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

In article <081220072320219637%plsperry@[EMAIL PROTECTED]
>,
Paul Sperry wrote:

> 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".

I see. I agree. This was a question I had a while ago, but I think it is
now answered. My question was merely this: if infinity is undefined, how
come we use it in x -> oo? So x approaches something we don't even know
what it is --- or approaches something that doesn't exist? But I believe
the answer to this question in particular is that the symbol is there to
merely allude to the fact that we're really saying: x can always get
larger than any upper bound you one may request.

>> > 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).

I see.

>> > 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. 

I'm interested in showing the two coincide; if you recall books from the
top of your head that do show it, or provide details about it, I'd like
to have their names; even if I don't currently have background to read
them, the day may come.

> 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.

Me too.