Some intuition behind o-minimal systems. I am following the definitions of an o-minimal system as defined in "Tame Topology and O-minimal Structures" by Lou van den Dries.
It is immediate from the definition that the graph of $\sin(x)$ is not a tame set (intersect it with $y=0$). But what about a slightly rotated one? Or one which is both rotated and translated. To me they look to be tame (unless rotated by $\pi/4$). Is it correct that these sets are contained in some o-minimal system? And how can I easily 'recognize' tame sets? E.g. my intuition is that a collection of sets in $\mathbb{R}^2$ are tame if they do not invalidate the minimality axiom ($S_1$ contains exactly finite unions of points and open intervals). If so I can just complete with whatever sets needed in order for it to be a o-minimal structure. 
And lastly, what about definable maps? I think that for semilinear sets the simplicial maps consititute a set of definable maps but there seems to be many more. How should I think of definable maps? (I am familiar with the monotonicity theorem). 
 A: This is a very late answer to this question, but yes, one can show directly from the definition that $\langle\mathbb{R},+,-,*,0,1,X\rangle$ is not o-minimal, where $X\subset\mathbb{R}^2$ is obtained by the graph of the sine function by rotation by $\theta$. Indeed, the same rotation moves the $x$-axis to a line $y = ax$ which intersects $X$ infinitely many times. The projection of this set down to the $x$-axis is an infinite discrete definable subset of $\mathbb{R}$ (defined by the formula $X(x,ax)$). 
In fact, if $S$ is the graph of the sine function and $X$ is the image of $S$ under any invertible linear transformation of $\mathbb{R}^2$, then $S$ is definable in $\langle\mathbb{R},+,-,*,0,1,X\rangle$ (this is another way of seeing that no such $\langle\mathbb{R},+,-,*,0,1,X\rangle$ is o-minimal). The point is that the invertible linear transformation itself is definable (by $f(x,y) = (ax + by, cx + dy)$ for some $a,b,c,d\in \mathbb{R}$), so $S$ is defined as the set of points $(x,y)$ such that $f(x,y)\in X$, i.e. by the formula $X(ax+by,cx+dy)$.  
Your intuition that you can just add any "tame-looking" set and then close under the definability operations is correct in the sense that this will give you a structure, but the o-minimality conditions might be violated, and it can be hard to know in advance whether they will be. For example, adding the graph of the exponential function to the o-minimal structure on the real field gives an o-minimal structure $\mathbb{R}_\mathrm{exp} = \langle \mathbb{R},+,-,*,0,1,e^x\rangle$, but this is not at all obvious: it's a very hard theorem due to Wilkie.
