# NavList:

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

**Re: Emergency Navigation**

**From:**Alexandre Eremenko

**Date:**2012 Jul 14, 04:32 -0400

Computing the sines of few angles is easy. The only thing which I recommend to memorize is pi. To memorize pi, I recommend this: http://www.math.purdue.edu/~eremenko/dvi/pi.pdf (for English, French or Russian speakers:-) Then you proceed as follows: if x is your angle in degrees, then y=2.pi.x/360 is your angle in radians. Then sin(y)=y-y^3/6+y^5/120-y^7/(1.2.3.4.5.6.7)+ etc. This is better to use for small angles. If your angle is not small use the doubling formulas like sin(2t)=2sin(t)cos(t), cos(t)=1-t^2/2+t^4/24-t^6/1.2.3.4.5.6+ etc. Dividing a 1-foot ruler into 1/10 of an inch divisions, without any instruments, is a much harder task:-) Alex. P.S. This is a late XVII century math. Ptolemy had to use a different method, much more sophisticated, starting from few angles (30, 45, 60, 90) for which sine and cosine can be found precisely, and then using division and addition formulas. To be more precise, he did not use our modern functions, sine and cos, but used the chord instead, chd(x)=2sin(x/2). His book contains the first table of chords that survived to out time. He also had to compute without having our decimal positional system, poor guy:-)