Table of values ​​of trigonometric functions

Note. This table of trigonometric function values ​​uses the √ sign to represent the 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 the degree measure of the angle. 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)

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”)

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 the corresponding angle values ​​in vradians. Of the trigonometric functions, the table shows sine, cosine, tangent, cotangent, secant and cosecant. For the convenience of solving school examples, the values ​​of trigonometric functions in the table are written in the form of a fraction while preserving the signs for extracting the square root of 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 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 there are formulas for reducing trigonometric functions.

The table of values ​​for the trigonometric sine function shows the values ​​for the following angles: sin 0, sin 30, sin 45, sin 60, sin 90, sin 180, sin 270, sin 360 in degrees, 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 degrees, which corresponds to cos 0 pi, cos pi by 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 tangent function 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 tangent functions are not defined tan 90, tan 270, tan pi/2, tan 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, tan 3 pi/2 in radian measure of angles. The following values ​​of the 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 make it easier to reduce fractions in school examples.

Three more trigonometry monsters. The first is the tangent of 1.5 one and a half degrees 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 values ​​of the functions sine and cosine visually represents the signs of sine and cosine depending on the magnitude of the angle. Especially for blondes, the cosine values ​​are underlined with a green dash to reduce confusion. The conversion of degrees to radians is also very clearly presented when radians are expressed in terms of pi.

This trigonometric table presents the values ​​of sine, cosine, tangent, and cotangent for angles from 0 zero to 90 ninety degrees at one-degree intervals. For the first forty-five degrees, the names of trigonometric functions should 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 the trigonometric functions at the bottom of the trigonometric table are different from the names at the top 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. Sine has positive values ​​from 0 to 180 degrees, or 0 to pi. Sine has negative values ​​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 ​​of 0 to 90 degrees and 180 to 270 degrees, corresponding to values ​​of 0 to 1/2 pi and pi to 3/2 pi. Negative values ​​of tangent and cotangent are from 90 to 180 degrees and from 270 to 360 degrees, or from 1/2 pi to pi and from 3/2 pi to 2 pi. When determining the signs of trigonometric functions for angles greater than 360 degrees or 2 pi, you should use the periodicity properties of these functions.

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 a negative angle will be positive. Sign rules must be followed when multiplying and dividing trigonometric functions.

1. The table of values ​​for the trigonometric sine function shows the values ​​for the following angles

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   There are reduction formulas on a separate page trigonometricfunctions. IN tablevaluesFortrigonometricfunctionssinusgivenvaluesForthe followingcorners: sin 0, sin 30, sin 45 ...

2. The proposed mathematical apparatus is a complete analogue of complex calculus for n-dimensional hypercomplex numbers with any number of degrees of freedom n and is intended for mathematical modeling of nonlinear

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   ... functions equals functions images. From this theorem should, What For finding the coordinates U, V, it is enough to calculate function... geometry; polynar functions(multidimensional analogues of two-dimensional trigonometricfunctions), their properties, tables and application; ...

3. In the fifth century BC, the ancient Greek philosopher Zeno of Elea formulated his famous aporias, the most famous of which is the “Achilles and the Tortoise” aporia. Here's what it sounds like:

   Let's say Achilles runs ten times faster than the tortoise and is a thousand steps behind it. During the time it takes Achilles to run this distance, the tortoise will crawl a hundred steps in the same direction. When Achilles runs a hundred steps, the tortoise crawls another ten steps, and so on. The process will continue ad infinitum, Achilles will never catch up with the tortoise.

   This reasoning became a logical shock for all subsequent generations. Aristotle, Diogenes, Kant, Hegel, Hilbert... They all considered Zeno's aporia in one way or another. The shock was so strong that " ... discussions continue to this day; the scientific community has not yet been able to come to a common opinion on the essence of paradoxes ... mathematical analysis, set theory, new physical and philosophical approaches were involved in the study of the issue; none of them became a generally accepted solution to the problem..."[Wikipedia, "Zeno's Aporia". Everyone understands that they are being fooled, but no one understands what the deception consists of.

   From a mathematical point of view, Zeno in his aporia clearly demonstrated the transition from quantity to . This transition implies application instead of permanent ones. As far as I understand, the mathematical apparatus for using variable units of measurement has either not yet been developed, or it has not been applied to Zeno’s aporia. Applying our usual logic leads us into a trap. We, due to the inertia of thinking, apply constant units of time to the reciprocal value. From a physical point of view, this looks like time slowing down until it stops completely at the moment when Achilles catches up with the turtle. If time stops, Achilles can no longer outrun the tortoise.

   If we turn our usual logic around, everything falls into place. Achilles runs at a constant speed. Each subsequent segment of his path is ten times shorter than the previous one. Accordingly, the time spent on overcoming it is ten times less than the previous one. If we apply the concept of “infinity” in this situation, then it would be correct to say “Achilles will catch up with the turtle infinitely quickly.”

