# 2026-09-24 - Under The Trig Table
       
       In Trig Tables for AWKeteers, i wrote an AWK script to generate a
       printable trig table.  My demo trig.pdf and trig.txt had 4 digits
       of precision.  AWK can easily give more.  But what if i wanted to
       calculate the table values by hand rather than using AWK?
       
       Wikipedia lists exact trigonometric values [1] [2].  These are
       defined using arithmetic and square roots.  OK, how does one
       calculate a square root by hand?  Heron's method [3] is named after
       Hero of Alexandria who described the method in his AD 60 work
       Metrica.
       
       For example, sin(45) is equal to half the square root of 2.
       Heron's method uses an iterative formula:
       
           X0 = S / 2
           X(n + 1) = (Xn + S/Xn) / 2
           
           n = 0:
               X = 1
           n = 1:
               X = (1 + 2 / 1) / 2 = 1.5
           n = 2:
               X = (1.5 + 2 / 1.5) / 2 = 1.41666666666666666666
           n = 3:
               X = (1.41666666666666666666 + 2 / 1.41666666666666666666) / 2
                 = 1.41421568627450980392
           n = 4:
               X = (1.41421568627450980392 + 2 / 1.41421568627450980392) / 2
                 = 1.41421356237468991062
           n = 5:
               X = (1.41421356237468991062 + 2 / 1.41421356237468991062) / 2
                 = 1.41421356237309504880
           
           sqrt(2) = 1.41421356237 
       
       After 5 iterations, this can be used to calculate sin(45) with
       6 digits of precision.
       
           sin(45) = sqrt(2) / 2 = .707107
       
       Exact trigonometric values using Heron's method cannot be calculated
       for angles that are not divisible by 3.  Traditionally, the sine of
       the missing angles can be interpolated from neighbors.  For example,
       (sin(3) - sin(0)) / 3 = 0.0174453, which is pretty close to sin(1).
       
       Wikipedia also has a Taylor series from the 1700's to calculate sine.
       The more terms calculated in the series, the more precise the result
       becomes.  This method works with all angles, not only multiples of 3.
       
           sin(x) = x - x^3 / 3! + x^5 / 5! - x^7 / 7! + ...
       
           x = 45 * pi / 180 = 0.78539816
           sin(0.78539816) = 0.78539816 - 0.78539816^3 / 3! +
               0.78539816^5 / 5! - 0.78539816^7 / 7! + 0.78539816^9 / 9!
           sin(0.78539816) = .707107
       
       The first four terms of the power series give 6 digits of precision. 
       That's far enough down this rabbit hole for me today.
       
       tags: science,technical
       
       # Footnotes
       
 (IMG) [1] Exact trigonometric values for multiples of 3 degrees
       
 (TXT) [2] Exact trigonometric values (wikipedia)
       
 (TXT) [3] Heron's method (wikipedia)
       
 (TXT) [4] Taylor series for sine (wikipedia)
       
       # Tags
       
 (DIR) science
 (DIR) technical