Ptolemy

Trigonometry … a mystery to most of us. The word comes from the Greek. « Gone » is the angle, as in polygon. The « tri » is the triangle. Metron is the art of measuring. The tri-gono-metry is the art of measuring the angles in the triangle.

The history of trigonometry

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Star map

In the earliest times, when at sea all terrestrial landmarks had disappeared, the last means available to mariners were the stars. Trigonometry was developed early in the history of mankind, because it allowed a reading of heaven essential to navigation.

As stated by Aristotle and Ptolemy, the stars hung in the sphere of « fixed ». It could therefore be relied upon to keep the North. Astronomers, sailors and architects were thus the first developers of this art of the angles of the triangle.

Trigonometry did not stay very long limited to navigation. Soon, architects, engineers and mathematicians quickly understood the interest of this new branch of geometry.

D’Alembert, the great french mathematician of the Encyclopedia, gave his definition of trigonometry in 1751: it is « the art of finding the unknown sides of a triangle by means of those (parts) that knows. « 

The history of trigonometry

Trigonometry – it all begins (as often) in Mesopotamia

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Babylonian dial base 60

It all began in Mesopotamia, specifically to Babylon, 2000 years before Christ. Astronomy was commonly used for interpreting messages from heaven: astronomers are also intermediaries between the king and gods.

The Babylonians developed their math in hexadecimal base. The circle was divided into four sections 90, or 360°.

This hexadecimal system has reached us: the dials of our clocks are still dimensioned according to a Babylonian base.

Archimedes
Archimedes

The Greeks

This  Babylonian legacy, enriched by the knowledge came from the Indus valley (the introduction of zero and the decimal system) has been synthesized and improved by the Greeks, Eudoxus of Cnidus, Archimedes, Euclid, in particular.

The latter were interested in the race of the planets, these celestial objects which, unlike the « fixed » (stars), described in the sky strange trajectories, which were neither circles or ellipses, sometimes accelerated, then slowed and even turned back.

As we could not measure the distance between the stars, the use of compass was compulsory: it defined the position of a star from its neighbor by the angle with the earth. The astronomer Hipparchus of Nicea had the idea of introducing the concept of rope, ancestor of the sinus to track the position of the planets:

rope
The string of Hipparchus of Nicea

It was connected by a line segment between the intersection:

  • the angular sector described by a planet
  • the circle of radius 1.

The length of this line segment representing the rope. Hipparchus was able to build strings tables . Knowing a rope, Hipparchus obtained by seeking in its tables, the angular sector. The rope, as given by Hipparchus, represented what is now called the sinus or, to be more exact, was double our sinuses.

geocentrism
Geocentric Ptolemaic

Ptolemy

Towards the end of the second century BC, this legacy was recovered by Ptolemy, an astrologer from Alexandria. Ptolemy did not believe in heliocentrism of Aristarchus of Samos or Pythagoras. He put the Earth at the center of the universe.

But with such an error, calculating the trajectory of the planets became very complicated. He had to multiply the number of spheres that his calculations « stick » more or less to reality.

It will take 1500 years for this error to be unmasked by Copernicus.

The history of trigonometry

Trigonometry – the dark days of Christianity

Altomonte, _Bartolomeo _-_-_ 1719 _ The_Immaculate_ConceptionChristendom was a dark period for mathematics in general and science in particular. The number of mathematicians was reduced to its bare minimum. The only authorized discussions should focus on purely theological concepts: the divinity of Christ; the immaculate conception or the Holy Trinity. The councils, such as Nicaea (325), were the bodies of Christendom where these essential topics were discussed and rediscussed. The earth was flat and trigonometry had no more interest. Divine Inquisition burned all those who by their questioning could undermine these simple principles.

Here is the kind of progress that allowed Christianity (at the First Council of Nicaea)

« We believe in one God, the Father Almighty, Creator of all things visible and invisible. And in one Lord Jesus Christ, the only Son of God, begotten of the Father, that is to say, the substance of the Father. God of God, Light from Light, true God from true God; begotten, not made, consubstantial with the Father; by whom all things were made in heaven and in earth. Who for us men and for our salvation came down from heaven, was incarnate and was made man; suffered and died crucified on a cross, rose again the third day, ascended into heaven, and come to judge the living and the dead. And the Holy Spirit. « 

Everything was said. However, trigonometry had not moved an inch.

The history of trigonometry

Trigonometry – the eastern side

Sinus Aryabhata

Christian land since science was bogged down in the sands of the Inquisition, the Greek heritage advantage to others. In the fifth century, India, Aryabhata, another astronomer, builds half ropes tables: the sine was born!

tri3

He took the opportunity to give an approximation of Π :

Multiply 104 by 8, then adding 62 000. You get the circumference of a circle of diameter twenty thousand. Aryabhata.

What gave Π = 62832/20000 = 3.1416.

Sheltered Christian wrath, he argued that Earth rotated on its axis and was not located in the center of the universe.

the tangent

Then, it was around the tangent, the tables were written by another astronomer Al-Battani, Persia in the eighth century.

sin5
The unit circle

The tangent is the ratio between the sine and cosine.

This invention of the tangent inspire mathematicians who will succeed him and will cause in the seventeenth century the calculus that we owe to French mathematician Fermat.

The history of trigonometry

Trigonometry – The Italian Renaissance

jacob bernoulli
Jean Bernoulli

Europe rediscovered mathematics thanks to Fibonacci in the twelfth century. He imported from North Africa, where his father was a businessman, zero, the Hindu-Arabic numerals and the decimal system. The Italian mathematics was first developed primarily for accounting and commercial reasons.

In seeking to calculate interest rates, Johann Bernoulli found the value of the constant ok logarythm (e).


