Number value(pronounced "pi") is a mathematical constant equal to the ratio

Denoted by the letter of the Greek alphabet "pi". old name - Ludolf number.

What is pi equal to? In simple cases, it is enough to know the first 3 characters (3.14). But for more

complex cases and where greater accuracy is needed, it is necessary to know more than 3 digits.

What is pi? The first 1000 decimal places of pi are:

3,1415926535 8979323846 2643383279 5028841971 6939937510 5820974944 5923078164 0628620899 8628034825 3421170679 8214808651 3282306647 0938446095 5058223172 5359408128 4811174502 8410270193 8521105559 6446229489 5493038196 4428810975 6659334461 2847564823 3786783165 2712019091 4564856692 3460348610 4543266482 1339360726 0249141273 7245870066 0631558817 4881520920 9628292540 9171536436 7892590360 0113305305 4882046652 1384146951 9415116094 3305727036 5759591953 0921861173 8193261179 3105118548 0744623799 6274956735 1885752724 8912279381 8301194912 9833673362 4406566430 8602139494 6395224737 1907021798 6094370277 0539217176 2931767523 8467481846 7669405132 0005681271 4526356082 7785771342 7577896091 7363717872 1468440901 2249534301 4654958537 1050792279 6892589235 4201995611 2129021960 8640344181 5981362977 4771309960 5187072113 4999999837 2978049951 0597317328 1609631859 5024459455 3469083026 4252230825 3344685035 2619311881 7101000313 7838752886 5875332083 8142061717 7669147303 5982534904 2875546873 1159562863 8823537875 9375195778 1857780532 1712268066 1300192787 6611195909 2164201989...

Under normal conditions, the approximate value of pi can be calculated by following the points,

below:

1. Take a circle, wrap the thread around its edge once.
2. We measure the length of the thread.
3. We measure the diameter of the circle.
4. Divide the length of the thread by the length of the diameter. We got the number pi.

Pi properties.

• pi- irrational number, i.e. the value of pi cannot be expressed exactly in the form

fractions m/n, where m and n are integers. This shows that the decimal representation

pi never ends and it is not periodic.

• pi is a transcendental number, i.e. it cannot be a root of any polynomial with integers

coefficients. In 1882, Professor Königsberg proved the transcendence pi, a

later, professor at the University of Munich Lindemann. Proof simplified

Felix Klein in 1894.

• since in Euclidean geometry the area of ​​a circle and the circumference of a circle are functions of pi,

then the proof of the transcendence of pi put an end to the dispute about the squaring of the circle, which lasted more than

2.5 thousand years.

• pi is an element of the period ring (that is, a computable and arithmetic number).

But no one knows whether it belongs to the ring of periods.

Pi formula.

• François Viet:

• Wallis formula:

• Leibniz series:

• Other rows:

What is the number pi we know and remember from school. It is equal to 3.1415926 and so on... To an ordinary person it is enough to know that this number is obtained by dividing the circumference of a circle by its diameter. But many people know that the number Pi appears in unexpected areas not only in mathematics and geometry, but also in physics. Well, if you delve into the details of the nature of this number, you can see a lot of surprises among the endless series of numbers. Is it possible that Pi hides the deepest secrets of the universe?

Infinite number

The number Pi itself arises in our world as the length of a circle, the diameter of which is equal to one. But, despite the fact that the segment equal to Pi is quite finite, the number Pi starts like 3.1415926 and goes to infinity in rows of numbers that never repeat. The first amazing fact is that this number, used in geometry, cannot be expressed as a fraction of whole numbers. In other words, you cannot write it as a ratio of two numbers a/b. In addition, the number Pi is transcendental. This means that there is no such equation (polynomial) with integer coefficients, the solution of which would be Pi.

The fact that the number Pi is transcendent was proved in 1882 by the German mathematician von Lindemann. It was this proof that became the answer to the question whether it is possible to draw a square with a compass and a ruler, whose area is equal to the area of ​​a given circle. This problem is known as the search for the squaring of a circle, which has troubled mankind since ancient times. It seemed that this problem had a simple solution and was about to be revealed. But it was an incomprehensible property of pi that showed that the problem of squaring a circle has no solution.

