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.

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