Given two vectors in $a, b\in{\mathbb R}^3$, the right hand rule axis-angle vector $\Theta$ that characterizes the rotation of $a$ into $b$ stands perpendicular to both and has the angle $\theta=\sqrt{\Theta\Theta}$ as it's norm, where $\Theta\Theta$ denotes the inner product of $\Theta$ with itself. The the normalized variant may be denoted $n_\Theta = \frac{1}{\theta}*\Theta$.
(See also first graphic. The map $t\in[0, \theta]\mapsto \exp(t*n_\Theta L)a$ then does the rotation job, where $L_i$ are the ${\mathrm{so}}(3)$ standard representation basis vectors.)
We know $ab=\sqrt{aa\cdot bb}\cdot \cos(\theta)$.
Let's restrict ourselves to cases $\theta\in(0,\frac{\pi}{2})$, then $$a\times b = \sqrt{aa\cdot bb}\cdot \sin(\theta) * n_\Theta = ab\cdot \frac{\tan(\theta)}{\theta}*\Theta$$
and so
$$\Theta = \frac{a\times b}{ab} /\ \frac{\tan(\theta)}{\theta}.$$
The known odd expansion of $\tan$ can be used to get
$$\frac{\tan(\theta)}{\theta} = \sum_{n=0}^\infty U_{2n+1} \frac{1}{(2n+1)!}\theta^{2n} = 1 + \frac{1}{3}\theta^2 + \frac{2}{15}\theta^4 + {\mathcal O}(\theta^6).$$
The fact that this ratio starts at $1$ and stays close to it for small angles is clear geometrically. (See also second graphic.)
Now given two vectors $a, b\in {\mathbb Q}^3\subset{\mathbb R}^3$, say presented as floating point and then treated as fractions in Mathematica, I'm interested in computing a converging sequence of rational axis-angle vector approximations $(\Theta_k)_k\in {\mathbb Q}^3$.
The norms in $\frac{a\times b}{ab}$ cancel out and I have this rational vector exactly. I also do already have the (also rational !) number $\cos(\theta)^2 = aa\cdot bb\,/\, (ab)^2$, and so I also have $\sin(\theta)^2$ and $\tan(\theta)^2$.
Question: What is a a good, say stable and fasting converging, sequence converging to $\frac{\tan(\theta)}{\theta}$ in available rationals? Might be a series or a product or an iteration scheme. Looking at the geometry on the unit circle, I'm almost expecting such a scheme to exist.
Analytically on the other hand, for example, just by pattern matching against $s_2=\sin(\theta)^2$, I can figure out that e.g.
$$\frac{\tan(\theta)}{\theta} = 1 + \frac{1}{3}s_2 + \left(\frac{2}{15}+\frac{1}{3^2}\right) s_2^2 + {\mathcal O}(s_2^3).$$
Of course I can write a routine which formalizes this pattern matching, but maybe there's some theory to lean on. I actually found I have a hard time finding a theory regarding the computation of one function value $f(x)$ in terms of the value $g(x)$ where $g^{-1}$ is known.
(It's clear that with a lot of effort, a sequence for the theta-fraction can be computed by concatenation of $k$-order approximated series for of four or five inverses of real valued functions, and evaluated at a known value like $s_2$. But hopefully there's a less pedestrian approach.)
