cosine of fraction of an angle in terms of the cartesian components

Question

Given,

$\cos\theta=\frac{x}{\sqrt{x^2+y^2}}$,

how can you write $\cos\frac{\theta}{n}$ (n an integer for simplicity) in terms of x and y? For example, one may say $\cos\frac{\theta}{n}=?\frac{x}{\sqrt{x^2+(y/n)^2}}$

Answer 1

Provided that $x$ and $y$ are suitable numbers (I think $y\geq 0$ and $(x,y)\neq(0,0)$ might be enough), you could write $$\cos \frac\theta n = \cos\left(\frac1n \arccos\left(\frac{x}{\sqrt{x^2+y^2}}\right)\right).$$

If there were a "nice" way to do it using multiplication, division, addition, subtraction, and square roots, then you could trisect an angle in classical geometry. But that cannot be done.