In Kalman filters why is it necessary to transform the systems dynamics matrix to the state transition matrix?
As I said in the comments, there should be no need at all to do such a transformation if all of your equations start in discrete-time.

The equation
$$
 (s \mathbf{I} - \mathbf{F} )^{-1}
$$
is just the Laplace transform of the solution for the state equation:
$$
\dot{\vec{x_t}} = \mathbf{F} \vec{x_t} + \vec{w_t}
$$

However, I really can't see how $\mathbf{F}$ works in:
$$
\vec{x_t} = \mathbf{F_t} \vec{x_{t-1}} + \mathbf{B_t} \vec{u_t} + \vec{w_t}
$$
because this is a discrete-time equation, and the equation you're looking at generating the sinewave is continuous-time.

Can you clarify?