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Discovery of Sine and Cosine of an angle, the intuition behind it is always intriguing. Apart from "that is the way they were defined", could someone explain how the discovery happened? I have read "What is Mathematics?" by R. Courant and related material on web. To this particular question, haven't found an answer for long.

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If I may expand on Alexandre’s answer:

It is correct that Ptolemy operated not with sines, but with chords. The use of sines (half-chords) was introduced (not “apparently” but definitely) by the Indian mathematicians, who used the Sanskrit word jīva- “bow string” (which is actually a translation of Greek chordē, but which Indians used not for the whole chord, but the half-chord). When Indian astronomical works were translated into Arabic the Indian term was adopted as jīb. This word is written identically, and pronounced almost identically, with the native Arabic word jayb “breast pocket of a garment”, and this is why it is (mis)translated by Latin sinus “breast”.

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  • $\begingroup$ Do you know the name of the Indian mathematician/astronomer who introduced sine? Did his work survive? Any references? $\endgroup$ – Alexandre Eremenko Apr 10 '15 at 17:45
  • $\begingroup$ It's in Wikipedia en.wikipedia.org/wiki/… $\endgroup$ – Conifold Apr 10 '15 at 17:52
  • $\begingroup$ @AlexandreEremenko. The earliest datable source seems to be Aryabhata (AD 476–550). $\endgroup$ – fdb Apr 10 '15 at 19:12
  • $\begingroup$ ...whose book is extant. $\endgroup$ – fdb Apr 10 '15 at 19:25
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Trigonometry was developed for the needs of astronomical computation. (The whole idea was foreign to the classical Greek mathematics (Euclides, Apollonius). They did not have even a fully developed number system, and did very little numerical computation, preferring pure geometric methods). Astronomical computations became possible after the adoption of the Babylonian sexagesimal notation for numbers.

Trigonometry is indispensable in most astronomical computations, the simplest one is passing from equatorial to ecliptic coordinates. This is needed for example when you compute such things as rising times, length of daylight, etc. Spherical trigonometry was developed simultaneously with the plane trigonometry, and the main emphasis was on the spherical one.

This happened in the the very end of the 1st century and is due to Menelaus. (His book Spherics can be dated by an observation made in the year 98). His trigonometric tables, if he made them, did not survive, and the earliest existing work where trigonometry is fully developed is Ptolemy's Almagest. Almagest contains the first known trigonometric tables, and Neugebauer thinks that Ptolemy was the first to make them.

At that time our usual trigonometric functions were not used. The only trigonometric function was the chord $\mathrm{chd}\ x=2\sin(x/2)$. This function is tabulated in Almagest. The role of cosine played $\mathrm{chd}(\pi-x)$, the cord of the complementary angle.

Sine was introduced by Abu'l Wafa in 8th century, as a more convenient function, and gradually spread first in the Muslim world, and then to the West. (But apparently it was used in India centuries before him), as a more convenient function. However this new notation was adopted very slowly, it took centuries. Copernicus (16th century) in fact used sine but calls it "half of the chord of double angle". But at the time before Copernicus there were already more advanced treatments of trigonometry, by J. Muller (Regiomontanus).

The best reference for all this information is the books

Neugebauer, History of ancient mathematical astronomy, and

Sverdlov and Neugebauer, Mathematical astronomy in Copernicus's De Revolutionibus.

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    $\begingroup$ Sine and cosine (named jya and kojya) already appear in Indian Siddhantas at the early sixth century. Abū al-Wafā lived in 10th century, and wasn't even the first Arab mathematician to use them, al-Khwārizmī produced tables of them a century earlier. But Arabs learned trigonometry from translations of Indian works, just like the decimal system. $\endgroup$ – Conifold Apr 10 '15 at 17:34
  • $\begingroup$ Both Arabs and Indians learned trigonometry from the Greeks. A slight change of notation should not be compared with introduction of trigonometry. $\endgroup$ – Alexandre Eremenko Apr 10 '15 at 17:43
  • $\begingroup$ The same goes for decimals vs sexagesimals, nonetheless Arabs chose Indian versions in both cases. $\endgroup$ – Conifold Apr 10 '15 at 17:51
  • $\begingroup$ What do you mean by "Indian version of sexagesimals" ? The version which all astronomers used and use nowadays is the Babilonian one. $\endgroup$ – Alexandre Eremenko Apr 10 '15 at 17:58
  • $\begingroup$ Indian version of a positional system was the decimal one, but both they and Arabs were aware of sexagesimals from Almagest, they just preferred decimals over them. The same with sines versus chords. $\endgroup$ – Conifold Apr 10 '15 at 18:02

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