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

In article <slrnfle9j7.h9u.dbast0s@[EMAIL PROTECTED]
>, Daniel C. Bastos
<dbast0s@[EMAIL PROTECTED]
> wrote:

> In ``Calculus'' by James Stewart, 4th edition, chapter 11, section 1 (if
> I recall correctly), there's the argument that the pair of equations
> above describes a circle because ~``we can eliminate t by noticing that
> 
>                   x^2 + y^2 = cos^2(t) + sin^2(t) = 1;
> 
> hence, this is a circle''~. Terminology: I use ~`` (...) ''~ to mean an
> approximate quote; that is, it's coming from my memory, and hopefully
> I'm not changing the meaning.
> 
> I believe that's a circle because --- so my argument goes --- a circle
> of radius r has points (cos(t)/r, sin(t)/r),

You mean (r*cos(t), r*sin(t))
 
> where r is the radius of
> some circle. So if r = 1, as is the case of the unit circle, then the
> pairs will be (cos(t), sin(t)). To justify, I can say that the
> definition of sine (of an angle t) is opposite side of a divided by
> hypothenuse, which is r. So if r = 1, sin(t) = opposite side. If the
> angle t is the angle formed by the radius of the circle in the first
> quadrant and the x-axis, then the opposite side is y = sin(t); reasoning
> similarly for cos(t), we're done.
> 
> Stewart doesn't talk much --- for understandable reasons. But I was
> hoping for an algebraic construction from the pair of the equations that
> would reach
> 
>                              x^2 + y^2 = 1,
> 
> which gives me a clear elimination of t; but I only have my argument
> above, if correct. 
> 
> I like algebra which doesn't need to allude to geometric pictures; it's
> much safer to work this way. Any thoughts on this matter that might
> improve my education will be appreciated.

Well, t = arcsin(y) = arccos(x) which, taking cosines gives 
x = cos(arc(sin(y))). By a "standard trick", 
cos(arc(sin(y))) = sqrt(1 - y^2). So, squaring, x^2 = 1 - y^2.

Of course the "standard trick" is geometric.

If you are trying to avoid "sin(t) = opposite/hypotenuse" etc you can
use the usual Calculus definition which says the ray which makes a
counterclockwise angle of t with the x-axis intersects the unit circle
centered at the origin, by definition, at the point (cos(t), sin(t)).
Not for nothing are the trig functions also called the "circular
functions".

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?"

If you want to while away a long winter's night, you can avoid geometry
altogether by defining sin(t) and cos(t) by their Taylor series and
show algebraically (and analytically) that sin^2(t) + cos^2(t) = 1.

-- 
Paul Sperry
Columbia, SC (USA)