Table of values of trigonometric functions
Note. This table of values for trigonometric functions uses the √ sign to denote the 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 we 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)
|
angle α (degrees) |
angle α (via pi) |
sin (sinus) |
cos (cosine) |
tg (tangent) |
ctg (cotangent) |
sec (secant) |
cause (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, 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")
| angle value α (degrees) | value of angle α 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 |
Each trigonometric function for a given angle corresponds to a certain value of this function. From the definitions of sine, cosine, tangent and cotangent, it is clear that the value of the sine of an angle is the ordinate of the point to which the initial point of the unit circle passes after it rotates through the angle , the value of the cosine is the abscissa of this point, the value of the tangent is the ratio of the ordinate to the abscissa, and the value of the cotangent is the ratio of the abscissa to the ordinate.
Quite often, when solving problems, it becomes necessary to find the values of the sines, cosines, tangents and cotangents of the indicated angles. For some angles, for example, at 0, 30, 45, 60, 90, ... degrees, it is possible to find the exact values of trigonometric functions, for other angles, finding the exact values is problematic and one has to be content with approximate values.
In this article, we will figure out what principles should be followed when calculating the value of the sine, cosine, tangent or cotangent. Let's list them in order.
Now let's consider each of the listed principles for calculating the values of sines, cosines, tangents and cotangents in detail.
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- Finding the values of sine, cosine, tangent and cotangent by definition. Lines of sines, cosines, tangents and cotangents. Values of sines, cosines, tangents and cotangents of angles of 30, 45 and 60 degrees. Flattening to an angle from 0 to 90 degrees. It is enough to know the value of one of the trigonometric functions. Finding values using trigonometric formulas. What to do in other cases?
Finding the values of sine, cosine, tangent and cotangent by definition
Based on the definition of sine and cosine, you can find the values \u200b\u200bof the sine and cosine of a given angle. To do this, you need to take a unit circle, rotate the starting point A (1, 0) by an angle, after which it will go to point A1. Then the coordinates of the point A1 will give, respectively, the cosine and sine of the given angle. After that, one can calculate the tangent and cotangent of the angle by calculating the ratios of the ordinate to the abscissa and the abscissa to the ordinate, respectively.
By definition, we can calculate the exact values of the sine, cosine, tangent and cotangent of the angles 0, ±90, ±180, ±270, ±360, … degrees (0, ±p/2, ±p, ±3p/2, ±2p, …radian). Let's break these angles into four groups: 360 z degrees (2p z radians), 90+360 z degrees (p/2+2p z radians), 180+360 z degrees (p+2p z radians), and 270 +360 z degrees (3p/2+2p z radians), where z is any integer. Let's depict in the figures where the point A1 will be located, which is obtained by rotating the starting point A by these angles (if necessary, study the material of the article the angle of rotation).
For each of these groups of angles, we find the values of the sine, cosine, tangent and cotangent using the definitions.
As for the other angles other than 0, ±90, ±180, ±270, ±360, … degrees, by definition we can only find approximate values of sine, cosine, tangent and cotangent. For example, let's find the sine, cosine, tangent and cotangent of the angle −52 degrees.
Let's build.
According to the drawing, we find that the abscissa of point A1 is approximately 0.62, and the ordinate is approximately −0.78. In this way, 
and 
. It remains to calculate the values of the tangent and cotangent, we have 
and 
.
It is clear that the more accurately the constructions are performed, the more accurately the approximate values of the sine, cosine, tangent and cotangent of a given angle will be found. It is also clear that finding the values of trigonometric functions, by definition, is not convenient in practice, since it is inconvenient to carry out the described constructions.
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Lines of sines, cosines, tangents and cotangents
Briefly, it is worth dwelling on the so-called lines of sines, cosines, tangents and cotangents. Lines of sines, cosines, tangents and cotangents are called lines depicted together with a unit circle, having a reference point and equal to unity in the introduced rectangular coordinate system, they clearly represent all possible values of sines, cosines, tangents and cotangents. We depict them in the drawing below.
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Values of sines, cosines, tangents and cotangents of angles of 30, 45 and 60 degrees
For angles of 30, 45 and 60 degrees, the exact values of sine, cosine, tangent and cotangent are known. They can be obtained from the definitions of sine, cosine, tangent and cotangent in a right triangle using the Pythagorean theorem.
