Inspired by Kenny Lau

By Euler's Formula,

$e^{i\theta} = \cos(\theta) + i\sin(\theta)$

$\Rightarrow e^{-i\theta} = \cos(\theta) - i\sin(\theta)$

Hence,

$e^{i\theta} + e^{-i\theta} = 2\cos(\theta)$

And similarly,

$e^{i\theta} - e^{-i\theta} = 2i\sin(\theta)$

Dividing these two equations,

$\dfrac{e^{i\theta} - e^{-i\theta}}{e^{i\theta} + e^{-i\theta}} = i\tan(\theta)$

$\tan(i) = i\dfrac{e - \frac{1}{e}}{e + \frac{1}{e}} \approx 0.762i$

On the Argand Plane, this is nothing but $(0,0.762)$ .

Hence the distance is: $\boxed{0.762}$ .