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    Re: Emergency Navigation
    From: Gary LaPook
    Date: 2012 Jul 14, 18:37 -0700
    or remember 113355. split inthe middle, 355/113 = 3.1415929 the last digit should be a six so 355/113ths gets it almost right to 8 significant figures.

    gl

    --- On Sat, 7/14/12, eremenko@math.purdue.edu <eremenko@math.purdue.edu> wrote:

    From: eremenko@math.purdue.edu <eremenko@math.purdue.edu>
    Subject: [NavList] Re: Emergency Navigation
    To: NavList@fer3.com
    Date: Saturday, July 14, 2012, 1:32 AM


    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:-)





       
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