Initially I want to add and subtract Latitudes and Longitudes.
Please also visit this thread of another forum. You will find an example file (by me) there, and additional hints.
Angles aren’t a specific type or subtype in spreadsheets. You thus have to work with ordinary numbers to represent angles. (“Angles in radians”.) The accuracy of the technical ‘Double’ format used for all numerical values in Calc is 52 bit mantissa + sign (exponent part not much significant for angles). This means (at least) 15 significant decimal digits. Since you need values up to an amount of Pi (=180°), and 1’’ is 1°/3600, you can distinguish angles(at least) to a resolution of 1E-8 ‘’. To calculate geographic coordinates this is by far accurate enough (about 600 nm along the equiator). GPS coordinates for everyday use are nowadays given either up to the 5th decimal place in degrees, e.g. (about 1m at the equator), up to three decimal places for the angular minutes (about 2m at the equator), or up to one decimal place for the angular seconds (about 3m at the equator), as is appropriate for the usual 10-digit-pocket calculators, and fully sufficient for everyday navigation.
Concerning your “adding and subtracting” of lattitudes and longitudes you first have to make your decision about the formats angles should be entered and displayed in. Try to avoid, if possible at all, the usage of more than one of the mentioned units (rad, °, ', ‘’). As customs will conflict with this advice, you may have to write a few user functions for conversion purposes. The remaining problem will be the input. Best dedicate at least three adjacent cells per row for the input, and a fourth one for the combined value. Once again: Stick to degrees only if actually possible.
For calculations going beyond adding and subtracting you will always have to rely on the mathematical standard (radians). For “human readability” you will have to introduce degrees. Try to avoid minutes and seconds.
If you exclusively use the degree as the unit for angles, you get a lot of advantages. mainly there are no problems with formatting as would be using ', ‘’. Only when calling a trigonomentric function you have to multiply the argument with PI()/180, and the result of a cyclometric function will be to devide by the same scaling constant.
If you cannot avoid using angular minutes (and seconds), please describe your reuirements to more detail.