For at least four and a half millennia, mankind has been trying to get an increasingly accurate value of pi. For example, in the Bible in the 1st Book of Kings (7:23), the number pi is taken equal to 3.

Remarkable in accuracy, the value of Pi can be found in the pyramids of Giza: the ratio of the perimeter and height of the pyramids is 22/7. This fraction gives an approximate value of Pi, equal to 3.142 ... Unless, of course, the Egyptians set such a ratio by accident. The same value already in relation to the calculation of the number Pi was received in the III century BC by the great Archimedes.

In the Ahmes Papyrus, an ancient Egyptian mathematics textbook that dates back to 1650 BC, Pi is calculated as 3.160493827.

In ancient Indian texts around the 9th century BC, the most accurate value was expressed by the number 339/108, which equaled 3.1388 ...

For almost two thousand years after Archimedes, people have been trying to find ways to calculate pi. Among them were both famous and unknown mathematicians. For example, the Roman architect Mark Vitruvius Pollio, the Egyptian astronomer Claudius Ptolemy, the Chinese mathematician Liu Hui, the Indian sage Ariabhata, the medieval mathematician Leonardo of Pisa, known as Fibonacci, the Arab scientist Al-Khwarizmi, from whose name the word "algorithm" appeared. All of them and many other people were looking for the most accurate methods for calculating Pi, but until the 15th century they never received more than 10 digits after the decimal point due to the complexity of the calculations.

Finally, in 1400, the Indian mathematician Madhava from the Sangamagram calculated Pi with an accuracy of up to 13 digits (although he still made a mistake in the last two).

Number of signs

In the 17th century, Leibniz and Newton discovered the analysis of infinitesimal quantities, which made it possible to calculate pi more progressively - through power series and integrals. Newton himself calculated 16 decimal places, but did not mention this in his books - this became known after his death. Newton claimed that he only calculated Pi out of boredom.

At about the same time, other lesser-known mathematicians also pulled themselves up, proposing new formulas for calculating the number Pi through trigonometric functions.

For example, here is the formula used to calculate Pi by astronomy teacher John Machin in 1706: PI / 4 = 4arctg(1/5) - arctg(1/239). Using methods of analysis, Machin derived from this formula the number Pi with a hundred decimal places.

By the way, in the same 1706, the number Pi received an official designation in the form of a Greek letter: it was used by William Jones in his work on mathematics, taking the first letter of the Greek word “periphery”, which means “circle”. Born in 1707, the great Leonhard Euler popularized this designation, which is now known to any schoolchild.

Before the era of computers, mathematicians were concerned with calculating as many signs as possible. In this regard, sometimes there were curiosities. Amateur mathematician W. Shanks calculated 707 digits of pi in 1875. These seven hundred signs were immortalized on the wall of the Palais des Discoveries in Paris in 1937. However, nine years later, observant mathematicians found that only the first 527 characters were correctly calculated. The museum had to incur decent expenses to correct the mistake - now all the numbers are correct.

When computers appeared, the number of digits of Pi began to be calculated in completely unimaginable orders.

One of the first electronic computers ENIAC, created in 1946, which was huge and generated so much heat that the room warmed up to 50 degrees Celsius, calculated the first 2037 digits of Pi. This calculation took the car 70 hours.

As computers improved, our knowledge of pi went further and further into infinity. In 1958, 10 thousand digits of the number were calculated. In 1987, the Japanese calculated 10,013,395 characters. In 2011, Japanese researcher Shigeru Hondo passed the 10 trillion mark.

Where else can you find Pi?

So, often our knowledge of the number Pi remains at the school level, and we know for sure that this number is indispensable in the first place in geometry.

