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

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

That should be t. Feel free to fix that in my quote and ignore this very
message. Thanks.

> 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.