To obtain the values of trigonometric functions for angles of 30 and 60 degrees, consider a right triangle with these angles, and take it such that the length of the hypotenuse is equal to one. It is known that the leg opposite the angle of 30 degrees is half the hypotenuse, therefore, its length is 1/2. We find the length of the other leg using the Pythagorean theorem: 
.
Since the sine of an angle is the ratio of the opposite leg to the hypotenuse, then 
and 
. In turn, the cosine is the ratio of the adjacent leg to the hypotenuse, then 
and 
. The tangent is the ratio of the opposite leg to the adjacent leg, and the cotangent is the ratio of the adjacent leg to the opposite leg, therefore, 
and 
, as well as 
and 
.
It remains to get the values of sine, cosine, tangent and cotangent for an angle of 45 degrees. Let's turn to a right triangle with angles of 45 degrees (it will be isosceles) and a hypotenuse equal to one. Then, by the Pythagorean theorem, it is easy to check that the lengths of the legs are equal. Now we can calculate the values of sine, cosine, tangent and cotangent as the ratio of the lengths of the corresponding sides of the considered right triangle. We have and 
.
The obtained values of the sine, cosine, tangent and cotangent of the angles of 30, 45 and 60 degrees will be very often used in solving various geometric and trigonometric problems, so we recommend that you remember them. For convenience, we will list them in the table of basic values of sine, cosine, tangent and cotangent.
To conclude this paragraph, we will illustrate the values of the sine, cosine, tangent, and cotangent of angles 30, 45, and 60 using the unit circle and lines of sine, cosine, tangent, and cotangent.
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Flattening to an angle from 0 to 90 degrees
Immediately, we note that it is convenient to find the values of trigonometric functions when the angle is in the range from 0 to 90 degrees (from zero to pi in half radians). If the argument of the trigonometric function, the value of which we need to find, goes beyond the limits from 0 to 90 degrees, then we can always use the reduction formulas to find the value of the trigonometric function, the argument of which will be within the specified limits.
For example, let's find the value of the sine of 210 degrees. By representing 210 as 180+30 or as 270−60, the corresponding reduction formulas reduce our problem from finding the sine of 210 degrees to finding the value of the sine of 30 degrees, or the cosine of 60 degrees.
Let's agree for the future when finding the values of trigonometric functions, always using the reduction formulas, go to angles from the interval from 0 to 90 degrees, unless, of course, the angle is already within these limits.
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It is enough to know the value of one of the trigonometric functions
Basic trigonometric identities establish relationships between the sine, cosine, tangent and cotangent of the same angle. Thus, with their help, we can use the known value of one of the trigonometric functions to find the value of any other function of the same angle.
Let's consider an example solution.
Determine what is the sine of the angle pi by eight, if 
.
First, find what the cotangent of this angle is: 
Now using the formula 
, we can calculate what the square of the sine of the angle pi by eight is equal to, and therefore the desired value of the sine. We have 
It remains only to find the value of the sine. Since the angle pi by eight is the angle of the first coordinate quarter, then the sine of this angle is positive (if necessary, see the section on the theory of the signs of sine, cosine, tangent and cotangent by quarters). In this way, 
.
.
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Finding values using trigonometric formulas
In the two previous paragraphs, we have already begun to cover the issue of finding the values of sine, cosine, tangent and cotangent using trigonometry formulas. Here we only want to say that it is sometimes possible to calculate the required value of a trigonometric function using trigonometric formulas and known values \u200b\u200bof sine, cosine, tangent and cotangent (for example, for angles of 30, 45 and 60 degrees).
For example, using trigonometric formulas, we calculate the value of the tangent of the angle pi by eight, which we used in the previous paragraph to find the value of the sine.
Find the value.
Using the formula for the tangent of a half angle, we can write the following equality 
. We know the values of the cosine of the angle pi by four, so we can immediately calculate the value of the square of the desired tangent: 
.
The angle pi by eight is the angle of the first coordinate quarter, so the tangent of this angle is positive. Consequently, 
.
.
An introductory lesson on trigonometry was presented in the previous presentation. Schoolchildren got acquainted with the concepts of sine, cosine and tangent, how they are denoted, how to find them. An acute angle of some right-angled triangle was considered. Also, they got acquainted with the basic trigonometric identity, which forms the basis for numerous formulas that students will get acquainted with a little later.
