I have $n$ unit vectors vectors $v_1...v_n$ in $\mathbb{R}^2$, each with respective angles $\theta_1...\theta_n$, which corresponds to $tan(\frac{y}{x})$. I want to find a vector with the average average angle of each of these vectors, $\frac{1}{n}\Sigma_{i=1}^n \theta_i$. The catch is that I'd like to do this without using trig functions, and not knowing any of the angles, only the $x$ and $y$ components (similarly I would not want an answer that uses infinite series since they would most likely basically be using trig functions).
All $x$ components of vectors will be positive to avoid some problems with infinities and zeros (implying $-\frac{\pi}{2}<\theta<\frac{\pi}{2}$).
This is trivially easy for $n = 2$, they just need to be added together. For example given $<1,0>$ and $<\frac{\sqrt{2}}{2},\frac{\sqrt{2}}{2}>$ of angles $0$ and $\frac{\pi}{4}$ respectively, I get $<1 + \frac{\sqrt{2}}{2},\frac{\sqrt{2}}{2}>$ with an angle of $\frac{\pi}{8}$. This can be extended to powers of two as well by just doing half of the vectors at a time.
I cannot figure out how to do this for other numbers of vectors. It seems like it ought to be possible, but I can't figure out a way to do it.
The specific operations allowed are those allowed in radicals (+,-,*,/,^) and logarithms. An answer without logarithms would be better, but I would still be interested and hearing them. Complex and imaginary numbers are not allowed.
Alternatively a proof that this is impossible would be a helpful answer if true.