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

In article <bqqdnbRHsZFYj8fanZ2dnUVZ_oOnnZ2d@[EMAIL PROTECTED]
>,
Darrell wrote:

> 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.

Hm, okay.

But I'm not actually trying to avoid geometry per se; I'm trying to
avoid geometric pictures. I like algebra theorems, in which from an
equation we derive another without any external help. I like this
because it is safe; it depends very little on human interpretation,
unlike pictures.

>> 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.

Okay.

> 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.

So it seems to me that the problem is whether we should let the word
``angle'' stand for measures between 0 and 180 or any number? I would be
happy with a definition that states that angles are equivalence cl*****
with the angles whose measures lie in [0,360) being the representatives
of the cl***** --- if this can be done as my intuition says so.