Does this define cos and sin in the real line uniquely? I was wondering if the following defines the functions $C$ and $S$ uniquely: There is a positive number $p$ such that$$\gamma (x)=(C(x),S(x))$$ is a path with period $2p$ that bijects $[0,2p)$ onto the unit circle in $\bf{R} ^2$, $\gamma (0)=(1,0)$, $\gamma (\frac{p}{2})=(0,1)$, and for all $x\in [0,2p)$ the length of $\gamma$ from $0$ to $x$ is $x$. Here ‘path’ simply means continuous map and the length from $a$ to $b$ is $$\sup{\sum_{i}{\lvert \gamma(x_{i})-\gamma(x_{i-1})\rvert}}$$ the sup being taken over all $x_0,\dotsc,x_n$ such that $a=x_0\le \dotsb\le x_n=b$. In other words, I am wondering if there are real valued functions $C$ and $S$ other than cosine and sine having these properties. The reason I'm interested in this is because it would define cosine, sine and $\pi$ somewhat ‘geometrically’ rather than the usual definitions involving power series.
 A: This question indicates why we might want to re-parameterise a curve by arclength; or, to say it differently, indicates why a curve that is already parameterised by arclength, like your $\gamma$, is particularly pleasant.
Since the length of the unit circle $\gamma([0, 2p))$ is both $2\pi$ and $\sup_{x \in [0, 2p)} \gamma([0, x]) = \sup_{x \in [0, 2p)} x = 2p$, we must have $p = \pi$.
Since $(\cos, \sin)$ is a continuous bijection from $\mathbb Z/2\pi\mathbb Z$ to the unit circle, we have that there is a unique function $\theta : [0, 2\pi) \to \mathbb R/2\pi\mathbb Z$ such that $\gamma(t) = (C(t), S(t))$ equals $(\cos(\theta(t)), \sin(\theta(t)))$ for all $t \in [0, 2\pi)$.  Since $\gamma$ is a bijection, so is $\theta$.  Since $\theta(0) = 0$, $\theta$ comes from either a continuous bijection $[0, 2\pi) \to [0, 2\pi)$ that is increasing, or a continuous bijection $[0, 2\pi) \to (-2\pi, 0]$ that is decreasing.  We'll denote this map (whichever one it is) again by $\theta$.
If you are willing to upgrade your continuity condition to the requirement that $\gamma : [0, 2\pi) \to S^1$ be $C^1$, then we have that $\gamma$ is rectifiable, and the length of $\gamma([0, x])$ is
$$
x = \int_0^x \sqrt{C'(t)^2 + S'(t)^2}\mathrm dt = \int_0^x \lvert\theta'(t)\rvert\mathrm dt = \lvert\theta(x)\rvert
$$
(since $\theta(0) = 0$ and $\theta$ is monotone).  In particular, $\theta(\pi/2)$ equals $\pm\pi/2$.  Since $\gamma(\pi/2)$ does not equal $(\cos(-\pi/2), \sin(-\pi/2))$, actually $\theta(\pi/2)$ equals $\pi/2$.  Thus, $\theta$ is an increasing, continuous bijection $[0, 2\pi) \to [0, 2\pi)$; and actually it is $C^1$, not just continuous, by our upgraded hypotheses.  Thus $\lvert\theta(x)\rvert$ actually equals $\theta(x)$, so $\theta(x)$ equals $x$, for all $x \in [0, 2\pi)$, as desired.