In addition to the formulas for the length and area of ​​a circle, the number Pi is used in the formulas for ellipses, spheres, cones, cylinders, ellipsoids, and so on: somewhere the formulas are simple and easy to remember, and somewhere they contain very complex integrals.

Then we can meet the number Pi in mathematical formulas, where, at first glance, geometry is not visible. For example, the indefinite integral of 1/(1-x^2) is Pi.

Pi is often used in series analysis. For example, here is a simple series that converges to pi:

1/1 - 1/3 + 1/5 - 1/7 + 1/9 - .... = PI/4

Among series, pi appears most unexpectedly in the well-known Riemann zeta function. It will not be possible to tell about it in a nutshell, we will only say that someday the number Pi will help to find a formula for calculating prime numbers.

And it is absolutely amazing: Pi appears in two of the most beautiful "royal" formulas of mathematics - the Stirling formula (which helps to find the approximate value of the factorial and the gamma function) and the Euler formula (which relates as many as five mathematical constants).

However, the most unexpected discovery awaited mathematicians in probability theory. Pi is also there.

For example, the probability that two numbers are relatively prime is 6/PI^2.

Pi appears in Buffon's 18th-century needle-throwing problem: what is the probability that a needle thrown onto a sheet of paper with a pattern will cross one of the lines. If the length of the needle is L, and the distance between the lines is L, and r > L, then we can approximately calculate the value of Pi using the probability formula 2L/rPI. Just imagine - we can get Pi from random events. And by the way Pi is present in the normal probability distribution, appears in the equation of the famous Gaussian curve. Does this mean that pi is even more fundamental than just the ratio of a circle's circumference to its diameter?

We can meet Pi in physics as well. Pi appears in Coulomb's law, which describes the force of interaction between two charges, in Kepler's third law, which shows the period of revolution of a planet around the Sun, and even occurs in the arrangement of electron orbitals of a hydrogen atom. And, again, the most incredible thing is that the Pi number is hidden in the formula of the Heisenberg uncertainty principle, the fundamental law of quantum physics.

Secrets of Pi

In Carl Sagan's novel "Contact", which is based on the film of the same name, aliens inform the heroine that among the signs of Pi there is a secret message from God. From a certain position, the numbers in the number cease to be random and represent a code in which all the secrets of the Universe are recorded.

This novel actually reflected the riddle that occupies the minds of mathematicians all over the planet: is the number Pi a normal number in which the digits are scattered with the same frequency, or is there something wrong with this number. And although scientists tend to the first option (but cannot prove it), Pi looks very mysterious. A Japanese man once calculated how many times the numbers from 0 to 9 occur in the first trillion digits of pi. And I saw that the numbers 2, 4 and 8 are more common than the rest. This may be one of the hints that Pi is not quite normal, and the numbers in it are really not random.

Let's remember everything that we have read above and ask ourselves, what other irrational and transcendental number is so common in the real world?

And there are other oddities in store. For example, the sum of the first twenty digits of Pi is 20, and the sum of the first 144 digits is equal to the "number of the beast" 666.

Main character The American TV series The Suspect, Professor Finch told students that, due to the infinity of pi, any combination of numbers can occur in it, from the numbers of your date of birth to more complex numbers. For example, in the 762nd position there is a sequence of six nines. This position is called the Feynman point, after the famous physicist who noticed this interesting combination.

We also know that the number Pi contains the sequence 0123456789, but it is located on the 17,387,594,880th digit.

All this means that in the infinity of the number Pi you can find not only interesting combinations of numbers, but also the encoded text of "War and Peace", the Bible, and even the main secret Universe, if it exists.

By the way, about the Bible. The well-known popularizer of mathematics Martin Gardner in 1966 stated that the millionth sign of the number Pi (still unknown at that time) would be the number 5. He explained his calculations by the fact that in the English version of the Bible, in the 3rd book, 14th chapter, 16 -m verse (3-14-16) the seventh word contains five letters. The million figure was received eight years later. It was number five.

Is it worth it after this to assert that the number pi is random?

