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

Bibliography.

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

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 from 0 to 90 degrees and from 180 to 270 degrees, corresponding to values from 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.

The table of values for the trigonometric sine function shows the values for the following angles
Document
There are reduction formulas on a separate page trigonometricfunctions. IN tablevaluesFortrigonometricfunctionssinusgivenvaluesForthe followingcorners: sin 0, sin 30, sin 45 ...
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
Document
... functions equals functions Images. From this theorem should, What For finding the coordinates U, V, it is enough to calculate function... geometry; polynarious functions(multidimensional analogues of two-dimensional trigonometricfunctions), their properties, tables and application; ...

Simply put, these are vegetables cooked in water according to a special recipe. I will consider two initial components (vegetable salad and water) and the finished result - borscht. Geometrically, it can be thought of as a rectangle, with one side representing lettuce and the other side representing water. The sum of these two sides will indicate borscht. The diagonal and area of such a “borscht” rectangle are purely mathematical concepts and are never used in borscht recipes.

How do lettuce and water turn into borscht from a mathematical point of view? How can the sum of two line segments become trigonometry? To understand this, we need linear angular functions.

You won't find anything about linear angular functions in math textbooks. But without them there can be no mathematics. The laws of mathematics, like the laws of nature, work regardless of whether we know about their existence or not.

Linear angular functions are addition laws. See how algebra turns into geometry and geometry turns into trigonometry.

Is it possible to do without linear angular functions? It’s possible, because mathematicians still manage without them. The trick of mathematicians is that they always tell us only about those problems that they themselves know how to solve, and never talk about those problems that they cannot solve. Look. If we know the result of addition and one term, we use subtraction to find the other term. All. We don’t know other problems and we don’t know how to solve them. What should we do if we only know the result of the addition and do not know both terms? In this case, the result of the addition must be decomposed into two terms using linear angular functions. Next, we ourselves choose what one term can be, and linear angular functions show what the second term should be so that the result of the addition is exactly what we need. There can be an infinite number of such pairs of terms. In everyday life, we get along just fine without decomposing the sum; subtraction is enough for us. But in scientific research into the laws of nature, decomposing a sum into its components can be very useful.

Another law of addition that mathematicians don't like to talk about (another of their tricks) requires that the terms have the same units of measurement. For salad, water, and borscht, these could be units of weight, volume, value, or unit of measurement.

The figure shows two levels of difference for mathematical . The first level is the differences in the field of numbers, which are indicated a, b, c. This is what mathematicians do. The second level is the differences in the field of units of measurement, which are shown in square brackets and indicated by the letter U. This is what physicists do. We can understand the third level - differences in the area of the objects being described. Different objects can have the same number of identical units of measurement. How important this is, we can see in the example of borscht trigonometry. If we add subscripts to the same unit designation for different objects, we can say exactly what mathematical quantity describes a particular object and how it changes over time or due to our actions. Letter W I will designate water with a letter S I'll designate the salad with a letter B- borsch. This is what linear angular functions for borscht will look like.

If we take some part of the water and some part of the salad, together they will turn into one portion of borscht. Here I suggest you take a little break from borscht and remember your distant childhood. Remember how we were taught to put bunnies and ducks together? It was necessary to find how many animals there would be. What were we taught to do then? We were taught to separate units of measurement from numbers and add numbers. Yes, any one number can be added to any other number. This is a direct path to the autism of modern mathematics - we do it incomprehensibly what, incomprehensibly why, and very poorly understand how this relates to reality, because of the three levels of difference, mathematicians operate with only one. It would be more correct to learn how to move from one unit of measurement to another.

Bunnies, ducks, and little animals can be counted in pieces. One common unit of measurement for different objects allows us to add them together. This is a children's version of the problem. Let's look at a similar problem for adults. What do you get when you add bunnies and money? There are two possible solutions here.

First option. We determine the market value of the bunnies and add it to the available amount of money. We got the total value of our wealth in monetary terms.

Second option. You can add the number of bunnies to the number of banknotes we have. We will receive the amount of movable property in pieces.

As you can see, the same addition law allows you to get different results. It all depends on what exactly we want to know.

But let's get back to our borscht. Now we can see what will happen for different angle values of linear angular functions.

The angle is zero. We have salad, but no water. We can't cook borscht. The amount of borscht is also zero. This does not mean at all that zero borscht is equal to zero water. There can be zero borscht with zero salad (right angle).

For me personally, this is the main mathematical proof of the fact that . Zero does not change the number when added. This happens because addition itself is impossible if there is only one term and the second term is missing. You can feel about this as you like, but remember - all mathematical operations with zero were invented by mathematicians themselves, so throw away your logic and stupidly cram the definitions invented by mathematicians: “division by zero is impossible”, “any number multiplied by zero equals zero” , “beyond the puncture point zero” and other nonsense. It is enough to remember once that zero is not a number, and you will never again have a question whether zero is a natural number or not, because such a question loses all meaning: how can something that is not a number be considered a number? It's like asking what color an invisible color should be classified as. Adding a zero to a number is the same as painting with paint that is not there. We waved a dry brush and told everyone that “we painted.” But I digress a little.

