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

In article <slrnfljjn8.rhh.dbast0s@[EMAIL PROTECTED]
>, Daniel C. Bastos
<dbast0s@[EMAIL PROTECTED]
> wrote:

> In article <061220071507321524%plsperry@[EMAIL PROTECTED]
>,
> Paul Sperry wrote:
> 
> > In article <slrnflg91u.no3.dbast0s@[EMAIL PROTECTED]
>, Daniel C. Bastos
> ><dbast0s@[EMAIL PROTECTED]
> wrote:
> >
> >> In article <051220072314256947%plsperry@[EMAIL PROTECTED]
>,
> >> Paul Sperry wrote:
> >> 
> >> > In article <slrnfle9j7.h9u.dbast0s@[EMAIL PROTECTED]
>, Daniel C. Bastos
> >> ><dbast0s@[EMAIL PROTECTED]
> wrote:
> >
> > [...]
> >
> >> > Another non-geometric approach is to define cos(t) = (u .
v)/(|u|*|v|)
> >> > where t is the angle between the vectors u and v.
> >
> > [...]
> >
> >> > Naturally one is left with the question "What is an angle?"
> >
> > [...]
> >
> >> What would you say yourself?
> >
> > [...]
> >
> > Well, since you asked...
> >
> > My intuitive idea of "angle" is, I suppose, pretty much that of
> > everyone else but intuition and rigor often don't play well together.
> >
> > 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?

I meant, obscurely, that in the same way we can avoid talking about
"infinity" in the limit example, we can avoid talking about an "angle"
in the dot product example. In other words, one can skirt the issue of
rigorously defining "angle".

> > There are ambiguities with the trig functions. For example, cos(x) is
a
> > perfectly well defined function of a real variable. Where we can run
> > into trouble is when we try to let that variable stand for an "angle".
> 
> Forgive my ignorance, but I don't see any trouble.

The trouble is in the argument of, say, cos( ). Is it the mysterious
"angle" and the cosine defined geometrically or is it just a real
number and the cosine defined analytically? Maybe, as Darrell suggests,
the argument is the size of a geometric object but then we not only run
into the problem of what the object is but also the problem of _how_ we
measure: radians - good; degrees - bad; grads - nobody knows what one
is (not really).
 
> > So, in my view, it is improper or, instead, meaningless to say "Let t
> > be angle between lines M and N" (although I'll confess to doing so at
> > every op****tunity). Also, it is awkward to properly state things like
> > "There are 2*Pi radians in a circle" - never mind the fact that there
> > are _no_ radians in a circle - or like "The sum of the angles of a
> > triangle is Pi radians".
> >
> > I guess I am asserting that there is no such thing as an angle (how
> > many angles can dance on the head of a pin?). At least I have never
> > seen a definition that I found satisfying. In any case, it doesn't
help
> > to swap off one word - "angle" - for another - "inclination".
> >
> > ...So there!
> 
> 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?

To me yes - in a way. Sometimes I want the geometric trig functions -
the usual ratios; sometimes I want them just to be particular functions
of real numbers. It is non-trivial to show that the two coincide. My
discussion of angles was mainly just to point out that there was more
there than meets the eye. 

None of this is a serious problem - mathematicians have been dealing
with trig functions for centuries with some success. My own preference
is an analytical definition and the rest are just interpretations of
those functions.

-- 
Paul Sperry
Columbia, SC (USA)