History of the Discovery and Use of Trigonometric Functions

Trigonometry was first used for surveying, cartography, astronomy and navigation. The trigonometric ratios were discovered and used for their practical usefulness in building pyramids, creating accurate maps, and the ancients did not really consider the trigonometric functions in terms of a geometrical science. They were merely a tool to use for everyday activities.

The Egyptians looked at trigonometric functions in terms of being properties of similar triangles. The Babylonians connected the idea of trigonometric functions to arcs of circles and the lengths of chords subtending those arcs. The Greeks built on the work of the Babylonians and created tables of chords.

The Rhind Papyrus (1650 B.C.) contained problems involving what we would now call the cotangent function to describe measurements of face slope angles of pyramids. A Babylonian clay tablet (1900-1600 B.C.) has a table of the secants of 15 angles from 30 to 45 degrees.

The Greeks worked with chords and the Hindus worked with sines. The Arabs, having contact with both civilizations, were influenced by both ways of viewing trigonometric functions. It was actually through the Arabs that Europe was introduced to the trigonometry of the sine function. This introduction was made through the work of al-Battani, On the Motion of the Stars around 900 B.C.

By 600 B.C. Thales of Miletos, a Greek philosopher had used similar triangles to measure the height of the Cheops pyramid using the ratio of a stick of known length and its shadow and the shadow of the pyramid. Around 200 B.C., Eratosthenes used the trigonometric functions and many different measurement techniques to make the first semi-close estimate of the circumference of the Earth. Hipparchos, a Babylonian mathematician, created one of the first extensive tables of trigonometric functions.

These tables were expressed in terms of 60th small parts of the radius for every one-half degree from 0 to 180 degrees. This table, with some expansions was printed in the first book of the Almagest, a mathematical treatise by Ptolemy, around 150 B.C. The tables in the Almagest contained chords for all arcs 0 to 180 degrees, which works out to be the equivalent to a table of modern day sines for angles 0 to 90 degrees in 15 arcminute intervals. That is 5400 sine values!

Since then, many mathematicians have worked to apply and expand our knowledge of the trigonometric functions. They started out as a tool for surveying and astronomy, then turned into a component of analytical geometry, and now are used for many things not even involving geometry.

Origins of the Names of the Trigonometric Functions

Sine - The original term for sine was ardha-jya, which means half-chord in the language of Indian mathematician Aryabhata. In translation to Arabic, this word became jiba, abbreviated jb. In translation to Latin, the translators assumed this was an abbreviation of jaib which means bulge, cove, or bay. This word translated to sinus, which eventually evolved to be sine today.

Cosine - In 1620, English astronomer and mathematician Edmund Gunter suggested that cosinus be used instead of the term sinus complementi that had previously been used. This term evolved to cosine.

Tangent - Originally the tangent of an angle was called the tangens of the angle.This term was developed in a more logical way than sine since the tangens of an angle is characterized by the length of the tangent segment.

Resources

Gullberg, J. (1997). Mathematics: From the birth of numbers. New York, NY: W.W. Norton & Company, Inc.6

Boyer, C. B. (1968). A history of mathematics. New York, NY: John Wiley & Sons, Inc.