The case of the disappearing angle units, or "the dangle of the angle"

[jlind]

(07-31-2019 07:34 PM)ijabbott Wrote:  I know angles are technically dimensionless, but lots of people like to give them units nonetheless. This leads to situations where the units magically disappear or reappear unless extra steps are taken.

Consider the following examples (images shamelessly borrowed from mathsisfun.com)...

Area of a sector of a circle



\(A=\theta \cdot \frac{r^2}{2} \)

(for \(\theta\) in radians). To make the math work when angles have units, the above expression for area technically needs to be divided by 1 radian:

\(A=\frac{\theta}{1\,\textrm{rad}} \cdot \frac{r^2}{2} \)

Area of a segment of a circle



\(A=(\theta - \sin{\theta})\cdot\frac{r^2}{2}\)

That's different - we have an angle (in radians) minus its sine. To make that work, we need to divide the plain angle by 1 radian:

\(A=(\frac{\theta}{1\,\textrm{rad}}-\sin{\theta})\cdot\frac{r^2}{2}\)

Arc length of a sector of a circle



\(L = \theta \cdot r\)

Again, we need to divide the angle by 1 radian to get rid of the angular units:

\(L = \frac{\theta}{1\,\textrm{rad}} \cdot r \)

Question

Is it better to ignore angular units and just treat angles as plain old numbers (as long as an angle of 1 corresponds to 1 radian), or does the angular aspect have some cosmic significance that shouldn't be casually discarded? I guess this is more of a philosophical question, inspired by the way the WP-43S project plans to handle angles.

ijabbot:

You are confusing dimensionless with unitless and equating them. They're not the same. A plane angle, which is dimensionless, is a scalar value with a unit to reflect a quantity, be it radians, degrees, grads, quadrants, sextants, turns, or some other unit of measure. Dimensionless and unitless are two very different things. There are an enormous number of dimensionless scalars with units of measure. The SI unit for a plane angle is the rad, the abbreviation for Radian. The unit for a solid angle is the sr, the abbreviation for Steradian.

Please see this discussion in Wikipedia regarding dimensions and units:
https://en.wikipedia.org/wiki/Physical_q...dimensions

John