First, I'm moving this out of "Linear and Abstact Algebra", since it has nothing to do with either, to "General Math".
Parametric equations are used when some geometrical (or physical) properties, inherent in the problem, as, for example, the x and y coordinates of a point, are given as functions of some extra parameter that is not inherent in the original problem.
There are two reasons for doing this. First, it often happens that we cannot write a curve in two dimensions in terms of a function y= f(x) because the has two different y values for a single x value. Any closed curve, like a circle, has that property. We can, in situations like that, write both x and y as a function of some other variable. If for example, you have a circle of radius 2 centered at (0,0) so it has equation x2+ y[/sup]2[/sup]= 4, you cannot solve for y as a function of x. You would have to write both y= \sqrt{4- x^2} and y= -\sqrt{4-x^2}. If, instead, you draw a radius from the point (x,y) to the center, you should see a right triangle, having legs of length x and y and hypotenuse of length 2. If we call the angle between the radius and the x-axis "t", then , by the definition of sine and cosine, sin(t)= y/2, cos(t)= x/2 so that x= 2cos(t), y= 2 sin(t).
A parameter doesn't have to have any geometric significance, like that t being an angle. The crucial point is the calculatin where you say you got "lost". (2 cos t)2+ (2 sin t)2= 4 cos2 t+ 4 sin2 t= 4(cos2 t+ sin2 t)= 4.
Are you not aware that cos2 t+ sin2 t= 1 for all angles t? That's easily derived from the Pythagorean theorem.
Another important reason for using parametric equations is that 3 dimensions, we can't write a curve in terms of a single equation. In three dimensions each point is given 3 coordinates, x, y, and z. A single equation restricts one of those but we are still free to choose the other two- that's a two dimensional equation, the equation of a surface not a curve. For example if 2x+ 2y- z= 4, I could take x and y to be any values I want and then solve for z. That's the equation of a plane. If I want a one dimensional curve, I might use two equations. For example, if I have both 2x+ 2y- z= 4 and x+ y- 2z= 3, I can select a value for anyone coordinate and have two equations left to solve for the other two- that's a one-dimensional object. (It is, in fact, the straight line at which the two planes intersect.) It is, however, much simpler to have equations that give x, y, z directly as a function of one variable- t. Since we are free to choose t and then can solve for x, y, z, that is "one dimensional"- a curve in 3 dimensions.
Another, non-mathematical reason for using parametric equations is that physicists can think of t as "time" and use the equations to repesent an object moving along the curve. That's often a good way to imagine parametric equations- an object moving along the curve so that it is at point (x(t), y(t), z(t)) at each time t.
As far as the change from "r(t)= 2 costi + 2sintj + tk" to "x=2cost, y=2sint and z=t" is concerned, that's a matter of definition- the "position vector" of the point with coordinates (x,y,z) is defined as xi+ yj+ zk, the vector from (0,0,0) to (x,y,z). One is the "vector equation", the other "parametric equation" but they are equivalent.
That curve is, by the way, called a "Helix"- it looks like a coiled spring. If you imagine looking straight down along the z-axis, so that you can't see the z-height, you see only (x,y)= (2 cos t, 2 sin t) and, as we saw, x(2 cos t)2+ (2 sin t)2= 4 cos2 t+ 4 sin2 t= 4(cos2 t+ sin2 t)= 4. We are seeing a circle of radius 2. Of course, looking "from the side" so we can see the z-height, we see the point (x,y,z), as t increases (think of it as "time") moving upward at the same time it goes around in a circle.