But if I work out $x^2 +y^2$ I get $t^2 + 2 t \sin(t) + \sin^2(t)$. This is always greater than one and so represents a circle of varying radius dependant on $t$. For a start why is the parametric equations not in one-to-one correspondence with the implicit function but also if I plot any of the implicit functions (for any $t$) then I do not get the same graph as the parametric graph.
I'm very confused. Would anyone be able to explain this?