Sinx y sin x cos y cos x sin y

where x = j/2, y = a/2. In a similar way, Ptolemy derived the equivalent of the formulas

Euclid Elements, Book I, 47 and Book III, 31, respectively.

sin(x + y) = sin x cos y + cos x sin y 2 sin2 x = 1 — cos 2x.

With these formulas and the chords already computed, it is possible to construct a table of chords in steps of 3° (or 3/2° or 3 /4°, etc.). However, Ptolemy's aim was to construct a table of chords in intervals of 1 /2° and he achieved this with an ingenious argument: that for acute angles a and j with ch a > ch j, ciia < ±1, (3.4)

a j a result known to Aristarchus and, since he had previously calculated ch 3° = 1; 34, 15, ch 4° = 0;47, 8, Eqn (3.4) implies that fchf° < ch1° < fchf

But to two sexagesimal places both f ch f° and f ch |° are equal to 1; 2, 50. Hence, Ptolemy knew ch 1° and could compute ch 1 ° = 0; 31, 25.

Ptolemy was now able to construct a table of chords from 1 /2° to 180° in steps of 1 /2° (equivalent to a table of sines from 1/4° to 90° in steps of 1/4°) accurate to two sexagesimal places.

With his table of chords, Ptolemy could use algorithms equivalent to the modern formulas (following the standard convention that the side opposite angle A is labelled a, etc.):

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