how can i show that (cost,sint,t) is not affine variety and find its zariski closure?

Question

how can i prove that the set X={(cost,sint,t): t belongs to R} a sub set of R^3 is not an affine variety? can i write X as (x,y,z) and then use nullestellensatz to find whole R^3 as zariski closure of X?

Answer 1

As for the first part, a semi-dirty trick is as follows: Consider a suitable line $L$ such that $X \cap L$ is infinite, but not the whole line $L$. Why does an existence of such a line prove that $X$ is not Zariski closed?

As for the second part: there is one nice relation that your set $X$ satisfies: we have $(\cos t)^2+(\sin t)^2=1$, so $X \subseteq V:=\{ (x, y, z) \; |\; x^2+y^2=1\}$. Try to show that $V$ is the Zariski closure of $X$. One approach is to mimic the line argument as above.