A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding
From: Frank Reed
Date: 2017 Nov 21, 08:10 -0800
Well, the first and fourth are obviously identical. That's just simple commutativity of addition: A+B=B+A. Neither form is preferred ever because they are identical. Incidentally, you have over-burdened the forms you have posted with parentheses. If you want a "canonical" (standard) form, then use:
h = asin(sin L · sin D + cos L · cos D · cos HA),
h = asin(cos L · cos D · cos HA + sin L · sin D),
where h is altitude, L is latitude, D is declination, and HA is hour angle (longitude difference between the sub-sun point or GP and the observer's position --sometimes called "t"). You don't need to wrap the products in parentheses because "order of operations" guarantees that the products are evaluated first.
As for the second, with +/- and the third with a tilde, ~, these are relics of an earlier era when you could not assume that a navigator understood enough basic algebra to handle positive and negative numbers (or, arguably, the available trig tables obscured the negative argument cases). Anyone in the 21st century should know that L and D can be positive or negative with negative values equivalent to coordinates south of the equator, and the trig functions can also be positive or negative depending on their arguments. Historically navigators were taught separate rules for handling different cases. This complicated navigational computations creating the appearance of different equations when, in fact, there's just one equation with cases handled fully --and easily-- by the algebraic signs. Also note that A~B is the "unsigned difference between A and B" equivalent to |A-B| (absolute value of A-B or B-A). As such, it's only a partial answer here --a single case pulled out of context.
In short, the +/- version isn't wrong --it's just old-fashioned. But the tilde version is old-fashioned and also wrong, unless it's accompanied by more rules. You should ignore both of these old rules unless you are specifically interested in studying the history of manual computation.