subexpressions (OT: math)

[Erik Max Francis]

Gary Herron wrote:

> No, not end-of-story.  Neither of us are being precise enough here.  To
> quote from your second link:
>     "Although the radian is a unit of measure, it is a dimensionless
> quantity."
> 
> But NOTE: Radians and degrees *are* units of measure., however those
> units are dimensionless quantities , i.e., not a length or a time etc.

They're both "unit of measure" in some general sense, I suppose, but 
only one is dimensionless.  pi/2 rad = pi/2.  90 deg != 90.  Of the 
plane angles, only radians are dimensionless, and of the solid angles, 
only steradians are dimensionless.  This is because they're defined as 
the ratio of two quantities with similar dimensions (length for radians 
and area for steradians).

> The arguments to sine and cosine must have an associated unit so you
> know whether to interpret sin(1.2) as sine of an angle measured in
> degrees or radians (or whatever else).

The problem with this reasoning is when angular-dimensioned quantities 
pop out of trigonometric functions, which happens routinely in the world 
of calculus.  When that happens, you use radians (or steradians) or 
you're screwed.

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