The angle 355 degrees gives us the same point on the unit circle as the angle -5 degrees. But one of these has three significant digits, the other has just one. (Relatedly, an absolute error of half a degree would yield a different relative error.) How can we resolve this?
Background: I am a mathematician, but numerical analysis is not my subject. I am teaching a course for students who will become sea captains and need math for navigation. I want to give them simple rules to follow for the number of significant digits in their answers when they solve problems, without diving deeper into error propagation. (Obviously error propagation for trigonometric functions is more complicated than for addition/subtraction/multiplication/division, but if they use those simple rules then at least they have a basic notion that crap in = crap out.) But I'm running into problems when it comes to angles, which of course are very relevant to my course. I already asked this question of the person who teaches numerical analysis at my university, but they didn't have an immediate answer.
A problem where this comes up is, for example: say a problem ends with calculating arcsin(-0.1). This would be negative, and we would round it off to whatever number of significant digits is dictated by our inputs (say, -5.7 degrees). I could stop here and have this as my answer. But if we want to express it as a positive angle, possibly the answer should then be 360 – 5.7 = 354.3 degrees, since addition preserves the number of decimals, and 360 should be seen as an exact number. But what if instead we'd waited until the end to do the rounding off? Then we'd have rounded it to 350 degrees if we wanted two significant digits…
Best Answer
The number of significant digits is a property of the number we use to represent (and approximate) an angle, not of the angle this number represents. I think it is sufficient for people to be aware of this difference, just as you explained.
But when working with angles in this context it really doesn't make sense to talk about significant digits. If you do want people to round properly you probably should use a fixed precision, saying you're rounding to e.g. n) decimal places ("n digits after the decimal point/comma"), which means always have the same absolute error.
The concept of significant digits only makes sense if you measure values across a large number of magnitudes where a the relative error is important, but this is not the case when you talk about e.g. bearings in navigation.