Mathematicians all over the world eat a piece of cake every year on March 14 - after all, this is the day of Pi, the most famous irrational number. This date is directly related to the number whose first digits are 3.14. Pi is the ratio of the circumference of a circle to its diameter. Since it is irrational, it is impossible to write it as a fraction. This is an infinitely long number. It was discovered thousands of years ago and has been constantly studied ever since, but does Pi have any secrets left? From ancient origins to an uncertain future, here are some of the most interesting facts about pi.

Memorizing Pi

The record for remembering numbers after the decimal point belongs to Rajveer Meena from India, who managed to remember 70,000 digits - he set the record on March 21, 2015. Before that, the record holder was Chao Lu from China, who managed to memorize 67,890 digits - this record was set in 2005. The unofficial record holder is Akira Haraguchi, who videotaped his repetition of 100,000 digits in 2005 and recently posted a video where he manages to remember 117,000 digits. An official record would only become if this video was recorded in the presence of a representative of the Guinness Book of Records, and without confirmation it remains only an impressive fact, but is not considered an achievement. Mathematics enthusiasts love to memorize the number Pi. Many people use various mnemonic techniques, such as poetry, where the number of letters in each word is the same as pi. Each language has its own variants of such phrases, which help to remember both the first few digits and a whole hundred.

There is a Pi language

Fascinated by literature, mathematicians invented a dialect in which the number of letters in all words corresponds to the digits of Pi in exact order. Writer Mike Keith even wrote a book, Not a Wake, which is completely written in the Pi language. Enthusiasts of such creativity write their works in full accordance with the number of letters and the meaning of the numbers. This has no practical application, but is a fairly common and well-known phenomenon in the circles of enthusiastic scientists.

Exponential Growth

Pi is an infinite number, so people, by definition, will never be able to figure out the exact numbers of this number. However, the number of digits after the decimal point has increased greatly since the first use of the Pi. Even the Babylonians used it, but a fraction of three and one eighth was enough for them. Chinese and creators Old Testament and was completely limited to three. By 1665, Sir Isaac Newton had calculated 16 digits of pi. By 1719, French mathematician Tom Fante de Lagny had calculated 127 digits. The advent of computers has radically improved man's knowledge of Pi. From 1949 to 1967 the number known to man numbers skyrocketed from 2037 to 500,000. Not so long ago, Peter Trueb, a scientist from Switzerland, was able to calculate 2.24 trillion digits of Pi! This took 105 days. Of course, this is not the limit. It is likely that with the development of technology it will be possible to establish an even more accurate figure - since Pi is infinite, there is simply no limit to accuracy, and only the technical features of computer technology can limit it.

Calculating Pi by hand

If you want to find the number yourself, you can use the old-fashioned technique - you will need a ruler, a jar and string, you can also use a protractor and a pencil. The downside to using a jar is that it has to be round, and accuracy will be determined by how well the person can wrap the rope around it. It is possible to draw a circle with a protractor, but this also requires skill and precision, as an uneven circle can seriously distort your measurements. A more accurate method involves the use of geometry. Divide the circle into many segments, like pizza slices, and then calculate the length of a straight line that would turn each segment into an isosceles triangle. The sum of the sides will give an approximate number of pi. The more segments you use, the more accurate the number will be. Of course, in your calculations you will not be able to come close to the results of a computer, nevertheless, these simple experiments allow you to understand in more detail what Pi is in general and how it is used in mathematics.

Discovery of Pi

The ancient Babylonians knew about the existence of the number Pi already four thousand years ago. The Babylonian tablets calculate Pi as 3.125, and the Egyptian mathematical papyrus contains the number 3.1605. In the Bible, the number Pi is given in an obsolete length - in cubits, and the Greek mathematician Archimedes used the Pythagorean theorem to describe Pi, the geometric ratio of the length of the sides of a triangle and the area of ​​\u200b\u200bthe figures inside and outside the circles. Thus, it is safe to say that Pi is one of the most ancient mathematical concepts, although the exact name of this number has appeared relatively recently.