The angle is greater than zero but less than forty-five degrees. We have a lot of lettuce, but not enough water. As a result, we will get thick borscht.

The angle is forty-five degrees. We have equal quantities of water and salad. This is the perfect borscht (forgive me, chefs, it's just math).

The angle is greater than forty-five degrees, but less than ninety degrees. We have a lot of water and little salad. You will get liquid borscht.

Right angle. We have water. All that remains of the salad are memories, as we continue to measure the angle from the line that once marked the salad. We can't cook borscht. The amount of borscht is zero. In this case, hold on and drink water while you have it)))

Here. Something like this. I can tell other stories here that would be more than appropriate here.

Two friends had their shares in a common business. After killing one of them, everything went to the other.

The emergence of mathematics on our planet.

All these stories are told in the language of mathematics using linear angular functions. Some other time I will show you the real place of these functions in the structure of mathematics. In the meantime, let's return to borscht trigonometry and consider projections.

Saturday, October 26, 2019

Wednesday, August 7, 2019

Concluding the conversation about, we need to consider an infinite set. The point is that the concept of “infinity” affects mathematicians like a boa constrictor affects a rabbit. The trembling horror of infinity deprives mathematicians of common sense. Here's an example:

The original source is located. Alpha stands for real number. The equal sign in the above expressions indicates that if you add a number or infinity to infinity, nothing will change, the result will be the same infinity. If we take the infinite set of natural numbers as an example, then the considered examples can be represented in this form:

To clearly prove that they were right, mathematicians came up with many different methods. Personally, I look at all these methods as shamans dancing with tambourines. Essentially, they all boil down to the fact that either some of the rooms are unoccupied and new guests are moving in, or that some of the visitors are thrown out into the corridor to make room for guests (very humanly). I presented my view on such decisions in the form of a fantasy story about the Blonde. What is my reasoning based on? Relocating an infinite number of visitors takes an infinite amount of time. After we have vacated the first room for a guest, one of the visitors will always walk along the corridor from his room to the next one until the end of time. Of course, the time factor can be stupidly ignored, but this will be in the category of “no law is written for fools.” It all depends on what we are doing: adjusting reality to mathematical theories or vice versa.

What is an “endless hotel”? An infinite hotel is a hotel that always has any number of empty beds, regardless of how many rooms are occupied. If all the rooms in the endless "visitor" corridor are occupied, there is another endless corridor with "guest" rooms. There will be an infinite number of such corridors. Moreover, the “infinite hotel” has an infinite number of floors in an infinite number of buildings on an infinite number of planets in an infinite number of universes created by an infinite number of Gods. Mathematicians are not able to distance themselves from banal everyday problems: there is always only one God-Allah-Buddha, there is only one hotel, there is only one corridor. So mathematicians are trying to juggle the serial numbers of hotel rooms, convincing us that it is possible to “shove in the impossible.”

I will demonstrate the logic of my reasoning to you using the example of an infinite set of natural numbers. First you need to answer a very simple question: how many sets of natural numbers are there - one or many? There is no correct answer to this question, since we invented numbers ourselves; numbers do not exist in Nature. Yes, Nature is great at counting, but for this she uses other mathematical tools that are not familiar to us. I’ll tell you what Nature thinks another time. Since we invented numbers, we ourselves will decide how many sets of natural numbers there are. Let's consider both options, as befits real scientists.

Option one. “Let us be given” one single set of natural numbers, which lies serenely on the shelf. We take this set from the shelf. That's it, there are no other natural numbers left on the shelf and nowhere to take them. We cannot add one to this set, since we already have it. What if you really want to? No problem. We can take one from the set we have already taken and return it to the shelf. After that, we can take one from the shelf and add it to what we have left. As a result, we will again get an infinite set of natural numbers. You can write down all our manipulations like this:

I wrote down the actions in algebraic notation and in set theory notation, with a detailed listing of the elements of the set. The subscript indicates that we have one and only set of natural numbers. It turns out that the set of natural numbers will remain unchanged only if one is subtracted from it and the same unit is added.

Option two. We have many different infinite sets of natural numbers on our shelf. I emphasize - DIFFERENT, despite the fact that they are practically indistinguishable. Let's take one of these sets. Then we take one from another set of natural numbers and add it to the set we have already taken. We can even add two sets of natural numbers. This is what we get:

The subscripts "one" and "two" indicate that these elements belonged to different sets. Yes, if you add one to an infinite set, the result will also be an infinite set, but it will not be the same as the original set. If you add another infinite set to one infinite set, the result is a new infinite set consisting of the elements of the first two sets.

