Definition of trignometric functions

Question

Why is the sine of an angle is the ratio of the length of the opposite side to the length of the hypotenuse? How this definition came? Similar question for other five trigonometric function? Why they are called circular functions? Any intuition or references or detailed answers will be welcomed. Thanks

Answer 1

The word sine comes down to us as something of a mistranslation from the Arabic jiba, itself a transliteration from Sanskrit of a word meaning "half a chord". If you draw a chord in a circle, then bisect it, that half of the chord subtends a central angle whose sine times the radius gives the half-length of the chord.

The word tangent is more easily connected to the geometric tangent to a circle. A point on a line tangent to a circle gives a line segment from there to the point of tangency whose length is the tangent function of the central angle subtended by that segment times the radius of the circle.

The other root trigonometric name secant comes from the Latin secare, to cut, and has a geometric interpretation tied to a line that intersects the circle twice (not being a tangent) and thus "cutting" it.

The mathematical definitions of these functions are justified by their usefulness, a utility born out by ancient understandings.