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

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|).

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".
Rudin, for example, goes to some trouble to show that cosine, as
defined by the exponentials, coincides with the intuitive geometric
definition.

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!

-- 
Paul Sperry
Columbia, SC (USA)