Significant Figures when dealing with bearings

Question

Significant figures are used to ensure that the value is precise, and fall in within error in the positive and negative direction.

327 degrees true can also be written as N33degreesW. As such, would this direction be 2 or 3 significant figures? It is understood that their total is 360 degrees, where addition and subtraction requires the lowest number of decimal points, not significant figures, but when using this in calculations, for example, with vectors, the question of the number of significant figures is apparent. It is important for rounding to minimise error.

This brings into the question whether the number of degrees in the 'hundred' place value actually is considered in precision, or is just an order of magnitude, as seen in p(O)H where the first value is an order of magnitude, not a significant figure.

Answer 1

Whenever you are in doubt about significant figures, it helps to remember that significant figures are not used by professional scientists. Significant figures are used for students to have a very rudimentary approach to handling uncertainty while they learn some other scientific topic. When you reach an "edge case" with significant figures then you should use an actual uncertainty analysis instead. In that case you would explicitly write $(327.0 \pm 0.5) \ ^\circ$ where the term after the $\pm$ is the expanded standard uncertainty of the bearing.

However, in this case there is no situation in which it matters. There are basically two cases where you get angles.

One is where you are looking at a difference between two vectors. In that case you are only ever subtracting two angles. So, as you mention, it is only based on the number of digits after the decimal point and not the number of significant figures. In this case it doesn't matter because the mathematical operations are not affected by the significant figures.

The other case is when you are looking at a frequency. Then the angle is a product of the frequency and time. In this case the mathematical operations are affected by the significant figures. However, in this case the bearings are not interchangeable. A positive frequency is physically distinct from a negative frequency, and confusing the two is an error called "aliasing". In this case the difference in the number of significant figures doesn't matter because only one is correct and that is the one that needs to be used.