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.
The Tate module allows you to compute, from an element of $\sigma \in Gal(\bar{\mathbb Q}/\mathbb Q)$, an element of $\hat{\mathbb Z}$, the trace, that if $\sigma=Frob_p$, is equal to $a_p$ everywhere except $\mathbb Z_p$. Since $\hat{\mathbb Z}$ is a compactification of an integers, there is a map to it from any notion of nonstandard integers. Thus any notion that sends automorphisms of $\bar{\mathbb Q}$, or $\mathbb C$, to angles or traces of Frobenius, should probably form a commutative diagram with the trace of the Tate module and that map.
The reason this cannot serve as a definition on its own is that as far as I know this map would not be injective, nor would the traces need to satisfy the relevant inequality.