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    Re: Cosine Method Revisited
    From: Chuck Taylor
    Date: 1996 Dec 15, 20:39 EST

    Gordon Talge, Oliver Jacobson, and perhaps others have commented
    that there is no ambiguity in the cosine formulas for sight
    reduction. At the risk of belaboring the point, I'd like to be a
    little more precise.
    Referring to the following form of the equations:
    sin Hc = (cos LHA * cos Lat * cos Dec) + (sin Lat * sin Dec)
    cos Z  = (sin Dec - (sin Lat * sin Hc)) / (cos Lat * cos Hc)
    Note first that I show LHA instead of t. There are two possible
    sign conventions to use for N/S Lat & Dec. If you choose North
    Lat and North Dec as + and South Lat and South Dec as -, the
    result is Zc, which can be converted to Zn = Azimuth by the
    following formulas:
            LHA > 180, Zn = Zc
            LHA < 180, Zn = 360 - Zc
    If, on the other hand, you choose a different sign convention,
    namely that Lat is always positive and that Dec is positive if it
    has the same name as Lat (both N or both S), and negative
    otherwise, then the result is Azimuth Angle Z, which is converted
    to Azimuth Zn by the following formulas:
            Lat N & LHA >180, Zn = Z
            Lat S & LHA >180, Zn = 180 - Z
            Lat S & LHA <180, Zn = 180 + Z
            Lat N & LHA <180, Zn = 360 - Z
    Using these conventions, your calculator will give you an
    unambiguous result. Why? It's largely a matter of happy
    circumstance. The signs of the various trig functions just happen
    to work out for the best. Check it out for yourself! Or perhaps
    someone else has a better explanation why it works....
    Chuck Taylor
    Everett, WA

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