While I was trying to read some Differential Geometry, I came upon this question:
Let $T(s)$ be the unit tangent vector field on a plane curve $\gamma(s)$ parametrized by arc length. Write $$T(s) =\begin{bmatrix}\cos \theta(s)\\ \sin \theta(s)\end{bmatrix}$$ where $\theta(s)$ is the angle of $T(s)$ with respect to the positive horizontal axis. Show that the signed curvature $\kappa$ is the derivative $d\theta/ ds$.
I can see how $\kappa$ will be $d\theta/ ds$ if $\theta(s)$ is smooth but I don't see how to prove the smoothness (or atleast differentiability) of $\theta(s)$.
My attempt : I tried to compose inverse trigonometric functions with $T(s)$ to get $\theta(s)$ but that didn't seem to work, as, in getting an actual smooth function that maps into $\mathbb{R}$ that gives the angle, there is some arbitrary choice as to what branch of angle is being choosed.
Can someone please help?