Comes from Latin tangens"what touches" from the verb tangere"play" from an Indo-European source tag-, "touch, handle". The definition of "tangent to a circle" appears in Book III of the book Elements, written by Euclid around 300 BC. He defines the tangent as the line that touches the circle so that it does not cut it when extended.
In 1583, Danish mathematician Thomas Fincke was the first to use the Latin term. linea tangens (touching line) in his work Geometry rotundi. After 14 years, there is the registration of the term in English “tangent line", at work Exercises Thomas Blundevil (1597), where he mentions:… Lines belonging to a Circle, called lines Tangent, and lines Secant"The Oxford Dictionary also cites a passage in which Blundevil refers to the tangent as a noun:"… the line AD is the Tangent, and the CD line is the Secant… ".
All these records relate to the geometric sense of tangent, but we know that there is still tangent in the trigonometric sense. The term was also first used in Latin (tangens) by Thomas Fincke in 1583 in his work Thomae Finkii Flenspurgensis Geometrye rotundi libri XIIII. At the time, the mathematician Viète (also known as Franciscus Vieta) did not like the term tangent, as it could be confused with the geometric concept of tangent. He preferred to use other words like sinus foecundarum, amsinus and prosinus.