I'm looking for a relatively simple algorithm that can quickly be done by hand to refine an initial estimate for the sine of an angle in degrees.
I've memorized a few landmark values for sine and come up with some simple techniques such that I can rapidly estimate the sine of any angle (in degrees) to within 10% error. What I'd like to do now is be able to take that estimate and refine it, presumably through some simple iterative algorithm that I can perform on a white board, for instance.
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Interpolating $\cos$ at $0$, $\pm{\pi\over2}$, and $\pm{\pi\over3}$ gives, expressed as a function of degrees, $$p(x):=1-{49\over324\,000} x^2+{1\over 291\,600\,000}x^4\ .$$ Drawing the two plots over the interval $\bigl[{-{\pi\over2}},{\pi\over2}\bigr]$ you cannot distinguish them with a naked eye.
You should know $\sin(x)$ and $\cos(x)$ for $x = 0$, $30$, $60$, $90, \ldots, 360$ degrees. Memorize $\sin$ and $\cos$ of $6$ degrees and $12$ degrees and you can use the addition formulas to calculate $\sin$ and $\cos$ for all multiples of $6$ degrees (e.g. for $18$ degrees, write $18 = 30 - 12$). Then memorize $\sin$ and $\cos$ of $1$, $2$ and $3$ degrees and you can use the addition formulas to calculate $\sin$ and $\cos$ for all multiples of $1$ degree.