Ptolemy - Part 4
MATHS 460 - BALL STATE UNIVERSITY - SUMMER 2000

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Trigonometric Identities

So far, we have calculated the sine and cosine functions of several values of θ - just like Ptolemy could calculate more Crd functions after he proved his famous theorem.  In order to calculate the rest of our sine and cosine functions, we need to distinguish some equations in order to compute the remaining functions.  These "equations" are referred to as trigonometric identities.  The basic and most famous identity is based upon the Pythagorean Theorem:  sin2 + cos2 = 1.

If we recall to part 2 of our discussion, in order to calculate Crd 12º, we took the difference between Crd 72º and Crd 60º using Ptolemy's Theorem.  In our trigonometry today, we have similar addition and subtraction identities for our sine and cosine functions.  For any angles α and β, the following identities appear in any trigonometry textbook:

sin (α + β) = sin α cos β + cos α sin β
sin (α - β) = sin α cos β - cos α sin β
cos (α + β) = cos α cos β - sin α sin β
cos (α - β) = cos α cos β + sin α sin β

 

From Trigonometric Identities to Trigonometric Tables

Let's now find sin 15º.  Using the second equation in the list above, we can say sin 15º = sin(45º - 30º).  Now using our formula, sin(45º - 30º) = sin 45º cos 30º - cos 45º sin 30º.  This is equivalent to:  √2/2(√3/2) - (√2/2)1/2.  This simplifies to √6/4 - √2/4 or √6 - √2)/4.  Now we can find the remaining sine and cosine functions, and create a sine and cosine trigonometric function table - similar to Ptolemy's chord table.  An online sine and cosine table can be found at: http://www.math.com.

 

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Up Ptolemy - Part 2 Ptolemy - Part 3 Ptolemy - Part 4 Ptolemy - Part 5

 

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