This lesson suggests considering certain angles: 45, 30 and 60 degrees. It is necessary to find their sine, cosine and tangent. All three of these angles are acute. It is assumed that we are working with right triangles, as in the previous lesson.
slides 1-2 (Presentation topic "The value of sine, cosine and tangent for angles of 30, 45 and 60 degrees", example)
The first slide of the presentation “The value of sine, cosine and tangent for angles of 30, 45 and 60 degrees” will show students some right-angled triangle, the acute angle of which is 30 degrees. Knowing that one of the angles is right, we can easily calculate the value of the third angle. The sum of all the angles of any triangle is 180 degrees. Eighth grade students should already know about this property. So, in order to find the third unknown angle, it is necessary to subtract 120 degrees from 180 and degrees, which is the sum of the other two sides. The third unknown angle is 60 degrees. This is marked on the drawing.
The author notes that the ratio of the legs of a right-angled triangle ABC is one-half. Where did the author get this number from? The fact is that the leg, which lies opposite the angle of 30 degrees, which can be seen in the figure, is equal to half the hypotenuse of this triangle. This is one of the important properties of right triangles. This ratio is the sine of an angle of 30 degrees. Thus, the sine of the angle of 30 degrees is found.
slides 3-4 (example, table of sines, cosines, tangents)
This ratio is also the cosine for the angle adjacent to the leg, that is, for an angle of 60 degrees. Further, based on the information that was obtained in the previous lesson, you can calculate the remaining tangent by dividing the found sine of a certain angle by the found cosine of the same angle.
The next slide similarly explores the sine, cosine, and tangent of a 45 degree angle. First, the third unknown corner is found. It turns out that the angles at the hypotenuse are equal, that is, the triangle, in addition to being rectangular, is also isosceles. By the Pythagorean theorem, we express the hypotenuse in terms of the legs. Since they are equal, as it turned out, it is possible to replace one leg with another and get a simple product of the number 2 by the square of one of the legs. Further, the author gets rid of irrationality and expresses legs. Thus, there are two legs. Further, using the studied formulas, you can find the sine, and cosine, and the tangent of an angle of 45 degrees.
The last slide shows these values in the form of a table. It is desirable that students write down a table for themselves from a notebook. We can say that it is an analogue of the multiplication table, only trigonometric. It is desirable that students know where these values \u200b\u200bcame from and remember the tables.
This article has collected tables of sines, cosines, tangents and cotangents. First, we give a table of basic values of trigonometric functions, that is, a table of sines, cosines, tangents and cotangents of angles 0, 30, 45, 60, 90, ..., 360 degrees ( 0, π/6, π/4, π/3, π/2, …, 2π radian). After that, we will give a table of sines and cosines, as well as a table of tangents and cotangents by V. M. Bradis, and show how to use these tables when finding the values of trigonometric functions.
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Table of sines, cosines, tangents and cotangents for angles 0, 30, 45, 60, 90, ... degrees
Bibliography.
- Algebra: Proc. for 9 cells. avg. school / Yu. N. Makarychev, N. G. Mindyuk, K. I. Neshkov, S. B. Suvorova; Ed. S. A. Telyakovsky.- M.: Enlightenment, 1990.- 272 p.: Ill.- ISBN 5-09-002727-7
- Bashmakov M.I. Algebra and the beginning of analysis: Proc. for 10-11 cells. avg. school - 3rd ed. - M.: Enlightenment, 1993. - 351 p.: ill. - ISBN 5-09-004617-4.
- Algebra and the beginning of the analysis: Proc. for 10-11 cells. general education institutions / A. N. Kolmogorov, A. M. Abramov, Yu. P. Dudnitsyn and others; Ed. A. N. Kolmogorova.- 14th ed.- M.: Enlightenment, 2004.- 384 p.: ill.- ISBN 5-09-013651-3.
- Gusev V. A., Mordkovich A. G. Mathematics (a manual for applicants to technical schools): Proc. allowance.- M.; Higher school, 1984.-351 p., ill.
- Bradis V. M. Four-digit mathematical tables: For general education. textbook establishments. - 2nd ed. - M.: Bustard, 1999.- 96 p.: ill. ISBN 5-7107-2667-2