A new take on Pi

Even before pi was related to circles, mathematicians already had many ways to even name this number. For example, in old mathematics textbooks one can find a phrase in Latin, which can be roughly translated as "the quantity that shows the length when the diameter is multiplied by it." The irrational number became famous when the Swiss scientist Leonhard Euler used it in his work on trigonometry in 1737. However, the Greek symbol for pi was still not used - it only happened in a book by the lesser-known mathematician William Jones. He used it as early as 1706, but it was long neglected. Over time, scientists adopted this name, and now this is the most famous version of the name, although before it was also called the Ludolf number.

Is pi normal?

The number pi is definitely strange, but how does it obey the normal mathematical laws? Scientists have already resolved many questions related to this irrational number, but some mysteries remain. For example, it is not known how often all digits are used - the numbers from 0 to 9 should be used in equal proportion. However, statistics can be traced for the first trillion digits, but due to the fact that the number is infinite, it is impossible to prove anything for sure. There are other problems that still elude scientists. It is quite possible that further development science will help to shed light on them, but for the moment this remains beyond the scope of human intelligence.

Pi sounds divine

Scientists cannot answer some questions about the number Pi, however, every year they understand its essence better. Already in the eighteenth century, the irrationality of this number was proved. In addition, it has been proved that the number is transcendental. This means that there is no definite formula that would allow you to calculate pi using rational numbers.

Dissatisfaction with Pi

Many mathematicians are simply in love with Pi, but there are those who believe that these numbers have no special significance. In addition, they claim that the number Tau, which is twice the size of Pi, is more convenient to use as an irrational one. Tau shows the relationship between the circumference and the radius, which, according to some, represents a more logical method of calculation. However, it is impossible to unambiguously determine anything in this matter, and one and the other number will always have supporters, both methods have the right to life, so it's just interesting fact, and not a reason to think that you should not use the number Pi.

π= 3,
1415926535 8979323846 2643383279 5028841971 6939937510 5820974944 5923078164 0628620899 8628034825 3421170679 8214808651 3282306647 0938446095 5058223172 5359408128 4811174502 8410270193 8521105559 6446229489 5493038196 4428810975 6659334461 2847564823 3786783165 2712019091 4564856692 3460348610 4543266482 1339360726 0249141273 7245870066 0631558817 4881520920 9628292540 9171536436 7892590360 0113305305 4882046652 1384146951 9415116094 3305727036 5759591953 0921861173 8193261179 3105118548 0744623799 6274956735 1885752724 8912279381 8301194912 9833673362 4406566430 8602139494 6395224737 1907021798 6094370277 0539217176 2931767523 8467481846 7669405132 0005681271 4526356082 7785771342 7577896091 7363717872 1468440901 2249534301 4654958537 1050792279 6892589235 4201995611 2129021960 8640344181 5981362977 4771309960 5187072113 4999999837 2978049951 0597317328 1609631859 5024459455 3469083026 4252230825 3344685035 2619311881 7101000313 7838752886 5875332083 8142061717 7669147303 5982534904 2875546873 1159562863 8823537875 9375195778 1857780532 1712268066 1300192787 6611195909 2164201989..

Didn't find it? Then look.

In general, it can be not only a phone number, but any information encoded using numbers. For example, if we represent all the works of Alexander Sergeevich Pushkin in digital form, then they were stored in the number Pi even before he wrote them, even before he was born. In principle, they are still stored there. By the way, curses of mathematicians in π are also present, and not only mathematicians. In a word, Pi has everything, even thoughts that will visit your bright head tomorrow, the day after tomorrow, in a year, or maybe in two. This is very hard to believe, but even if we pretend to believe it, it will be even more difficult to get information from there and decipher it. So instead of delving into these numbers, it might be easier to approach the girl you like and ask her for a number? .. But for those who are not looking for easy ways, well, or just interested in what the number Pi is, I offer several ways to calculations. Count on health.

