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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?

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It's easiest to think of the trig functions on a circle- this is how they were constructed before calculators

enter image description here

  • 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

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    +1 Very nice answer, although co- could be explained better. The cosine is the sine of the complement of the angle. If a is an angle, (90-a) is its complement, and cos(a)=sin(90-a), so the sine of the complement of an angle is the cosine. Same for cotangent and cosecant.
    – Peter Shor
    Commented Sep 2, 2011 at 4:18
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    Note the spelling of centre.For completion, to get the secant and the tangent you also need to divide by the radius.
    – Bogdan Lataianu
    Commented Sep 2, 2011 at 6:08
  • Could you combine your other answer with this one? It is rather unintuitive that "bowstring" would become "bay" without knowing that it's just a misinterpretation.
    – Ryan Reich
    Commented Sep 2, 2011 at 18:21
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    @Ryan - it's more of a linguistic aside, the explanation of how the mistranslation came about. And if I merge them it loses the thread of other people's comments
    – mgb
    Commented Sep 2, 2011 at 18:24
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    All the etymologies for sine that I ran across said the spelling comes down from the Latin "sinus" including the meaning "curved surface" and this was a mistranslation of the Arabic. This might be more useful than "bay"...
    – rschwieb
    Commented Dec 3, 2014 at 11:26
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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.

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    This does not address the words as trig functions.
    – Daniel
    Commented Sep 1, 2011 at 20:34
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    E.g. Why does "secant" in the trig sense come from "cutting"? Why does "tangent" in the trig sense come from "touching"? (Sine makes sense to me now, thanks to Martin Beckett's comment.)
    – Daniel
    Commented Sep 1, 2011 at 21:15
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    This is explained somewhat better on this site
    – Peter Shor
    Commented Sep 2, 2011 at 2:59
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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."
(http://www.etymonline.com/index.php?term=sine)

ps. Please leave this as a separate answer to preserve the comment thread below

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  • This shows why the chord-of-an-arc sine is called a sine, but not the trigonometric function sine.
    – Daniel
    Commented Sep 1, 2011 at 20:36
  • The length of a chord is how you calculate a sin en.wikipedia.org/wiki/Chord_(geometry)
    – mgb
    Commented Sep 1, 2011 at 20:46
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    Could you expound in your answer? That's not an obvious/ubiquitous fact.
    – Daniel
    Commented Sep 1, 2011 at 21:09
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    Take a circle, draw two lines form the centre to the circumference and join them with a chord. Half of the length of the chord divided by the radius is the sin of half of the angle between the two lines. Half angles are used because there are a bunch of useful formulae for half angles. Remember sine is defined as the opposite/hypotenuse for a right angle triangle
    – mgb
    Commented Sep 1, 2011 at 21:14
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    Then it will get migrated to maths!
    – mgb
    Commented Sep 1, 2011 at 21:16

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