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I remember reading that in a couple of places that ancient trigonometry, particularly Ancient Greek trigonometry, used the chord as the primitive concept instead of sine and cosine. I can't tell whether the chord or half-chord was treated as primitive in the Indian mathematical tradition.

Here are two couple equivalent definitions of the chord function with an argument anachronistically written in terms of radians, written $k(t)$ . This function also technically assigns negative lengths to chords when the angle $t$ goes around the circle an odd number of complete revolutions.

$$ k(t) = 2\sin\left(\frac{t}{2}\right)$$

$$ k'' = -\frac{k}{4} \;\;\;\text{where}\;\;\; k(0) = 0 \;\;\text{and}\;\; k'(0) = 1 $$

The chord is also the length of the line segment between two points that are $t$ radians apart on the unit circle.

If you are primarily working with circles and don't have a notion of Cartesian coordinates, then it makes sense that chord would be the primitive trigonometric function. It seems to me that chord is easier to explain than sine if you're accustomed to thinking of geometric drawings up to a rigid motion.

So my question is:

  • Which mathematical traditions used the chord as one of their primitive trigonometric functions?
  • When did those traditions, or derivatives thereof, move away from using the chord function? Did this shift correspond to a key discovery?
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    $\begingroup$ All of this is already detailed on Wikipedia. Greeks used chords, Indians switched to sines and cosines, maybe because they were focused on numerical computations more than geometry. There was no "key discovery". Arabs and medieval Europeans imported them along with the decimals. $\endgroup$ – Conifold May 27 at 23:04
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The chord function appears in Greek astronomical works. The most accessible is Ptolemy’s Almagest; the first chapter explains the theory and how the table is computed. The switch to half-chords happened in India, again in astronomical works.

The details can be found in Glen van Brummelen’s history of early trigonometry, The Mathematics of the Heavens and the Earth. Already in Ptolemy one can find passages in which he connects chords to the angle at the circumference instead of the central angle. That is half the central angle, of course.

One can also see in the Almagest contexts in which the key function is half the chord (on pp. 95-96 van Brummelen gives an example from the solar theory). So the move from chords to sines is motivated by computational convenience: it saves factors of two in various formulas.

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