Can one make $\sin nx=I$ for all $n\ne 0$

Question

This is a generalization of another question which was quite straight forward to refute.

Let's start with the definition for $\sin$:

Assume that $E$ is a linear space over $\mathbb Q$, ring and with compatible topology. Then we define $\sin$ on $E$ as: $$\sin x = \sum_{k=0}^\infty (-1)^k{x^{2k+1}\over (2k+1)!}$$ whenever the series converges.

Now for the questions ($I$ is the multiplicative identity):

1. Is there an $E$ such that there exists an $x\in E$ such that $\sin nx=I$ for all $n\in\mathbb Z_+$?
2. Is there an $E$ such that there exists an $x\in E$ such that $\sin nx=\pm I$ for all $n\in\mathbb Z_+$?
3. Is there a normed $E$ such that there exists an $x\in E$ such that $|\sin nx|=1$ for all $n\in\mathbb Z_+$?

It's clear that $\mathbb C$ does not fulfill the requirements for $E$. I think that spaces of square matrices don't either (I think one can use Jordan normal form and use the result from $\mathbb C$ to prove that).