Who and when for the first time opened the number π, still remains a mystery. It is known that the builders of ancient Babylon have already used them with them when designing. On clinical signs that thousands of years, even the tasks that were proposed to be solved with π are preserved. True, then it was believed that π is equal to three. This is evidenced by the sign found in the city of Suza, in two hundred kilometers from Babylon, where the number π was indicated as 3 1/8.

In the process of calculations, the Babylons found that the radius of the circumference as chord enters it six times, and shared a circle of 360 degrees. And at the same time they made the same with the orbit of the sun. Thus, they decided to believe that in the year 360 days.

In ancient Egypt π was 3.16.
In Ancient India - 3,088.
In Italy, at the turn, the eras believed that π is 3,125.

In antiquity, the earliest mention of π belongs to the famous challenge of the square of the circle, that is, it is impossible to build a square with a circulatory and ruler, the area of \u200b\u200bwhich is equal to the area of \u200b\u200ba certain circle. Archimeda equated π to fraction 22/7.

The closer to the exact value of π came in China. It was calculated in the V century n. e. The famous Chinese astronomer Czu Chun Zhi. It was calculated π quite simple. It was necessary to write the odd numbers twice: 11 33 55, and then, dividing them in half, put the first to the denomoter, and the second in the numerator: 355/113. The result coincides with modern calculations π up to the seventh mark.

Table values \u200b\u200bof trigonometric functions

Note. In this table, the values \u200b\u200bof trigonometric functions use a sign √ to designate a square root. For the designation of the fraction - the symbol "/".

see also Useful materials:

For definitions of trigonometric function, Find it at the crossing line indicating the trigonometric function. For example, a sine of 30 degrees - we are looking for a column with the sin header (sinus) and we find the intersection of this table column with a string "30 degrees", you read the result on their intersection - one second. Similarly found cosine 60. degrees sinus 60. degrees (again, in the intersection of SIN column (sinus) and lines 60 degrees, we find the value Sin 60 \u003d √3 / 2), etc. Similarly, there are values \u200b\u200bof sinuses, cosine and tangents of other "popular" corners.

Sinus pi, cosine pi, tangent pi and other corners in radians

The cosine table below, sinuses and tangents are also suitable for finding the value of trigonometric functions, the argument of which set in radians. To do this, use the second column of the corner values. Due to this, you can translate the value of popular corners from degrees to radians. For example, we will find an angle of 60 degrees in the first row and read its value in radians under it. 60 degrees are equal to π / 3 radians.

The number Pi unambiguously expresses the dependence of the circumference length from the degree of the angle. Thus, Pi radians are 180 degrees.

Any number expressed through PI (radians) can be easily translated into a degree measure, replacing the number Pi (π) to 180.

Examples:
1. Sinus p..
SIN π \u003d SIN 180 \u003d 0
Thus, sinus pi is the same as the sinus is 180 degrees and it is zero.

2. Cosine P..
COS π \u003d COS 180 \u003d -1
Thus, cosine pi is the same as the cosine of 180 degrees and it is equal to minus one.

3. Tangent P.
TG π \u003d TG 180 \u003d 0
Thus, Tangent Pi is the same as the tangent 180 degrees and it is zero.

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

If the table of trigonometric functions is values \u200b\u200binstead of the value of the function, the duct is specified (Tangent (TG) 90 degrees, Cotangent (CTG) 180 degrees) means with this value of the degree of the angle the function does not have a certain value. If the dummy is not - the cell is empty, then we have not yet made the desired value. We are interested in how users come to us and complement the table with new values, despite the fact that current data on cosine values, sinuses and tangents of the most common angles' values \u200b\u200bis quite enough to solve most of the tasks.

Table of values \u200b\u200bof trigonometric functions SIN, COS, TG for the most popular corners
0, 15, 30, 45, 60, 90 ... 360 degrees
(digital values \u200b\u200b"as on Brady's tables")

People passionate about mathematics around the world every year eat on a piece of the fourteenth cake of March - after all, this is the day of the number Pi, the most famous irrational number. This date is directly related to the number, the first numbers of which 3.14. Pi is the ratio of the circumference of the circle to the diameter. Since it is irrational, write it in the form of a fraction is impossible. This is an infinitely long number. He was discovered thousands of years ago and since then they are constantly studying, but did they have any secrets? From ancient origin to an indefinite future, here are some of the most interesting facts about the number Pi.

Memorization of p.

The record in the memorization of the number after the comma belongs to Rajvir mine from India, which was able to remember 70,000 digits - he set the record twenty-first of March 2015. Before that, the record holder was Chao Lu from China, which was able to remember 67,890 digits - this record was delivered in 2005. The unofficial record holder is Akira Haragii, who recorded its repetition of 100,000 digits in the video in 2005 and not so long ago, which he managed to remember 117,000 digits. The official record would only be in the event that 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 learn the pi digit. Many people use various mnemonic techniques, for example, poems, where the number of letters in each word coincides with the pi numbers. In each language, there are similar options for these phrases that help remember both the first few digits and a complete hundred.

