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

In article <EIKdnQETnvkdF8banZ2dnUVZ_ruqnZ2d@[EMAIL PROTECTED]
>,
Darrell wrote:

> Daniel C. Bastos wrote:
>
>> 
>> Okay. I was having trouble with your notation, so I wrote x + 3 = t and
>> then y = 3(x + 3) - 11 = 3x + 9 - 11 = 3x - 2. So these two are
>> equivalent. They must also have the same graph. I still follow you.
>
> The parametric equations are obviously not equivalent.
>
> The graphs, although they may look the same after completion of a 
> sketch, are not the same.  How were they traced?  Or to ask it more 
> properly, what are the orientations of the graphs?
>
> The graph of a set of parametric equations is defined as the set of 
> points (x,y) obtained as the parameter varies over some interval.
>
> Consider the same parametric equations and the interval [0,1].
>
> a) x = t
>     y = 3t - 2
>
> For t = 0, x = 0
>             y = 3(0) - 2 = -2  ...the point (0,-2)
> For t = 1, x = 1
>             y = 3(1) - 2 = 3 - 2 = 1   ...the point (1,1)
>
> b) x = t - 3
>     y = 3t - 11
>
> For t = 0, x = 0 - 3 = -3
>             y = 3(0) - 11 = -11  ...the point (-3,-11)
> For t = 1, x = 1 - 3 = -2
>             y = 3(1) - 11 = 3 - 11 = -8   ...the point (-2,-8)

Hm, I see. They are traced differently.

> Note that all four points satisfy the rectangular equation y = 3x - 2. 

What do you mean? All the points above satisfy y = 3x - 2. I don't know
what a rectangular equation is, though; what is it?

> Now reconsider what were were discussing about the equivalence relation 
> between 360deg, and -360deg.  Specifically, think of how this applies to

> a circle traced clockwise from a certain point vs. the same circle 
> traced counterclockwise from the same point.

They'd be traced differently. Is this a problem to have 360, -360, 0 all
be the same angle? Because we would trace them differently? If you say
so, then I guess you're saying that an angle in itself is the tracing of
its degrees. If you agree with this conditional, then I agree with you.

I have been thinking of an angle as its measure; that's true. In
practice, it really is its measure that matters for the ``day-to-day''
computations, so that's my habit. But sure, the angle in itself seems to
be something more subtle.

Thanks for leading the way.