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Can someone explain to me what a parametric equation is exactly? Why it is used (instead of a normal function)? In other words, what is the significance of it?

Second, to be more specific, in my book, there is an example where

r(t) 2 costi + 2sintj + tk t>0.

Then what they say is that the "parametric equations of the curve are x=2cost, y=2sint and z=t". How did they get this? How did they know to extract a parametric equation from this?

Then from that (and this is where i got really lost) they said that

x^2 + y^2 = (2cost)^2 + (2sint)^2 = 2^2

??? Can someone please explain this train of thought? I'm pretty sure this is a basic concept but i just dont understand the fundamental idea behind it. Thanks so much.