There is a language p.

The mathematics enthusized to the literature invented a dialect in which the number of letters in all words corresponds to the pi numbers in exact order. The writer Mike Keith even wrote a book not a wake, which is fully created in the PI language. Enthusiasts of such creativity write their works in full compliance with the number of letters value of numbers. It has no applied application, but is a fairly common and known phenomenon in circles of enthusiastic scientists.

Exponential growth

Pi is an infinite number, so people by definition will never be able to establish accurate numbers of this number. However, the number of numbers after the comma has greatly increased since the first use of PI. We used Babylonians, but they had enough fraction in three integers and one eighth. The Chinese and the creators of the Old Testament were limited to the top three. By 1665, Sir Isaac Newton calculated 16 digits pi. By 1719, the French mathematician Tom Pante de Lanya calculated 127 digits. The appearance of computers radically improved human knowledge about Pi. From 1949 to the 1967th, the number of famous numbers a number rapidly increased from 2037 to 500,000. Not so long ago, Peter Trourb, a scientist from Switzerland, was able to calculate 2.24 trillion digits Pi! It took 105 days. Of course, this is not the limit. It is likely that with the development of technologies it will be possible to establish an even more accurate figure - since pi is infinite, the accuracy limit simply does not exist, and only the technical features of computing techniques can limit it.

Hit Manual calculation

If you want to find a number yourself, you can use old-fashioned technique - you will need a ruler, bank and rope, you can also use the transport and pencil. The minus of the use of banks is that it should be round, and the accuracy will be determined how well the person can wind up the rope around it. You can draw a circle with transport, but it also requires skills and accuracy, since the uneven circle can seriously distort your measurements. A more accurate method involves the use of geometry. Divide the circle into a plurality of segments as pizza on pieces, and then calculate the length of the straight line, which would turn each segment into an anoscele triangle. The sum of the parties will give an approximate number Pi. The more segments you use, the more accurate the number will be. Of course, in its calculations you will not be able to get closer to the results of the computer, however, these simple experiments make it possible to understand more detail that in general is the number Pi and how it is used in mathematics.

Opening P.

Ancient Babylonians knew about the existence of the number of Pi already four thousand years ago. Babylonian signs are calculated by PI as 3,125, and in the Egyptian mathematical papyrus there is a number 3,1605. In the Bible, the PP number is given in an outdated length - in the elbows, and the Greek mathematician Archimedes used to describe the PI Theorem of Pythagora, the geometric ratio of the length of the side of the triangle and the area of \u200b\u200bfigures inside and outside the circles. Thus, it is safe to say that PI is one of the most ancient mathematical concepts, at least the exact name of this number and appeared relatively recently.

New look at pi

Even before the number of PI has become correlated with circles, mathematicians have already many ways even for the name of this number. For example, in ancient textbooks in mathematics, you can find a phrase on Latin, which can be roughly translated as "the amount that shows the length when the diameter is multiplied with it." The irrational number became famous when the Swiss scientist Leonard Euler used him in his works on trigonometry in 1737. Nevertheless, the Greek symbol for Pi was still not used - this happened only in the book of the less famous Mathematics William Jones. He used him already in 1706, but this remained without attention. Over time, scientists have accepted such a name, and now it is the most famous version of the name, although it was also called a ludolphic number.

Is the number of pi?

The number Pi is definitely strange, but how much does it obey normal mathematical laws? Scientists have already allowed many issues related to this irrational number, but some riddles remain. For example, it is not known how often all the numbers are used - the numbers from 0 to 9 should be used in equal proportion. However, according to the first trillions, the figures are traced, but due to the fact that the number is infinite, it is impossible to prove it. There are other problems that still elude scientists. It is possible that the further development of science will help shed light on them, but at the moment it remains outside the human intelligence.

Pi sounds divine

Scientists cannot respond to some questions about the number of pi, nevertheless every year they are all better understanding his essence. Already in the eighteenth century, the irrationality of this number was proved. In addition, it was proved that the number is transcendental. This means that there is no certain formula that would calculate PI with rational numbers.

Dissatisfaction with the number of p.

Many mathematics are just in love with Pi, but there are those who believe that these numbers have no particular importance. In addition, they assure that the number of Tau, which is two times more pi, more convenient to use as irrational. Tau shows the link of the length of the circle and radius, which, according to some, is a more logical calculus method. However, it is impossible to unambiguously determine anything in this matter, and one and another number will always have supporters, both methods have the right to life, so it's just an interesting fact, and not a reason to think that you should not use the number of pi.

