Wisdom of the Ancients

Q1) How did the ancients derive the trigonometric sin relations? 

A1) The ancients knew how to circumscribe various polygons inside the circle such as the triangle, square, pentagon, and hexagon.  This allowed them to know the sin relations for certain angles (360/3 = 120 degrees for the triangle, 360/4 = 90 degrees for the square, 360/5 = 72 degrees for the pentagon, 360/6 = 36 degrees for the hexagon).  The sin relations were obtained for these angles by treating the circumscribed polygons’ sides as chords.  Specifically half of such a chord’s length divided by the radius of the circumscribing circle yields the sin relation.  However the ancients often expressed such relations not as the sin of half the angle subtending the chord, but as the radius times the sin of half the subtending angle.  Thus in the book “On the Shoulders of Giants” by Stephen Hawking, in the chapter on Nicolaus Copernicus it provides a copy of Copernicus’s “Table of Chords in a Circle”.  This table assumes a circle of diameter 200,000 units and therefore with a radius of 100,000 units.  Thus for an angle of 36 degrees (36 degrees 0 minutes) the table displays a value of 58779 units in the “Halves of the chords subtending twice the arcs” column.  This is simply the radius multiplied by sin(36). Typing in sin(36) on a modern calculator yields and rounding to 6 decimal places yields 0.587785.  Multiplying this value by 100,000 yields 58778.5 which is 58779 after rounding up to the nearest integer.

How did the ancients obtain chord lengths for angles other than those provided by circumscribing regular polygons?  Ancient mathematicians like Ptolemy discovered geometrical relations that allow one to obtain lengths of chords related to known chords in simple arithmetic relationships such as sums and differences.  In this way differences and sums of known chords and their subtending angles could be obtained.  The fundamental geometrical law known as Ptolemy’s theorem is the basis for obtaining these sums and differences: see http://en.wikipedia.org/wiki/Ptolemy’s_theorem and http://hypertextbook.com/eworld/chords.shtml.

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