Re: The prosthapheresis formulae

In article <p2ch941lah7a3j4kkug5rlprccg7bnqea9@xxxxxxx>,
... <...@xxxxxxxxxxxxxx> wrote:

On Tue, 5 Aug 2008 12:25:50 -0700 (PDT), Andrew Usher
<k_over_hbarc@xxxxxxxxx> wrote:

I just discovered the identity

sin a cos b = (sin(a+b) + sin(a-b))/2

and a few relatives

and I looked it up and found that they are the group called
prosthapheresis. Does anyone know how these were originallly
discovered? Probably the same way as I just did, by projecting
spherical circles onto a plane.

Andrew Usher

sin(a + b) = sin a cos b + cos a sin b

sin(a ? b) = sin a cos b ? cos a sin b

Now, add these two identities and compare with what you posted.




So what about the origin of these underlying formulae? They go
back to the trigonometry used for ancient Greek astronomy.

The earliest surviving version is Ptolemy's from the 2nd century
A.D. His "Almagest" Book I section 10 explains how he calculated his
table (section 11) listing the chord length AB for each angle AOB in a
standard circle with centre O. Later the chord function was replaced by
the modern sine (half the chord of twice the angle), but that alters the
addition formula in only minor ways.

That formula came from a theorem now often called "Ptolemy's
Theorem" about a cyclic quadrilateral ABCD:
AC.BD = AB.DC + AD.BC.
The special case where AD is a diameter easily gives the modern addition
formula for the sine as above. Ptolemy used it to calculate chords of
angles up to 90 degrees step by step from a few basic cases.

Ken Pledger.
.

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