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A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding

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Re: Logarithms by Hand
From: Frank Reed
Date: 2014 Jun 6, 16:09 -0700

There's one other angle for which you know the sine and cosine: any very small angle. The usual choices would be one second of arc or one minute of arc. To four significant figures, the sine of 1' is 1/3438. To six significant figures, the sine of 1" is 1/206265 (or to the same six sig figs, the sine of 1' is 60/206265). These magic numbers are easily calculated: 3438=60·180/pi and 206265=3600·180/pi. Meanwhile the cosines of such small angles are equal to 1 to six digits. Now we can work angle addition formulas and step out from any of the values known from basic geometry. For example, if you want the sine of 45°23', you can use

$\sin\left ( 45^{\circ} 23'\right )=\sin\left ( 45^{\circ})\cdot\cos\left (23'\right )+\cos\left ( 45^{\circ})\cdot\sin\left (23'\right )$

We know the sine and cosine of 45°, and we can replace cos(23') by 1 and sin(23') by 23/3438, and when you work it out, the result to four figures is 0.7118, which is correct. So we can easily fill in the gaps in a table this way. No practical value to any of this in the real world or even in hypothetical modern world scenarios, but it's good, clean fun!

-FER

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