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

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?

> 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.

> 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?