The trace of Frobenius of an elliptic curve and its the number of points over the finite field it is over are both integers, $a_p$ and $p$, satisfying the relation $a_p^2\leq 4p$. It seems to me that a nonstandard angle should come from a pair of nonstandard numbers, $a_u$ and $u$, satisfying $a_u^2\leq 4u$. Then with the natural map from nonstandard integers to nonstantard reals, $a_u/2\sqrt{u}$ would be a nonstandard real number in the interval $[-1,1]$, thus a real number, and the $\cos$ of some real number.

In particular you're working over an ultrafilter over the set of primes, so you know what you want $u$ to be already. So the question is for ways to find nonstandard numbers $a_u$ satisfying this inequality.

Do you want different ways to compute the same number from an elliptic curve, or entirely different ways to compute the number? Do you want to compute it from an elliptic curve over an ultra-finite field? 

I'm not sure how an automorphism of the complex numbers relates to this.