# successive approximation for sine, using pencil and paper

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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So you have a white board but no pocket calculator? If the angle is given somehow, why not just measure the sine? – Hagen von Eitzen Sep 27 '12 at 17:53
What is your usual desired accuracy? You can use Bhaskara I's sine approximation formula to get within 2% relative error (and that is worst case scenario, most of the time it is less than 0.3%). – Ragib Zaman Sep 27 '12 at 17:55
Yes, I often have a whiteboard, or a scrap of paper and a pencil, but no calculator. By measure do you mean use the calculator, or actually measure the hypotenuse and height? The latter isn't generally possible (think for instance phase angles in AC waveforms) and the former is what I'm trying not to have to do. – sh1ftst0rm Sep 27 '12 at 17:56
@Ragib: Thanks, that will be a useful formula to learn; I was not familiar with it. This is quite a bit more accurate than what I currently have, and would probably be more than sufficient for the kind of back of the envelope calculations where this is useful. However, I still wonder if there's a simple iterative approach that could refine the estimate to any arbitrary level of accuracy. – sh1ftst0rm Sep 27 '12 at 18:05

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.
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.