The meanings of these words are very similar: the sine of an angle in a right triangle is the ratio of the opposite side to the hypotenuse; the secant is the ratio of the hypotenuse to the adjacent side, and the tangent is the ratio of the opposite side to the adjacent side.
Since they have such similar functions, I wondered why sine comes from the Arabic word for pocket, secant comes from the Latin word for cut, and tangent comes from the Latin for to touch. What do the etymologies have to do with the current meaning?
It's easiest to think of the trig functions on a circle- this is how they were constructed before calculators
sine - is from the Latin for bay and/or the Arabic for bowstring. Picture the chord (segment from A-B) and the circle A-D-B as a bow. If you measure the length of A-C and divide by the radius of the circle (O-A) you get the sine of the angle theta
cosine is from "complement sin". It's the complement to the angle, so if you measure the distance from the centre to the chord (O-C) and divide by the radius you get the cos of the angle.
tangent is from the Latin for touch. A tangent is a line that touches the circle once. By definition this meets a line to the centre at right angle, so you always have a right angle triangle and so an easy definition of the tan of the angle
secant is from cut (Latin again). It cuts the tangent from O-E
cosecant, cotangent etc are like cosine, the complements to their respective functions, but unless you do a lot of maths you probably won't meet them
Chamber's 20th Century Dictionary (and a 1980-ish version of it, at that) says:
sine - (math.) n. orig. the perpendicular from one end of an arc to the diameter through the other: now (as a function of an angle) the ratio of the side opposite (or its supplement) in a right triangle to the hypotenuse ... [L. sinus, a bay]
secant - adj. cutting ... [L. secans, -antis, pr.p. of secare, to cut.]
tangent - adj. touching without intersecting. ... [L. tangens, -entis, pr.p. of tangere, to touch.]
So, basically, the words are derived from Latin (that's the 'L.'). The original definition of sine more closely resembles a 'bay' than the current trigonometric definition does.
Sine: from L. sinus "fold in a garment, bend, curve." Used mid-12c. by Gherardo of Cremona > in M.L. translation of Arabic geometrical text to render Arabic jiba "chord of
an arc, sine" (from Skt. jya "bowstring"), which he confused with jaib
"bundle, bosom, fold in a garment."
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