I'm trying to calculate the arc length of the circle using a metric, but I get stuck on how to find the velocity of my curve in the tangent space. So the circle $\mathbb{S}^1$ is defined by $x^2+y^2=1$ for $(x,y)\in \mathbb{R}^2$ and my path is given by $\gamma (t)=(\cos (t), \sin (t))$ (with domain $(0,\pi)$).
Now my first thought was that if I have a chart $\pi_x(x,y)=x$ which maps the upper half of the circle onto the $x$-axis, then the tangent vector should be given by: $\frac{d(\pi_x\circ\gamma)}{dt}(t)=-\sin (t)$. I see that this is obviously false since the curve travels at uniform speed and the chart deleted part of the necessary information.
My second thought was to look at $\gamma$ as a map between manifolds and try to use the definition of the push forward. Now, using the geometric definition of the tangent space, the only thing I get is that for a curve $p$ on $(0,\pi)$ the vector $p'(0)\in T_t (0,\pi)$ is mapped to $(\gamma\circ p)'(0)\in T_{\gamma (t)}\mathbb{S}^1$. I'm however clueless as to what the actual value now in my case.
Could somebody help me out with this?
(Note that I am trying to avoid embedding the tangent space directly in $\mathbb{R}^2$ and using the algebraic definition of the tangent space.)