TRIG-i used to hate the most in school

Trigonometry was originally created by the Greeks to aid in the study of astronomy. Hipparchus of Bithynia (190-120 B.C.) tabulated trigonometric ratios, to enable the calculation of a planet's position as formulated by Apollonius. Angles were also defined, taking the Babylonian measure of 360 degrees. The chord was defined, and the cosine and sine loosely defined. The results sin² x + cos² x = 1 and the half-angle formulae were also derived, geometrically.

Claudius Ptolemy worked further on Hipparchus' chord table and came up with a more complete one. He used Euclid's propositions to aid in his work and developed a method of calculating square roots, though he never explained how. Using his theorem (for a quadrilateral inscribed in a circle, the product of the diagonals equals the sum of the products of the opposite sides) and the half-angle formula, he derived the sum and difference (addition) formulae.

Ptolemy then proceeded to work on plane triangles. In this process, he developed the idea of inverse trigonometric functions. He also derived, in modern terms, the Sine and Cosine Rules.

Medieval Trigonometry

The Chinese, in the medieval times, studied astronomy, and hence, trigonometry. They introduced the tangent function. However, most of their work are in the field of astronomy, and many of their trigonometric advancements were not continued.

The Indians were the next to advance the study of trigonometry. They developed their own sine tables, using the Greek half-angle formula. Later, the cosine table was also constructed. Techniques of approximation to a relatively high accuracy were also introduced.

The Indian works were translated and read by the Islamic mathematicians, who also worked on trigonometry. Similar to the Greeks and Indians, they related trigonometry and astronomy. The Indian sine was used, as well as the chord. The cosine was also formally introduced, by Abu Abdallah Muhammad ibn Jabir al-Battani.

The tangent function resurfaced; and the cotangent, cosecant and secant functions were introduced. Although their definitions were initially geometric, it was soon realised that they were the reciprocal functions of tangent, sine and cosine respectively. Highly accurate tables were developed for the trigonometric functions. The triple-angle formulae, already derived, was used for this.

European Developments

Trigonometry reached Europe in the medieval times. Richard of Wallingford wrote a text on trigonometry, Quadripartium. He related the Indian sine to the ancient chords. He used Euclid's Elements as a basis for his arguments in plane trigonometry. Levi ben Gerson worked on plane trigonometry, particularly the laws of sines and cosines.

In the 16th century, trigonometry was incorporated into geography and navigation. Knowledge of trigonometry was used to construct maps, determining the position of a land mass in relation to the longitudes and latitudes.

Johannes Muller, or more popularly known as Regiomontanus, wrote a text On Triangles. He studied plane trigonometry, including results for solving triangles. He expanded on Levi ben Gerson's work. He proved the Sine Rule, and also considered the ambiguous case in using the rule.

Later works improved the tables of sines, which has been worked on extensively; as well as included tables for the other functions. Thomas Finck was the first to use the modern terms "tangent" and "secant".

The workd so far applied trigonometric concepts in astronomy. It was only until Bartholomew Pitiscus when there was a text considering the solving of a plane triangle on earth. He invented the word "trigonometry", in his title Trigonometriae sive, de dimensione triangulis, Liber (Book of Trigonometry, or the Measurement of Triangles). He developed his own sine and tangent tables. However, like all the tables that had been calculated before, the values are actually the lengths of certain lines in a fixed circle.

Later developments in trigonometry are mainly the use of trigonometric ratios in calculus; analysis, differential equations and integration, just to name a few.