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    Re: First sine table (Ptolemy)
    From: Frank Reed
    Date: 2009 Jan 23, 23:23 -0800

    Jim, you asked:
    "Were infinite series used in the initial creation of tables?"
    No, not the early tables.
    One way to generate tables is to use the trigonometric identities for sums and 
    differences of angles. Consider this: we know some exact values, e.g. sine 
    and cosine of 0�, 30�, 45�, 60�, 90�. If you have arbitrarily good values for 
    the square root of two and the square root of three, then you have 
    arbitrarily good values for the trig functions of those angles. Now we can 
    also calculate the sine of one degree to fairly good accuracy using a variety 
    of methods (see below). That's our anchor. And the cosine of one degree is 
    sqrt(1-sin^2(1)) so we calculate that one by hand as accurately as possible. 
    Then how do we get the sine and cosine of 2 degrees? Use the identities:
    sin(a+b) = sin(a)*cos(b) + sin(b)*cos(a),
    cos(a+b) = cos(a)*cos(b) - sin(a)*sin(b), 
    which in this case would be
    sin(2�) = sin(1�)*cos(1�) + sin(1�)*cos(1�),
    cos(2�) = cos(1�)*cos(1�) - sin(1�)*sin(1�).
    And then the sine and cosine of 3 degrees? Just build on the previous result:
    sin(3�) = sin(2�)*cos(1�) + sin(1�)*cos(2�),
    cos(3�) = cos(2�)*cos(1�) - sin(2�)*sin(1�).
    And so on... 
    And note that you really only need to calculate from 0 to 45 degrees. After 
    that, you just re-label the table using sin(x) = cos(90-x) etc.
    Next we look up all of these values in our table of logarithms (let's assume 
    someone has already provided us with that) and then we can get a table of 
    tangents rather quickly from tan(a) = inverselog[logsin(a)-logcos(a)].
    These are the general sorts of calculations that were used. There are lots of 
    tricks for speeding things up and reducing error. For example, we can get 
    exact values for the sine and cosine of 15 degrees using half-angle 
    identities. And from there, the sine/cosine of 7.5 degrees. Then going back 
    to the sum identities, we get "exact" values for any angle that is a multiple 
    of 7.5 degrees. 
    Finally, up at the top I mentioned that we need the sine of 1 degree. At some 
    level of accuracy the sine of a small angle is equal to the arc. That is, 
    sin(x degrees) = x*pi/180, for a small angle. For a low accuracy table, you 
    could assume that this holds for x=1�. For a higher accuracy table, you would 
    assume that this holds for x=1'. Then you build up arcminute by arcminute 
    until you get a better value for the sine of 1 degree and proceed again as 
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