The history of pi, sine, cosine

Call it the effect of reading 'The man who knew infinity – A life of the genius Ramanujan'. Of late I have been reading a lot of mathematics and trying to find out the origin of certain constants and functions ( like pi, sine, cosine etc) that we have been brought up to accept as a standard.

And the discovery was amazing. Most of the constants are not a result of some complex scientific assumptions or have not been born as a need to compute any complex mathematical equation. In fact these constants have always existed in nature and have been discovered by mathematicians over the years.

The History of Pi

Everybody knows that the value of pi is 3.14…er, something, but how many people know where the ratio came from? Actually, the ratio came from nature--it's the ratio between the circumference of a circle and its diameter, and it was always there, just waiting to be discovered. But who discovered it?

It's hard to pinpoint who, exactly, first became conscious of the constant ratio between the circumference of a circle and its diameter, as human civilizations seem to have been aware of it as early as 2550 BC. The Great Pyramid at Giza, which was built between 2550 and 2500 BC, has a perimeter of 1760 cubits (one cubit is about 18 inches, though it was measured by a person's forearm length and thus varied) and a height of 280 cubits, which gives it a ratio of 1760/280 or approximately 2 times pi. Egyptologists believe these proportions were chosen for symbolic reasons, but, of course, we can never be too sure.

The earliest textual evidence of pi dates back to 1900 BC; both the Babylonians and the Egyptians had a rough idea of the value. The Babylonians estimated pi to be about 25/8, while the Egyptians estimated it to be about 256/81.

The Ancient Greek mathematician Archimedes of Syracuse (287-212 BC) is largely considered to be the first to calculate an accurate estimation of the value of pi. Archimedes didn't calculate the exact value of pi, but rather came up with a very close approximation of 3.1485.

Pi, or, at least, the approximate ratio, also appears in the Bible:

"And he made a molten sea, ten cubit from the one brim to the other: it was round all about, and his height was five cubits: and a line of thirty cubits did compass it round about." --1 Kings 7:23

Etymology of 'pi' 

The Greek letter pi (p) was first used to denote the constant ratio in 1707, by Welsh mathematician William Jones, though the use of the symbol was not popularized until 1737 by Swiss mathematician and physicist, Leonhard Euler. Pi was taken from the Greek word for perimeter.

The History of Sine , Cosine

For centuries, mathematicians have tried to find the length of a chord given the length of an arc of a circle. This study resulted in the forming of the "chord function" which gave the result mentioned above. The chord function was discovered by Hipparchus of Nicaea (180–125 BC) and Ptolemy of Roman Egypt (90–165 AD).

The evolution of sine and cosine function was the result of the study on the same line.

Aryabhata(476–550 CE) formed a table giving the values of "half-chords"( length of chord/2 ) on arcs formed on a circle of a radius of 3438 units. Values of half-chords for a given angle of arc is what is today called a sine function of the angle. The Sine was called Jiya, Cosine was called CotiJya.

Āryabhaṭa's table was the first sine table ever constructed in the history of mathematics.The now lost tables of Hipparchus (c.190 BC – c.120 BC) and Menelaus (c.70–140 CE) and those of Ptolemy (c.AD 90 – c.168) were all tables of chords and not of half-chords.  Āryabhaṭa's table remained as the standard sine table of ancient India.

Etymology of Sine, Cosine

Etymologically, the word sine derives from the Sanskrit word for chord, jiva*(jya being its more popular synonym). This was transliterated in Arabic as jiba جــيــب, abbreviated jb جــــب . Since Arabic is written without short vowels, "jb" was interpreted as the word jaib جــيــب, which means "bosom", when the Arabic text was translated in the 12th century into Latin by Gerard of Cremona. The translator used the Latin equivalent for "bosom", sinus (which means "bosom" or "bay" or "fold") [7][8] The English form sine was introduced in the 1590s.

Sources :
http://www.techhive.com/article/191389/a_brief_history_of_pi.html
http://en.wikipedia.org/wiki/Aryabhata%27s_sine_table
http://nrich.maths.org/6908