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

In article <051220072314256947%plsperry@[EMAIL PROTECTED]
>,
Paul Sperry wrote:

> 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))

True. Thanks.

>> 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.

This one looks good.

>
> Naturally one is left with the question "What is an angle?"

I would have to go with something like Euclid's definition: ``the
inclination of two right lines extending out from one point in different
directions is called a rectilineal angle.'' Except that if we say this,
then how do we classify an angle of zero radians? The lines don't extend
out in different directions, in this case. What would you say yourself?

> 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.

Hopefully I'll be able to work on some of this soon enough. Thanks for
your thoughts and everyone else's.