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

Daniel C. Bastos wrote:


<...>

 > 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.

Get used to pictures together with algebra.  Analytic geometry wasn't 
invented for nothing, and it won't go away in your studies.  You are 
beginning to study curves in the plane, which are by definition a set of 
parametric equations _along with_ a graph.

x = cos(t)
y = sin(t)

Goal:  write as a rectangular equation.

Process:

Solve for x and y in terms of t (already done).

Apply the identity:  sin^2(t)+cos^2(t)=1.  Substitute:

y^2 + x^2 = 1   ...rectangular equation

Seems your question boils down to the existence/nonexistence of a non 
geometric based proof of that identity. I do not personally know of any 
proof of the identity using absolutely no geometry somewhere along the 
way, since it is Pythagorean in nature, but others may.  But the bigger 
question is, why would you not want to invoke Pythagorus geometrically?

-- 
Darrell