NavList:

Frank you wrote: For example, if they want to know the angular height of a lighthouse, they'll take the height in feet, divide by the distance in feet, and then look up or calculate the inverse sine. Possibly that number then has to be converted to minutes of arc. This is the long way around, and it's an abuse of trigonometry. 

I think you’re being rather hard here.  To me this is trigonometry.  In the UK the first thing you learn about trig functions is tangent = opposite side over adjacent side (or perpendicular over base at my school).  The near equality of sine and tangent and treating a small piece of circumference as a straight line when tiny angles are involved is useful gymnastics once the student gains experience and confidence but remembering “Some people have coal black hair through perpetual brushing” will always get you out of a hole whatever the size of the angle, as does remembering “All silly tabby cats” for the sign of trig functions of angles greater than 90.

Yes  angular size(minutes of arc) = radius of Earth(nautical miles)·length of ship(feet)/distance to ship(feet) is a good formula, but if you use it with an angle and ship’s length to measure distance to the ship, make sure the ship is fully side on to you or it’ll be nearer than you think.  DaveP