   How to avoid this logical trap? Remain in constant units of time and do not switch to reciprocal units. In Zeno's language it looks like this:

   In the time it takes Achilles to run a thousand steps, the tortoise will crawl a hundred steps in the same direction. During the next time interval equal to the first, Achilles will run another thousand steps, and the tortoise will crawl a hundred steps. Now Achilles is eight hundred steps ahead of the tortoise.

   This approach adequately describes reality without any logical paradoxes. But this is not a complete solution to the problem. Einstein’s statement about the irresistibility of the speed of light is very similar to Zeno’s aporia “Achilles and the Tortoise”. We still have to study, rethink and solve this problem. And the solution must be sought not in infinitely large numbers, but in units of measurement.

   Another interesting aporia of Zeno tells about a flying arrow:

   A flying arrow is motionless, since at every moment of time it is at rest, and since it is at rest at every moment of time, it is always at rest.

   In this aporia, the logical paradox is overcome very simply - it is enough to clarify that at each moment of time a flying arrow is at rest at different points in space, which, in fact, is motion. Another point needs to be noted here. From one photograph of a car on the road it is impossible to determine either the fact of its movement or the distance to it. To determine whether a car is moving, you need two photographs taken from the same point at different points in time, but you cannot determine the distance from them. To determine the distance to a car, you need two photographs taken from different points in space at one point in time, but from them you cannot determine the fact of movement (of course, you still need additional data for calculations, trigonometry will help you). What I want to draw special attention to is that two points in time and two points in space are different things that should not be confused, because they provide different opportunities for research.

   Wednesday, July 4, 2018

   The differences between set and multiset are described very well on Wikipedia. Let's see.

   As you can see, “there cannot be two identical elements in a set,” but if there are identical elements in a set, such a set is called a “multiset.” Reasonable beings will never understand such absurd logic. This is the level of talking parrots and trained monkeys, who have no intelligence from the word “completely”. Mathematicians act as ordinary trainers, preaching to us their absurd ideas.

   Once upon a time, the engineers who built the bridge were in a boat under the bridge while testing the bridge. If the bridge collapsed, the mediocre engineer died under the rubble of his creation. If the bridge could withstand the load, the talented engineer built other bridges.

   No matter how mathematicians hide behind the phrase “mind me, I’m in the house,” or rather, “mathematics studies abstract concepts,” there is one umbilical cord that inextricably connects them with reality. This umbilical cord is money. Let us apply mathematical set theory to mathematicians themselves.

   We studied mathematics very well and now we are sitting at the cash register, giving out salaries. So a mathematician comes to us for his money. We count out the entire amount to him and lay it out on our table in different piles, into which we put bills of the same denomination. Then we take one bill from each pile and give the mathematician his “mathematical set of salary.” Let us explain to the mathematician that he will receive the remaining bills only when he proves that a set without identical elements is not equal to a set with identical elements. This is where the fun begins.

   First of all, the logic of the deputies will work: “This can be applied to others, but not to me!” Then they will begin to reassure us that bills of the same denomination have different bill numbers, which means they cannot be considered the same elements. Okay, let's count salaries in coins - there are no numbers on the coins. Here the mathematician will begin to frantically remember physics: different coins have different amounts of dirt, the crystal structure and arrangement of atoms is unique for each coin...

   And now I have the most interesting question: where is the line beyond which the elements of a multiset turn into elements of a set and vice versa? Such a line does not exist - everything is decided by shamans, science is not even close to lying here.

   Look here. We select football stadiums with the same field area. The areas of the fields are the same - which means we have a multiset. But if we look at the names of these same stadiums, we get many, because the names are different. As you can see, the same set of elements is both a set and a multiset. Which is correct? And here the mathematician-shaman-sharpist pulls out an ace of trumps from his sleeve and begins to tell us either about a set or a multiset. In any case, he will convince us that he is right.

   To understand how modern shamans operate with set theory, tying it to reality, it is enough to answer one question: how do the elements of one set differ from the elements of another set? I'll show you, without any "conceivable as not a single whole" or "not conceivable as a single whole."

   Sunday, March 18, 2018

   The sum of the digits of a number is a dance of shamans with a tambourine, which has nothing to do with mathematics. Yes, in mathematics lessons we are taught to find the sum of the digits of a number and use it, but that’s why they are shamans, to teach their descendants their skills and wisdom, otherwise shamans will simply die out.

   Do you need proof? Open Wikipedia and try to find the page "Sum of digits of a number." She doesn't exist. There is no formula in mathematics that can be used to find the sum of the digits of any number. After all, numbers are graphic symbols with which we write numbers, and in the language of mathematics the task sounds like this: “Find the sum of graphic symbols representing any number.” Mathematicians cannot solve this problem, but shamans can do it easily.

   Let's figure out what and how we do in order to find the sum of the digits of a given number. And so, let us have the number 12345. What needs to be done in order to find the sum of the digits of this number? Let's consider all the steps in order.

   1. Write down the number on a piece of paper. What have we done? We have converted the number into a graphical number symbol. This is not a mathematical operation.

   2. We cut one resulting picture into several pictures containing individual numbers. Cutting a picture is not a mathematical operation.