If you compare the circles of different sizes other than each other, then you can see the following: the dimensions of different circles are proportional. This means that with an increase in the diameter of the circle, the length of this circle increases at the same number of times. Mathematically write it as possible:

where C1 and C2 are the length of two different circles, and D1 and D2 are their diameters.
This ratio works in the presence of a proportionality coefficient - the constant already familiar to us. From the relationship (1) we can conclude: the length of the circle C is equal to the product of the diameter of this circle to the proportionality non-circumference:

C \u003d π d.

Also, this formula can be written in other form, expressing the diameter D through the radius R of this circle:

C \u003d 2π R.

Just this formula is a conductor into the world of circles for seventh graders.

Since more antiquity, people tried to establish the value of this constant. For example, the residents of Mesopotamia calculated the area of \u200b\u200bthe circle by the formula:

Where π \u003d 3.

In ancient Egypt, the value for π was more accurate. In 2000-1700 BC, the scribe, called Akhmes, was a papyrus, in which we find recipes for solving various practical problems. So, for example, to find the area of \u200b\u200bthe circle, it uses the formula:

What considerations did he get this formula? - Unknown. Probably, based on its observations, however, as other ancient philosophers did.

In the footsteps of Archimedes

Which of the two number is more 22/7 or 3.14?
- They are equal.
- Why?
- Each of them is equal to π.
A. A. Vlasov. From the exam ticket.

Some believe that the fraction 22/7 and the number π is identically equal. But this is a delusion. In addition to the above incorrect response to the exam (see Epigraph), you can also add one very entertaining puzzle. The task says: "Put one match so that equality becomes faithful."

The solution will be such: you need to form a "roof" for two vertical matches on the left, using one of the vertical matches in the denominator on the right. It turns out a visual image of the letter π.

Many know that the approximation π \u003d 22/7 determined the ancient Greek mathematician Archimedes. In honor of this, such an approximation is called "Archimedean" number. Archhimeda managed not only to establish an approximate value for π, but also to find the accuracy of this approximation, namely, find a narrow numerical gap to which the value of π belongs. In one of his works, Archimedes proves the chain of inequalities, which would look like this in modern way:

it can be written easier: 3,140 909< π < 3,1 428 265...

As you can see from inequalities, Archimedes found a rather accurate value with an accuracy of 0.002. The most surprising thing is that he found the first two signs after the comma: 3.14 ... It is exactly that we most often use in simple calculations.

Practical use

Two ride in the train:
- Here look, the rails are straight, the wheels are round.
Where is the knock?
- How come from? Wheels are round, and the area
Circle Pi Er Square, here is a square and knocks!

As a rule, get acquainted with this awesome number in 6-7 grades, but more thoroughly they are engaged in the end of the 8th grade. In this part of the article, we present the main and most important formulas that will use you in solving geometric tasks, only to begin with, we agree to take π for 3.14 for the convenience of counting.

Perhaps the most famous formula among schoolchildren in which π is used, this is the formula of the length and area of \u200b\u200bthe circle. The first - the formula of the area of \u200b\u200bthe circle - is written as follows:

where S is the area of \u200b\u200bthe circumference, R is its radius, D is the diameter of the circle.

The length of the circle, or, as it is sometimes called, the perimeter of the circle is calculated by the formula:

C \u003d 2. π R \u003d π D,

where C is the circumference length, R radius, D is the circle diameter.

It is clear that the diameter D is equal to two radius R.

From the circumference length formula, you can easily find a circle radius:

where D is the diameter, C - the length of the circle, R is the circle radius.

These are the basic formulas that each student must know. It also sometimes has to calculate the area of \u200b\u200bnot the entire circumference, but only its part sectors. Therefore, we present it to you - a formula for calculating the area of \u200b\u200bthe circle sector. She looks like this:

where S is the sector area, R is the radius of the circle, α is the central angle in degrees.

Such a mysterious 3,14

True, it mysteriously. Because in honor of these magical numbers, holidays are arranged, films take off, spend public actions, they write poems and much more.

For example, in 1998, the film of the American director Darren Aronofski called "Pi". The film received many awards.

Every year on March 14 at 1:59:26 Persons interested in mathematics, celebrate the "day of Pi". By the holiday, people prepare a round cake, sit down for a round table and discuss the number Pi, solve problems and puzzles associated with PI.

Attention, this amazing number did not go and poets, an unknown wrote:
It is only necessary to try and remember everything as it is - three, fourteen, fifteen, ninety-two and six.

Let's entertain!

Your attention is offered interesting rebuses with the number of Pi. Sold down the words that are encrypted below.

1. π R

2. π L.

3. π K.

Answers: 1. PIR; 2. inscribed; 3. Pick.