The set of natural numbers is used for counting in the same way as a ruler is for measuring. Now imagine that you added one centimeter to the ruler. This will be a different line, not equal to the original one.

You can accept or not accept my reasoning - it is your own business. But if you ever encounter mathematical problems, think about whether you are following the path of false reasoning trodden by generations of mathematicians. After all, studying mathematics, first of all, forms a stable stereotype of thinking in us, and only then adds to our mental abilities (or, conversely, deprives us of free-thinking).

pozg.ru

Sunday, August 4, 2019

I was finishing a postscript to an article about and saw this wonderful text on Wikipedia:

We read: "... the rich theoretical basis of the mathematics of Babylon did not have a holistic character and was reduced to a set of disparate techniques, devoid of a common system and evidence base."

Wow! How smart we are and how well we can see the shortcomings of others. Is it difficult for us to look at modern mathematics from the same perspective? Slightly paraphrasing the above text, I personally got the following:

The rich theoretical basis of modern mathematics does not have a holistic character and is reduced to a set of disparate sections, devoid of a common system and evidence base.

I won’t go far to confirm my words - it has a language and conventions that are different from the language and conventions of many other branches of mathematics. The same names in different branches of mathematics can have different meanings. I want to devote a whole series of publications to the most obvious mistakes of modern mathematics. See you soon.

Saturday, August 3, 2019

How to divide a set into subsets? To do this, you need to enter a new unit of measurement that is present in some of the elements of the selected set. Let's look at an example.

May we have plenty A consisting of four people. This set is formed on the basis of “people.” Let us denote the elements of this set by the letter A, the subscript with a number will indicate the serial number of each person in this set. Let's introduce a new unit of measurement "gender" and denote it by the letter b. Since sexual characteristics are inherent in all people, we multiply each element of the set A based on gender b. Notice that our set of “people” has now become a set of “people with gender characteristics.” After this we can divide the sexual characteristics into male bm and women's bw sexual characteristics. Now we can apply a mathematical filter: we select one of these sexual characteristics, no matter which one - male or female. If a person has it, then we multiply it by one, if there is no such sign, we multiply it by zero. And then we use regular school mathematics. Look what happened.

After multiplication, reduction and rearrangement, we ended up with two subsets: the subset of men Bm and a subset of women Bw. Mathematicians reason in approximately the same way when they apply set theory in practice. But they don’t tell us the details, but give us the finished result - “a lot of people consist of a subset of men and a subset of women.” Naturally, you may have a question: how correctly has the mathematics been applied in the transformations outlined above? I dare to assure you that essentially everything was done correctly; it is enough to know the mathematical basis of arithmetic, Boolean algebra and other branches of mathematics. What it is? Some other time I will tell you about this.

As for supersets, you can combine two sets into one superset by selecting the unit of measurement present in the elements of these two sets.

As you can see, units of measurement and ordinary mathematics make set theory a relic of the past. A sign that all is not well with set theory is that mathematicians have come up with their own language and notation for set theory. Mathematicians acted as shamans once did. Only shamans know how to “correctly” apply their “knowledge.” They teach us this “knowledge”.

In conclusion, I want to show you how mathematicians manipulate .

Monday, January 7, 2019

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.
I'll show you the process with an example. We select the “red solid in a pimple” - this is our “whole”. At the same time, we see that these things are with a bow, and there are without a bow. After that, we select part of the “whole” and form a set “with a bow”. This is how shamans get their food by tying their set theory to reality.

Now let's do a little trick. Let’s take “solid with a pimple with a bow” and combine these “wholes” according to color, selecting the red elements. We got a lot of "red". Now the final question: are the resulting sets “with a bow” and “red” the same set or two different sets? Only shamans know the answer. More precisely, they themselves do not know anything, but as they say, so it will be.

This simple example shows that set theory is completely useless when it comes to reality. What's the secret? We formed a set of "red solid with a pimple and a bow." The formation took place in four different units of measurement: color (red), strength (solid), roughness (pimply), decoration (with a bow). Only a set of units of measurement allows us to adequately describe real objects in the language of mathematics. This is what it looks like.

The letter "a" with different indices denotes different units of measurement. The units of measurement by which the “whole” is distinguished at the preliminary stage are highlighted in brackets. The unit of measurement by which the set is formed is taken out of brackets. The last line shows the final result - an element of the set. As you can see, if we use units of measurement to form a set, then the result does not depend on the order of our actions. And this is mathematics, and not the dancing of shamans with tambourines. Shamans can “intuitively” come to the same result, arguing that it is “obvious,” because units of measurement are not part of their “scientific” arsenal.

Using units of measurement, it is very easy to split one set or combine several sets into one superset. Let's take a closer look at the algebra of this process.

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)

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.