Regiomontanus

Mathematics diffused throughout Europe. In Germany in the fifteenth century, Regiomontanus (an astronomer!), admirer of Ptolemy gave to trigonometry its credentials. He « detached » trigonometry from Astronomy and made it a branch of wholly mathematics. The use of the term « sinus » ( « fold » in Latin) became final. The rope was dropped.

Back in India

In India, found the relation between the sine of an angle (its projection on the y-axis) and cosine (its projection on the x-axis).

cos² + sin² = 1

A basic application of the Pythagorean theorem on the triangle formed by the cosine, sine and radius of the circle (which is also (the hypotenuse of the triangle) to easily find this simple but important result.

The history of trigonometry

Trigonometry and Pi

We can not talk about Mathematics without mentioning Pi , the constant that has turned the head of the great thinkers of antiquity. The Indians, early, used to characterize Pi angles within a circle of radius 1.tri5

  • The half-circle radius of R = 1 measured  Π radians (180 °);
  • The right angle (quadrant) measured  Π / 2 radians (90 °);
  • The complete tour worth (and still is) 2 Π radians (360 °).

Varahamihira

From the sixth century, Varahamihira, another Indian mathematician, astronomer of course, had established a relationship between the sine and cosine of an angle α:

sin α = cos (Π / 2-α).

Sine and cosine were clearly linked. The sine of angle α is equal to the cosine of the angle at which law would have cut this angle α. Many other trigonometric formulas has been discovered, the best known are those of the angles:

  • sin ( x + y ) = sin x cos y + cos x · sin y
  • sin ( xy ) = sin x cos y – cos x · sin y
  • cos ( x + y ) = cos x cos y – sin x · sin y
  • cos ( xy ) = cos x cos y + sin x · sin y

The history of trigonometry

Trigonometry and imaginary world

In Italy, in the sixteenth century, Bombelli had solved the cubic equation by inventing imaginary numbers. These numbers had the distinction of having a negative square. We must understand the importance of daring  Italian mathematician.

For the Greeks, a number was an object of nature. A negative number, for example, made no sense: nothing could indeed measure – 5 cubits! The Indians of the Indus valley was not embarrassed by such considerations. They made a first giant step inventing negative numbers. Bombelli made the second one. He invented the imaginary number i = √-1 with i² = -1. That number had no physical meaning! What could be a basket i apples? Yet this magic number allowed blazing advanced mathematics. When falling on difficulty, calculations in imaginary wolrd were facilitated.

We define the complex numbers as follow :

A complex number z is a number which has:

  • a real part, characterized by a real has ;
  • and an imaginary part of the type b × i, or simply bi , where b is a real and i is the imaginary number .

It is noted in the form z = a + bi. But what’s the link with trigonometry?

A point M can be represented by its coordinates a and b in the Cartesian coordinate modified complex plane:

  • the x axis is the axis of the real part and
  • the y-axis is the axis of the part imaginary .
Tri6
complex plane

We say that M is the point of affix z = a + bi. It can also be written z = cos x + i sin x , if x is the angle in radians that is OM with the real axis.

The module (the length of the segment OM) of the complex number z = cos x + i sin x is equal to √ (x + cos² sin²x). The complex plane is a plane in which a point can be defined with sizes from trigonometry.

The history of trigonometry

Trigonometry formula MOIVRE

louis14aftrigaud_1638-1715lrg36k
Louis XIV

In 1685, Louis IV revoked the Edict of Nantes by Henry IV desired. The persecution of Protestants started up again. It was in this context agitated that Abraham MOIVRE, a Protestant French mathematician, enabled a breakthrough. He sets in the complex plane affix a point M z = a + bi, registered with the circle of radius 1. He put z = cos x + i sin x. He showed that the point N of affix z was also on the circle and formed with the axis of real numbers an angle of nx radians , or more generally:

(cos x + i sin x) n = cos nx + i sin nx –  MOIVRE Formula

A small demonstration by recurrence of the form MOIVRE

It is true for n = 1. Assume it is true to the rank n:

(cos x + i.sin x) n = cos nx + nx i.sin

Were successively:

  •  (cos x + i.sin x) n + 1 = (cos x + i. sin x). (cos x + i. sin x) n
  • = (X + cos i.sin x). (Cos nx + i sin nx.) – (Induction hypothesis)
  • = Cos x. nx cos – sin x. sin nx + i (sin x cos nx + cos x sin nx..) – (developing)
  • = Cos ((n + 1) x) + i.sin ((n + 1) x) – (with the trigonometric formulas cos (a + b) and sin (a + b))

So :

(cos x + i.sin x) n + 1 = cos ((n + 1) x) + i.sin ((n + 1) x)

This is also true at rank n + 1 and this is true all the time.

The history of trigonometry

Trigonometry, Euler and most beautiful formula of the world

euler2
Euler

When we look at the history of mathematics, whatever the subject, you come one time or another on Euler. This is the case also for trigonometry: The Devil Swiss mathematician establishes the famous formula:

e i x = cos x + i sin x – Euler

This expression can greatly facilitate calculations. Putting x = Π, that is to say a semicircle, we find:

e – 1 = 0

formula that was recently voted as the best ever.

The series TAYLOR to find Euler’s formula


  • sin x = x¹ / 1! – X³ / 3! + X 5 /5 Stars! – X7 / 7 ..!
  • cos x = 1 – x 2 /2! + X 4 /4! – X 6 /6! + X 8 /8! ..
  • e (x) = 1 + x + x 2 /2! + X 3 /3 puzzles! + X 4 /4! …

So :

  • cos x + i sin x = 1 + ix / 1! + (Ix) ² / 2! + (Ix) ³ / 3! + …

and

  • e i x = 1 + ix / 1! + (Ix) ² / 2! + (Ix) ³ / 3! + …

so

  • e i x = cos x + i sin x
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