Re: parametric: x = cos(t), y = sin(t)

Daniel C. Bastos wrote:
> In article <061220071507321524%plsperry@[EMAIL PROTECTED]
>,
> Paul Sperry wrote:

>> We occasionally define whole phrases. For example "lim(f(x) ; x -> oo)
>> = L" is defined as a whole; "oo", by itself, is not (or should not be)
>> used without some context to go with it. Not even in things like a one
>> point compactification (such as the Reals with oo appended) the "oo" is
>> just the name for some point not in the original space - we could just
>> as well append "@[EMAIL PROTECTED]
".
>>
>> In the same spirit "the cosine of the positively oriented angle between
>> the non-zero vectors u and v" can be defined _as a whole_ to 
>> be (u . v)/(|u|*|v|).
> 
> What would be the cosine of the posivitively oriented angle between the
> non-zero vectors u and v not defined as a whole? I understand what you
> mean above with the limit example; and I agree, but what piece of
> 
>                        cos(t) = (u . v)/(|u|*|v|)
> 
> cannot be taken separately?

For example, the pieces |u| and |v|.  They represent lengths of vectors 
and you are trying to avoid geometry.  In the same spirit, the statements:

lim x-> oo = L
or
lim x-> c = oo

when taken as _wholes_ as was already mentioned, need not define 
specifically what "oo" is as a piece of those wholes.  Likewise, when 
taking the definition of:

cos(t) = (u . v)/(|u|*|v|)

....as a _whole_ one need not define what |u| and |v| are individiually. 
  Hence, you avoid the geometry.

> 
> My intuitive idea of an angle is how open two intersecting lines are. I
> don't know how to define that mathematically, though, but what I haven't
> seen very well is the problem between the concept of an angle with a
> function such as cosine. I look at cosine as a function of one variable;
> the ``angle'' is merely a variable. Do you say that this is
unsatisfying?

When defining angle as the (lesser) measure between two intersecting 
rays, as is usually done in the beginning, one need consider the fact 
that the _most_ that measure can be is 180 degrees.

Extending the definition, in the usual trig fa****on, both rays coincide 
along the positive x-axis and have vertex at the origin.  One ray, 
called the initial side, remains fixed in place.  The other ray, called 
the terminal side, is then rotated.  This rotation can be 
counterclockwise (positive) or clockwise (negative) and can rotate any 
number of times and stop at any position in the plane.  This definition 
of angle allows for the measure to be _any_ real number of degrees, not 
just numbers between 0 and 180 degrees.

The concept is further generalized as the "circular functions," i.e. the 
trigonometric functions.  For example, by stating:

f(x) = cos(x)

.....we are not necessarily talking about an angle x, but a _real number_ 
x.  Of course, depending upon the application, x can be anything.  But 
the problem with the limited definition of angle would necessarily 
restrict the domain of this function to 0<x=<pi.  This function is too 
im****tant to be so restricted.

-- 
Darrell