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    Re: Azimuth Equation
    From: George Huxtable
    Date: 2005 Oct 19, 21:42 +0100

    Todd Shanklin wrote-
    
    
    I've been recently programming a TI-81 Calculator to do my sight reductions.
      Everything has been going swimmingly, getting closer and closer with my
    sights and shooting more stars- until the past week or so when the LHA of
    Altair at twilight became closer to 360 degrees.  With all my fiddling, I've
    narrowed the problem and any accompanying possibly useful troubleshooting
    information down to this:
    
    The calculator will find Hc, but when it tries to calculate the azimuth, it
    sends back the error message "Math 04 Error".  When I enter the equation and
    values manually as one equation I get the same message after pressing enter.
    
    Strangely it seems, I have found that when the Declination is South (that
    is- the dec value is negative) the error message does not occur and
    calculator works just fine.
    
    The error message only shows up when the LHA is 345 degrees or greater.
    
    I'm using the azimuth equation out of the Nautical Almanac:
    cos-1[(sinDec cosLat)-(cosDec cosLHA sinLat)/cosHc]
    
    
    To find Hc: sin-1[(sinDec sinLat)+(cosDec cosLHA cos Lat)]
    In entering the equation, if the Declination or latitude is South, or the
    Longitude is West, the value is negative.
    
    I am unable to figure this out; I've reached the end of my mathematical
    tether at this point.  When I work it out manually one step at a time
    everything works out fine (e.g. calculate the sin Dec*cosLat, save that
    value, calc. cos Dec*cosLHA...)  Does anyone have any clue as to why this is
    happening?  Like I said, I had no problems whatsoever until the LHA began to
    approach >345 degrees.
    
    One example from my recent sights during which the problem occured:
    Altair-10/13/05- 0157:28 GMT
    LHA= 349.9983333
    Dec= N8.885
    Lat=N38.563
    
    Todd Shanklin
    
    
    ==============================
    From George:
    
    There seems to be a problem here with the bracketing. The Nautical almanac
    defines the angle A as arc cos (inverse cos; same meaning as cos-1) of X,
    where X = (sin dec cos lat - cos dec cos LHA sin lat) / cos Hc.
    
    This is different from the way Todd has written the expression for X, as-
    
    X= [(sinDec cosLat)-(cosDec cosLHA sinLat)/cosHc]
    
    In the first case, cos Hc divides the whole expression; as it should. In the
    second case, it divides only the second term. That's true in most notations,
    but my memories of TI machines are very distant ones.
    
    You would get the wrong answer in the second case. However, for the numbers
    that Todd quotes, I think you should get some sort of answer in each case,
    not a refusal.
    
    I don't know why Todd's calculator refuses when it comes to the jump.
    There's only a limited range of LHA that will give a valid answer, with
    azimuth in the range 0 to 180. If the machine tries to calculate an angle
    from its cos, when that cos is outside the range -1 to +1, it will balk. If
    Hc has been correctly calculated according to the formula
    
    Hc = sin-1[(sinDec sinLat)+(cosDec cosLHA cos Lat)]
    
    as Todd quotes, then cos A should not be outside the allowed range. But (of
    course) one isn't allowed to tinker with any of the quantities Dec, Lat,
    LHA, in the equation for A, without first recalculating the new value for
    Hc.
    
    Yes, there's another way (and I think a better one) of calculating azimuth
    using an arc tan formula, but the arc cos formula, as in the Nautical
    Almanac, ought to work, so the first priority is to discover why it isn't
    doing so, for Todd. It might be helpful if he tells us his deduced value for
    Hc.
    
    I think he will find that somewhere his programmed expression differs from
    what was intended, in some additional way beside the error explained above.
    
    George.
    
    =================================
    
    
    

       
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