This angle is the complement of the elevation angle (e) of the object as measured from the horizontal plane, and in some cases a slight change to the formulas will determine the value of e directly, or the zenith angle (zenith distance) can be calculated and then subtracted from 90 degrees to determine the elevation angle.
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Sight Reduction using The Law of cosines - (ref page C1)When observing an astronomical object from point one (L1, LO1) t = DLO = the difference in longitude, or local hour angle. (L.H.A.) Dec = Declination of object C = Azimuth angle. e = the elevation angle of the astronomical object (degrees) sin e = sin L1 * sin Dec + cos L1 * cos Dec * cos t cos C = (sin Dec - sin L1 * sin e ) / ( cos L1 * cos e ) or sin C = ( cos Dec * sin t ) / cos e (caution - ambiguity)
Sight Reduction using the Adgeton method (inverted) - (ref page C2)sin R = sin t * cos Dec (R is less than 90 degrees ) sin K = sin Dec / cos R (note 1) sin e = cos R * cos (K-L1) (e is less than 90 degrees ) sin C = sin R / cos e (note 2) (note 1) If t is greater than 90 degrees then K is greater than 90 degrees. (note 2) If L1 is greater than K then C is greater than 90 degrees.
Sight Reduction using the no-name method - (ref. page C4)Q = distance from pole to the perpendicular, measured along the meridian of point one. t = DLO = difference in longitude, or local hour angle (L.H.A.) Dec = Declination of object C = Azimuth angle e = elevation angle (degrees) tan Q = (cos t) / tan Dec tan C = (tan t) * (sin Q) / cos (L1+Q) tan e = (cos C) * tan (L1+Q)