The formula you use to calculate this is:
sin(el) = sin(l)sin(dec) - cos(l)cos(dec)cos(HA)
cos(az) = [ sin(dec) - sin(l)sin(el) ] / cos(l)cos(el)
where el = elevation angle
az = azimuth angle
l = latitude of Washington DC.................... +38.9 degrees
dec = average declination of Gemini................ +20.0 degrees
HA = hour angle of Gemini at the date and time
The hour angle measures how far to the east or west of the local meridian ( line connecting north and south which crosses through your local zenith) Gemini is on February 29 at 10 PM Eastern Standard Time. This is determined from: HA = LST - RA where LST is the Local Sidereal Time and RA is the Right Ascension of Gemini which is about 7:30. Now, to calculate the Local Sidereal Time, you need to know that when it is 10:00 PM EST or 22:00 EST, it is 22:00 + 5:00 or 3:00 on March 1, Greenwich Mean Time because we are 5 hours earlier than the local time at Greenwich England. You now have to find what the Local Sidereal Time is for 3:00 March 1. To do this, you have to use a simple formula:
GST = GMT x C + d x A - B
where C = 1.002738
B = 17.403 for 1996
A = 0.065709
d = 31 + 29 + 1 = 61 days from January 1 to March 1.
from which you get that GST = 3:00 - 13:23 = -10:23 or adding 24:00
you get 13:37 GST. You now have to correct for the fact that the observations
are made in Washington DC at this time and date, not in Greenwich England.
The longitude of Washington DC is 77 West or Greenwich, so this means
you have to subtract 77/15 = 5.1333 hours from the GST to get the Local
Sidereal Time for Washington. This means that LST = 13:37 - 5:08 = 8:29.
In other words, at 10 PM from Washington DC, the Right Ascension that
passes through your local meridian is RA = 8 hours 29 minutes. Now, Gemini is
at a Right Ascension of about 7hours 29minutes, so this means that the Hour Angle
is HA = 8:29 - 7:29 = 1:00, and since there are 15 degrees per hour, this
means that Gemini is 15 degrees to the East of the local meridian.
We can now resume with the final calculation of
the azimuth and elevation, and get that the elevation angle is 67 degrees, and
the azimuth is 141 degrees. At least this is what I predict it should
be. Judging from where it seems to be by direct observation at 10PM
last night, I do not think I am too far off!! The hardest part of the
calculation is getting the Local Sidereal Time correct for your observing
site.