EMT635 History of Mathematics

February 25, 1997

Geometric Proof of a Trigonometric Identity from Ptolemy's Almagest.

sin(x-y)=sinx*cosy-cosx*siny

The purpose of this activity is to experience Ptolemy's intuitive, geometric proof of a trigonometric identity which we would treat algebraically. As prerequisite knowledge, you must recall three facts:

1. An inscribed angle which intersects the diameter of a circle is a right angle. (i.e. angles ACD and ABD are right angles).

2. An inscribed angle is equal to one-half the central angle intersecting the same arc or chord.

3. The sum of the products of opposite sides of an inscribed quadrilateral is equal to the product of the diagonals. (i.e. BC*AD + AB*CD = AC*BD).

Step I

Label AB, AC, AD, and BD in terms of x and y using trigonometric functions.

Step II

Label BC in terms of x and y using trigonometric functions.

Step III

Show that the relationship BC*AD + AB*CD = AC*BD is equivalent to

sin(x-y) = sinx*cosy - siny*cosx.