Trigonometry means the study of trigons (triangles). The term was first used by Bartholomew Pitiscus in 1595. If anyone is not familiar with the definitions for sine, cosine, or tangent, go to the "Right Triangle Definitions" here.
Trigonometry itself emerged from astronomy. The Babylonians, for example, showed evidence of using trigonometric functions and it is from them that we have the concept of a circle consisting of 360 degrees.
The first known person to publish a table of trigonometric functions was the Greek mathematician Hipparchus around 140 B.C. Hipparchus used these functions to calculate the size of chords in a circle from a given angle. Today, Hipparchus is known as the father of trigonometry.
All of Hipparchus's major works have been lost. Most of our information about Hipparchus comes from Claudius Ptolemy. Ptolemy's work, the Almagest, stands one of history's most influential books. Almagest was not its original name. Almagest comes the Arab translation "al-majisti" which means the greatest. It presents an earth-centered universe (from Aristotle) where planetary orbits can be understood through trigonometic functions.
The first use of the sine function comes around 500 AD in the Hindu work Aryabhata. It includes a table of half chords which are called jya. This half chord table is referenced by Brahmagupta in 628 and Bhaskara in 1150.
The Arab mathematicians used the word "jiba" and "jaib" to refer to the half chord table from the Aryabhata. The term "jaib" is Arabic for "fold". So, when the Europeans began to translate the Arab texts, they used the term "sinus" (which is Latin for "fold") as the name of the half chord table. Fibonacci, for example, used the term "sinus rectus arcus". The term "cosinus" was introduced by Edmund Gunter in 1620.
The concept of tangents was well known at the time of Thales and was associated with the measurement of shadows. The first known table of tangents (referred to as the table of shadows) was presented by an Arab mathematician in 860. The term tangent itself was first used by Thomas Finke in 1583. The term cotangent was first used by Gunter in 1620.
The secant and cosecant were not used in tables until the 15th century. Copernicus for example used the concept of the secant which he called hypotenusa.