Re: Latitude by law of cosines
From: Frank Reed
Date: 2018 Feb 6, 12:59 -0800
Last month, Chris Caswell asked about some formulae labeled "for latitude" in John Karl's book:
sin(coL+G)=sin H sin G / sin d
where: tan G = tan d / cos LHA
Here coL is the co-latitude, H is the corrected altitude (in other words, Ho), d is the declination, G is an intermediate quantity,and LHA is the Sun's hour angle (for folks who worry about the distinction between LHA and "t", no, it doesn't matter here, and it would be reasonable to write HA rather than LHA). Although it's written up in a confusing form here, treating this as a simple calculating "machine", it's a thing that takes the Sun's hour angle and its declination, combines it with its observed altitude, and yields the latitude. It's the opposite of a traditional "time sight" calculation. I tried to drive home the point last month that this is never an important tool. Navigators don't need this "machine". Of course, there is one small exception...
I figure it's worth talking about that exception to see why this pair of equations would be useful in an exotic case. It's a rather old-fashioned sight-taking scenario for which there are simple well-established equations and tables. These equations are simply a different way of working up that specialty sight. Have enough clues? What's it called? What sort of sight do you have when you know the Sun's hour angle, or to put it differently, you know with some accuracy how many minutes to/from meridian passage, and you have measured the Sun's altitude, and you want to get the latitude only, setting aside the normal option of using the altitude for a common line of position? What do we call such a sight? Need more clues?? Greg Rudzinski is a rather big fan of working such sights by the traditional approach.