Throughout most of my mathematics education, the cosine function has been defined formally either using its series expansion, as $\mathfrak{Re}(\exp i\theta)$, or as the unique solution to $y+y''=0$ with $y(0)=1$ and $y'(0) = 0$.
Although these definitions make the mathematics more convenient, I've always been interested to see what it would be like to try to formalise it directly with the geometric intuition from the unit circle, assuming only the notion of distance (i.e. that we have a complete metric space with metric $d\colon\mathbb R^2 \times \mathbb R^2 \to \mathbb R$).
Since, by geometric intuition, we have that the cosine function is the $x$-coordinate obtained by walking along the unit circle, my idea was to take a line segment from the point $(1,0)$ to another point on the circle such that the distance is $\theta$, and split it into two line segments whose sum of lengths is also $\theta$. We then split into 3 line segments, and so on, in a limiting manner,
in such a way that the sum of lengths is always $\theta$, as illustrated below with $\theta = 2\pi/5$.
When this procedure converges, the $x$- and $y$-coordinates of the limiting point should be the cosine and sine of $\theta$ respectively. The problem is I can't find an easy way to express the $n$th coordinate of this procedure (assuming we are splitting the line into $n-1$ segments) so that I can take the limit.
I was going to try and do things the other way round, that is, define $\cos^{-1}\colon [-1,1] \to [0,\pi]$ by approaching the circle from below and finding the length. I did manage to do this, and got that \begin{align*} \cos^{-1}(x) &= \sum_{k=1}^\infty d\left[\left(\frac{(k-1)x}{n}, \sqrt{1-\left(\frac{(k-1)x}{n}\right)^2}\right), \left(\frac{kx}{n}, \sqrt{1-\left(\frac{kx}{n}\right)^2}\right)\right]\\[4pt] &= \sum_{k=1}^\infty \sqrt{2 \left(1-\sqrt{1-\frac{(k-1)^2 x^2}{n^2}} \sqrt{1-\frac{k^2 x^2}{n^2}}\right)-\frac{2 k(k-1) x^2}{n^2}}, \end{align*}
and I suppose one can extrapolate from here to obtain the $\cos$ and $\sin$ functions and extend them appropriately.
But it feels like the other way, that is, finding that limiting point, should be possible. I appreciate any assistance with this.