Trigonometry developed from the study of right-angled triangles by applying their relations of sides and angles to the study of similar triangles. The word trigonometry comes from the Greek words "trigonon" which means triangle, and "metria" which means measure. The term trigonometry was first invented by the German mathematician Bartholomaeus Pitiscus, in his work, Trigonometria sive de dimensione triangulea, and first published 1595. The primary use of trigonometry is for operation, cartography, astronomy and navigation, but modern mathematicians has extended the uses of trigonometric functions far beyond a simple study of triangles to make trigonometry indispensable in many other areas. Especially astronomy was very tightly connected with trigonometry, and the first presentation of trigonometry as a science independent of astronomy is credited to the Persian Nasir ad-Din in the 13 century.

Trigonometric functions have a varied history. The old Egyptians looked upon trigonometric functions as features of similar triangles, which were useful in land surveying and when building pyramids. The old Babylonian astronomers related trigonometric functions to arcs of circles and to the lengths of the chords subtending the arcs.

Centuries later, trigonometric functions acquired a geometric interpretation when they came to be looked upon as the lengths of specific line segments related to the central angle.

It is quite difficult to describe with certainty the beginning of trigonometry... In general, one may say that the emphasis was placed first on astronomy, then shifted to spherical trigonometry, and finally moved on to plane trigonometry.

An early Hindu work on astronomy, the Surya Siddhanta, gives a table of half-chords based on Ptolemy’s table. But the first work to refer explicitly to the sine as a function of an angle is the Aryabhatiya of Aryabhata (ca. 500 AD).

Now begins an interesting etymological evolution that would finally lead to our modern word "sine". When the Arabs translated the Aryabhatiya into their own language, they retained the word jiva without translating it’s meaning.

When the Arabic version was translated into Latin, jiva, or jaib was translated into sinus, which means curve. We find the word sinus in the writings of Gherardo of Cremona, who translated many of the old Greek works. Other writers followed and soon the word sinus or sine became common in mathematical texts throughout Europe.

The remaining two main functions have a more recent history. The cosine function, which we today regard as equal in importance to the sine, first arose from the need to compute the sine of the complementary angle. The name cosinus originated with Edmund Gunter in 1620: he wrote co.sinus, which was modified to cosinus by John Newton (not related to Isaac Newton).

The word "tangent" comes from the Latin tangere, to touch. It was introduced by the Dane Thomas Fincke in 1583.

Thales of Miletos (625 – 547 BC.) is said to be the first man to determine the height of the Cheops pyramid by comparing the length of its shadow with the length of the shadow cast by a rod of known length. He also used trigonometry to determine the distances offshore at sea.

Aristarchus of Samos (310 – 250 BC), a mathematician and astronomer was the first man to propound a heliocentric theory of the universe – eighteen centuries before Copernicus. He made an attempt to compare the distance from Earth to the Sun and to the Moon. His reasoning was perfectly sound but instrument he used to determine the angle of sight between the sun and the half moon failed him by fault calibration. He found the distance to the sun to be about eighteen - twenty times that to the moon, instead of the correct figure of approximately 390 times.

The first known attempt to calculate circumference of the Earth was made by Eratosthenes of Alexandria. Having heard that, at the summer solstice, the sun was at zenith at Aswan, he decided to determine the height of the sun also at Alexandria. From his measurement he deduced, correctly, that the distance between Alexandria and Aswan must equal 1/50^th of the Earth’s circumstances, but all his other data were inaccurate or pure guesswork. But remember that this was approximately 1700 years before Columbus tried to find India, and found America.

Hippoarchus of Nicea (? – 127 BC) is also famous and most determination of the length of the lunar month ( the time the moon use in one revolution around the Earth), to within one second of today’s accepted value. As a mathematician, Hippoarchus introduced to Greek the Babylonian method of dividing the circle into 360 degrees.

The most important work in the history of trigonometry and astronomy is Almagest, which means the greatest of the great. It was written by the great mathematician and astronomer Ptolemy of Alexandria in the second century AD. It is a thirteen-book mathematical collection, and it contains over 1000 pages in modern edition. In the first book, there are tables of chords for all arcs 0° – 180° , at 0,5° intervals to at least 5 places of decimals. The Almagest also contains theorems corresponding to the present day law of sines. That so much of early Greek work on astronomy has been lost, could be a result of the completeness and elegance of presentation of the Almagest. The Almagest had become the basic textbook in astronomy for more than a thousand years.

Building further on the Almagest the Persian Abu al-Wafa systematised theorems and proves of trigonometry and prepared extensive trigonometric tables of sines and tangens.

Among the estimated 500.000 tablets found, some 300 deal with mathematical issues. These are of two kinds: table texts and problem texts, the latter dealing with algebraic and geometric problems.

One of the oldest and most interesting tablets is the one known as Plimpton 322. It dates from the Old Babylonian period of the Hammurabi dynasti, roughly 1800-1600 BC. Plimpton 322 deals with Pythagorean triples; c² = a² + b². It seems that the Babylonians were not only familiar with the Pythagorean Theorem a thousand years before Pythagoras, but they also knew the rudiments of number theory and had the computational skills to put the theory into practice.

Sources:

Mathematics – From the birth of numbers, Jan Gullberg

Trigonometric delights, Eli Maor

Some Internet-pages.