subexpressions (OT: math)

[Erik Max Francis]

Gary Herron wrote:

> Of course not!  Angles have units, commonly either degrees or radians.
	...
> I don't know of any name for the units of "sqrt of angle", but that
> doesn't invalidate the claim that the value *is* a dimensioned
> quantity.  In lieu of a name, we'd have to label such a quantity as
> "sqrt of degrees" or "sqrt of radians".      After all, we do the same
> thing for measures of area.  We have some units of area like "acre", but
> usually we label areas with units like  "meters squared" or "square
> meters".    That's really no stranger than labeling a quantity as "sqrt
> of degrees".

What he (probably) means is that the only proper unit of angle is 
dimensionless, i.e., the radian.  All the rest are different ways of 
measuring things that technically violate rules of dimensional analysis.

In practice, it doesn't much matter, since angles are usually nicely 
stuffed inside trigonometric functions, e.g., r = A sin theta.  The 
problem occurs when, via calculus, angles get pulled out of 
trigonometric substitution.  This can happen via the chain rule easily 
enough.  For the example r(t) = A sin theta(t) with A constant, the 
derivative

	v(t) = dr/dt = A (d/dt) sin theta(t) = A dtheta/dt cos theta.

dtheta/dt is the time rate of change of an angle, and since it appears 
outside of a trigonometric function, must be expressed in units of 
radians per unit time or you'll get the wrong answers for v(t).  Using 
alternate units (deg, grad, gon, rev, etc.) for expressing the angle 
here is not an option; you must use radians, which are technically just 
another name for a dimensionless figure.

The radian is defined as the length of the subtended arc divided by the 
length of the radius, which is a length divided by a length, and thus 
dimensionless.  (Like in a lot of dimensional analysis, you write "rad" 
when desired to give a helpful hint, not because it's necessary; the 
radian has the same units as 1.)

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Erik Max Francis && max at alcyone.com && http://www.alcyone.com/max/
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