pi, tg pi / 6, tg pi / 4, tg pi/3, tg pi, tg 2 pi in radian measure of angles. The following values ​​of the trigonometric functions of the tangent are not defined tg 90, tg 270, tg pi/2, tg 3 pi/2 and are considered equal to infinity.

For the trigonometric function cotangent in the trigonometric table, the values ​​​​of the following angles are given: ctg 30, ctg 45, ctg 60, ctg 90, ctg 270 in degree measure, which corresponds to ctg pi / 6, ctg pi / 4, ctg pi / 3, tg pi / 2, tg 3 pi/2 in radian measure of angles. The following values ​​of trigonometric cotangent functions are not defined ctg 0, ctg 180, ctg 360, ctg 0 pi, ctg pi, ctg 2 pi and are considered equal to infinity.

The values ​​of the trigonometric functions secant and cosecant are given for the same angles in degrees and radians as sine, cosine, tangent, cotangent.

The table of values ​​of trigonometric functions of non-standard angles shows the values ​​of sine, cosine, tangent and cotangent for angles in degrees 15, 18, 22.5, 36, 54, 67.5 72 degrees and in radians pi/12, pi/10, pi/ 8, pi/5, 3pi/8, 2pi/5 radians. The values ​​of trigonometric functions are expressed in terms of fractions and square roots to simplify the reduction of fractions in school examples.

Three more monsters of trigonometry. The first is the tangent of 1.5 degrees and a half, or pi divided by 120. The second is the cosine of pi divided by 240, pi/240. The longest is the cosine of pi divided by 17, pi/17.

The trigonometric circle of the values ​​of the sine and cosine functions visually represents the signs of the sine and cosine depending on the magnitude of the angle. Especially for blondes, the cosine values ​​​​are underlined with a green dash to be less confused. The conversion of degrees to radians is also very clearly presented, when radians are expressed through pi.

This trigonometric table presents the values ​​of sine, cosine, tangent and cotangent for angles from 0 zero to 90 ninety degrees in one degree intervals. For the first forty-five degrees, the names of trigonometric functions must be looked at at the top of the table. The first column contains degrees, the values ​​of sines, cosines, tangents and cotangents are written in the next four columns.

For angles from forty-five degrees to ninety degrees, the names of the trigonometric functions are written at the bottom of the table. The last column contains degrees, the values ​​of cosines, sines, cotangents and tangents are written in the previous four columns. You should be careful, because the names of trigonometric functions in the lower part of the trigonometric table are different from the names in the upper part of the table. Sines and cosines are interchanged, just like tangent and cotangent. This is due to the symmetry of the values ​​of trigonometric functions.

The signs of trigonometric functions are shown in the figure above. The sine has positive values ​​from 0 to 180 degrees or from 0 to pi. The negative values ​​of the sine are from 180 to 360 degrees or from pi to 2 pi. Cosine values ​​are positive from 0 to 90 and 270 to 360 degrees, or 0 to 1/2 pi and 3/2 to 2 pi. Tangent and cotangent have positive values ​​from 0 to 90 degrees and from 180 to 270 degrees, corresponding to values ​​from 0 to 1/2 pi and from pi to 3/2 pi. Negative tangent and cotangent values ​​are 90 to 180 degrees and 270 to 360 degrees, or 1/2 pi to pi and 3/2 pi to 2 pi. When determining the signs of trigonometric functions for angles greater than 360 degrees or 2 pi, the periodicity properties of these functions should be used.

The trigonometric functions sine, tangent and cotangent are odd functions. The values ​​of these functions for negative angles will be negative. Cosine is an even trigonometric function - the cosine value for negative angle will be positive. When multiplying and dividing trigonometric functions, you must follow the rules of signs.

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