What is the value of Pi? Methods for its calculation:

1. Experimental method. If pi is the ratio of a circle's circumference to its diameter, then perhaps the first and most obvious way to find our mysterious constant would be to manually take all measurements and calculate pi using the formula π=l/d. Where l is the circumference of the circle and d is its diameter. Everything is very simple, you just need to arm yourself with a thread to determine the circumference, a ruler to find the diameter, and, in fact, the length of the thread itself, and, well, a calculator if you have problems with dividing into a column. A saucepan or a jar of cucumbers can act as a measured sample, it doesn’t matter, the main thing? so that the base is a circle.

The considered calculation method is the simplest, but, unfortunately, it has two significant drawbacks that affect the accuracy of the resulting Pi number. Firstly, the error of measuring instruments (in our case, this is a ruler with a thread), and secondly, there is no guarantee that the circle we measure will have correct form. Therefore, it is not surprising that mathematics has given us many other methods for calculating π, where there is no need to make accurate measurements.

2. Leibniz series. There are several infinite series that allow you to accurately calculate the number of pi to a large number of decimal places. One of the simplest series is the Leibniz series. π = (4/1) - (4/3) + (4/5) - (4/7) + (4/9) - (4/11) + (4/13) - (4/15) ...
It's simple: we take fractions with 4 in the numerator (this is the one on top) and one number from the sequence of odd numbers in the denominator (this is the one on the bottom), sequentially add and subtract them with each other and get the number Pi. The more iterations or repetitions of our simple actions, the more precisely the result. Simple, but not effective, by the way, it takes 500,000 iterations to get the exact value of Pi to ten decimal places. That is, we will have to divide the unfortunate four as many as 500,000 times, and in addition to this, we will have to subtract and add the results obtained 500,000 times. Want to try?

3. The Nilakanta series. No time fiddling around with Leibniz next? There is an alternative. The Nilakanta series, although it is a bit more complicated, allows us to get the desired result faster. π = 3 + 4/(2*3*4) - 4/(4*5*6) + 4/(6*7*8) - 4/(8*9*10) + 4/(10*11 *12) - (4/(12*13*14) ... I think if you carefully look at the given initial fragment of the series, everything becomes clear, and comments are superfluous. On this we go further.

4. Monte Carlo method A rather interesting method for calculating pi is the Monte Carlo method. Such an extravagant name he got in honor of the city of the same name in the kingdom of Monaco. And the reason for this is random. No, it was not named by chance, it's just that the method is based on random numbers, and what could be more random than the numbers that fall out on the Monte Carlo casino roulettes? The calculation of pi is not the only application of this method, as in the fifties it was used in the calculations of the hydrogen bomb. But let's not digress.

Let's take a square with a side equal to 2r, and inscribe in it a circle with a radius r. Now if you randomly put dots in a square, then the probability P that a point fits into a circle is the ratio of the areas of the circle and the square. P \u003d S cr / S q \u003d πr 2 / (2r) 2 \u003d π / 4.

Now from here we express the number Pi π=4P. It remains only to obtain experimental data and find the probability P as the ratio of hits in the circle N cr to hit the square N sq.. AT general view the calculation formula will look like this: π=4N cr / N sq.

I would like to note that in order to implement this method, it is not necessary to go to the casino, it is enough to use any more or less decent programming language. Well, the accuracy of the results will depend on the number of points set, respectively, the more, the more accurate. I wish you good luck 😉

Tau number (instead of conclusion).

People who are far from mathematics most likely do not know, but it so happened that the number Pi has a brother who is twice as large as it. This is the number Tau(τ), and if Pi is the ratio of the circumference to the diameter, then Tau is the ratio of that length to the radius. And today there are proposals by some mathematicians to abandon the number Pi and replace it with Tau, since this is in many ways more convenient. But so far these are only proposals, and as Lev Davidovich Landau said: "A new theory begins to dominate when the supporters of the old one die out."

March 14 is declared the day of the number "Pi", since this date contains the first three digits of this constant.

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)

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")