   3. Convert individual graphic symbols into numbers. This is not a mathematical operation.

   4. Add the resulting numbers. Now this is mathematics.

   The sum of the digits of the number 12345 is 15. These are the “cutting and sewing courses” from shamans that mathematicians use. But that's not all.

   From a mathematical point of view, it does not matter in which number system we write a number. So, in different number systems the sum of the digits of the same number will be different. In mathematics, the number system is indicated as a subscript to the right of the number. With the large number 12345, I don’t want to fool my head, let’s consider the number 26 from the article about. Let's write this number in binary, octal, decimal and hexadecimal number systems. We won't look at every step under a microscope; we've already done that. Let's look at the result.

   As you can see, in different number systems the sum of the digits of the same number is different. This result has nothing to do with mathematics. It’s the same as if you determined the area of ​​a rectangle in meters and centimeters, you would get completely different results.

   Zero looks the same in all number systems and has no sum of digits. This is another argument in favor of the fact that. Question for mathematicians: how is something that is not a number designated in mathematics? What, for mathematicians nothing exists except numbers? I can allow this for shamans, but not for scientists. Reality is not just about numbers.

   The result obtained should be considered as proof that number systems are units of measurement for numbers. After all, we cannot compare numbers with different units of measurement. If the same actions with different units of measurement of the same quantity lead to different results after comparing them, then this has nothing to do with mathematics.

   What is real mathematics? This is when the result of a mathematical operation does not depend on the size of the number, the unit of measurement used and on who performs this action.
   

   Sign on the door He opens the door and says:

   Oh! Isn't this the women's restroom?
   - Young woman! This is a laboratory for the study of the indephilic holiness of souls during their ascension to heaven! Halo on top and arrow up. What other toilet?

   Female... The halo on top and the arrow down are male.

   If such a work of design art flashes before your eyes several times a day,

   Then it’s not surprising that you suddenly find a strange icon in your car:

   Personally, I make an effort to see minus four degrees in a pooping person (one picture) (a composition of several pictures: a minus sign, the number four, a designation of degrees). And I don’t think this girl is a fool who doesn’t know physics. She just has a strong stereotype of perceiving graphic images. And mathematicians teach us this all the time. Here's an example.

   1A is not “minus four degrees” or “one a”. This is "pooping man" or the number "twenty-six" in hexadecimal notation. Those people who constantly work in this number system automatically perceive a number and a letter as one graphic symbol.

   
   

   This article contains tables of sines, cosines, tangents and cotangents. First, we will provide a table of the basic values ​​of trigonometric functions, that is, a table of sines, cosines, tangents and cotangents of angles of 0, 30, 45, 60, 90, ..., 360 degrees ( 0, π/6, π/4, π/3, π/2, …, 2π radian). After this, 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.

   Page navigation.

   Table of sines, cosines, tangents and cotangents for angles of 0, 30, 45, 60, 90, ... degrees

   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
   • 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.
   • 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.
   • Gusev V. A., Mordkovich A. G. Mathematics (a manual for those entering technical schools): Proc. allowance.- M.; Higher school, 1984.-351 p., ill.
   • Bradis V. M. Four-digit math tables: For general education. textbook establishments. - 2nd ed. - M.: Bustard, 1999.- 96 p.: ill. ISBN 5-7107-2667-2

   In the article, we will fully understand what it looks like table of trigonometric values, sine, cosine, tangent and cotangent. Let's consider the basic meaning 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 values ​​of trigonometric functions.
   First let's look at table of cosine, sine, tangent and cotangent from an angle of 0, 30, 45, 60, 90,... degrees. The definition of these quantities allows us to determine the value of the functions of angles of 0 and 90 degrees:

   sin 0 0 =0, cos 0 0 = 1. tg 00 = 0, cotangent from 00 will be undefined
   sin 90 0 = 1, cos 90 0 =0, ctg90 0 = 0, tangent from 90 0 will be uncertain

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

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

   Let us represent all the obtained values ​​in the form trigonometric table:

   Table of sines, cosines, tangents and cotangents!

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

   

   Also, based on the properties of periodicity, the table can be increased if we replace the angles with 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 look at how to use the table in a solution.
   Everything is very simple. Since the value we need lies at the intersection point of the cells we need. For example, take the cos of an angle of 60 degrees, in the table it will look like:

   

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

   

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

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

   Sine and cosine

   
   
   

   tg of the angle starting from 00 ending with 760, ctg of the angle starting from 140 ending with 900.

   
   
   

   tg up to 900 and ctg of small angles.

   
   

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

   Let's find the designation sin (designation in the column on the left edge) 42 minutes (designation is on the top line). By intersection we look for the designation, it = 0.3040.
   
   The minute values ​​are indicated with an interval of six minutes, what to do if the value we need falls exactly within this interval. Let's take 44 minutes, but 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 amendment and add to 0.3040 + 0.0006 we get 0.3046.
   
   With sin 47 minutes, we take 48 minutes 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
   
   The values ​​of tg 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 13min = 25.83
   
   

   Well, we've looked at the basic trigonometric tables. We hope this information was extremely useful to you. If you have any questions about the tables, be sure to write them in the comments!

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