Angles
Part of my units pages
Angles are dimensionless, so choosing a base angle is the same as choosing how many divisions should go into a full turn. An obvious choice is one, which lets us use units like Hz for angular velocity.
SI makes a different choice: the radian, whose sine and cosine functions have slopes oscillating between ±1. Larger divisions, like full turns, give a steeper slope; smaller divisions, like degrees, give a shallower slope. Sine and cosine are derivatives of each other (modulo a minus sign), scaled by this maximum slope; hence this slope acts as a constant of proportionality (like those discussed in the coherence page). Forcing this slope to be 1 gives us the radian as our unit of angle. (Radians can be defined in other ways, e.g. the angle subtended by an arc whose length equals its radius; I just think this sine/cosine relationship fits the ‘constant of proportionality’ template nicely). Using radians as our unit for angles gives derived units like radians per second (AKA rad/s) for angular velocity.
There are a little over six radians in a full turn; the exact ratio is irrational, around 6.283…. This number is so ubiquitous in geometry, and mathematics more broadly, that we denote it with the symbol τ; hence one turn equals τ radians. If we use radians as our base angle, certain formulae become quite simple (e.g. the length of an arc is the angle in radians multiplied by the radius), but others require τ (e.g. angular frequency in Hertz is the angular velocity in radians per second divided by τ). If we use turns as our base angle, these sets of formulae switch around, e.g. converting turns to Hertz requires no conversion factor, but the length of an arc is the angle in turns multiplied by the radius multiplied by τ.
I’m undecided as to which is preferable as a base/default, since each have their merits. Current practice is to use radians by default, and use a factor of τ when talking about turns, e.g. “3τ” is three turns. This makes sense, but this irrational multiplier makes many “everyday” situations more complicated than if we use turns (e.g. we can get surprisingly far without irrationals!). This, along with the 1-to-1 conversion between units like the Hertz and Becquerel, makes me favour turns as our unit of angle.
As for other approaches:
Gradians should be avoided, since their definition as one four-hundredth of a turn is completely arbitrary and redundant. (Remember, metric isn’t about powers of ten!)
The use of π to denote a half-turn is likewise redundant and arbitrary
The division of a turn into 360 degrees gives many nice, round numbers for easing mental arithmetic; indicating that a generic trecentosexagesimal prefix is a good idea. Repurposing the word “degree” for this would be easy for angles, as in “90 degree turn” for a quarter turn (AKA a “right angle”); it would be less familiar in other contexts, like “2 degree Volts” for 1/180th of a Volt; and it would unfortunately be ambiguous in others, like “3 degree Kelvin” for 1/120th of a Kelvin (not 3 Kelvin!)
The subsequent sexagesimal divisions of degrees into arcminutes (sixtieths of a degree turn) and arcseconds (sixtieths of an arcminute) are inconsistent and confusing, and hence should be avoided.
The subsequent decimal divisions into e.g. milliarcseconds are simply absurd.