Zitat von [email protected]:
Dale Tronrud wrote:
While it is true that angles are defined by ratios which result in
their values being independent of the units those lengths were measured,
common sense says that a number is an insufficient description of an
angle. If I tell you I measured an angle and its value is "1.5" you
cannot perform any useful calculation with that knowledge.
I disagree: you can, for instance, put this number x = 1.5 (without
units) into the series expansion for sin X :
x - x^3/(3!) + x^5/(5!) - x^7/(7!) + ...
and compute the value of sin(1.5) to any desired degree of accuracy
(four terms will be enough to get an accuracy of 0.0001). Note that
the x in the series expansion is just a real number (no dimension, no
unit).
... However you get this Taylor expansion under the assumption that
sin'(0)=1 sin''(0)=0, sin'''(0)=-1, ...
this only holds true under the assumption that the sin function has a
period of 2pi and this 'angle' is treated as unitless. Taking e. g.
the sine function with a 'degree' argument treated properly as 'unit'
will result in a Taylor expansion showing terms with this unit
sticking to them.
Clemens
