I'm learning parametric equations in this section. Although I understand why the following works, I'm having difficulty understanding why the method employed for solving it is the correct one.
I'm instructed to derive a Cartesian equation for the parametric equations given and describe the path the particle traverses. Describing the path amounts to simply graphing the path, i.e. the Cartesian function or parametric equation, and superimpose arrows indicating direction.
So, without further ado, the parametric equations give are:
$$ \begin{array}{rcl} x & = & \cos2t \\ y & = & \sin2t \end{array} $$ Over the interval: $0 \le t \le \pi$
The solution involves observing that $x^2 + y^2 = 1 => \cos^2(2t) + \sin^2(2t) = 1$
Squaring these functions to make them produce something convenient seems like "fixing" the problem to do what I'd like it to do. Why is this a valid approach? I hope I'm asking my question well. Squaring both of these functions seems like turning the problem into something it isn't, but yet, it's the right thing to do. Why?
Thanks, Andy
It is the right approach.
Take a candidate cartesian equation, replace x and y withe their expression x(t), y(t) and massage the formula until the parameter t disappears.
Then you are sure that every point of your parametric curve is also on the curve with the candidate cartesian equation.
You may have points of the parametric curve which are not on the cartesian curve. Like the equations $(x,y) = (e^t, e^{-t})$ which gives only one branch of the hyperbola $xy=1$, the one for which $x,y>0$.
A part of that, finding the cartesian equation is some kind of a guess work.