If a closed plane curve has turning number zero how to prove the continuous argument of the unit tangent field has the same period as the plane curve

Question

Consider a closed plane curve $\gamma:\mathbb{R}\mapsto\mathbb{R^2}$, of period $L$ with unit tangent field $T$.

The unit tangent field has continuous argument $\theta$ such that

$$T = (cos(\theta(s)),sin(\theta(s)))$$

If a closed plane curve $\gamma$ has turning number zero, how can we prove that the continuous argument $\theta$ of the unit tangent field has the same period as the curve $\gamma$?

We know that to have turning number zero implies that $ \theta (L)$ = $ \theta(0)$

Why does this mean

$ \theta (L +x)$ = $ \theta(x)$ for all $x$ ?