Ancient Indians in the Gupta Empire invented the sine and other trigonometric functions. They called this the jaya, which meant "chord". Once the Gupta declined and the Muslims emerged as the leading intellectuals of the world, they borrowed the sine, and with it, the Sanskrit word for "sine". However, they only kept the phoneme, which was phonetically transcribed to jiba. This meant that the word had no real meaning, but was assigned to the sine function nonetheless. Later still, when Europeans wanted to borrow the word, they messed things up and confused jiba with jaib, the Arabic word for "bosom". So, while intending to borrow a term for a mathematical function, they mistranslated it, and ended up using sinus, which was the Latin word for "bosom". This easily became "sine" when it was borrowed into English in the 1590s. Sinus, which has nothing to do with the nose, is of unknown Proto-Indo-European origin. The co- in cosine was correctly translated from kotiya as meaning "complementary", so a cosine is a "complementary sine" or "complementary bosom".
Adam Aleksic has been described as the internet's sixth most famous etymologist. He loves writing about himself in the third person, he's a freshman at Harvard University, and he has disturbing interests in linguistics, vexillology, geography, board games, limericks, and law.
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