> [!NOTE] Theorem (Unit speed parametrisation of unit circle) > The parametrisation $\underline{r}(t)=(\cos t,\sin t), \quad \forall t\in\mathbb{R}$has [[Unit speed parametrisation|unit speed]]. *Proof.* Let $L(t)$ be the [[Arc length of regular parametrized curve|arc length function]] of the arc starting at $t=0$. STS $L'(t)=1$ at each point. Consider a point $(\cos t, \sin t)$ and a nearby point $(\cos(t+h), \sin(t+h))$. When $h$ is very small, the straight line distance between these two points is $\sqrt{ (\cos(t+h) - \cos t)^{2} + (\sin(t+h)-\sin t )^{2} } = 2 \sin(h/2)$by repeated use of addition formulae. Now $\lim_{ h \to 0^{+} } \frac{2\sin\left( \frac{h}{2} \right)}{h} = \lim_{ h \to 0^{+} } \frac{\sin\left( \frac{h}{2} \right)}{\frac{h}{2}} = \lim_{ p \to 0^{+} } \frac{\sin p}{p} = 1$since the last expression is the derivative of $\sin$ at $0$.