Let $x,y:[a,b]\to\mathbb{R},\ a<b, a,b\in\mathbb{R}$ be two smooth functions ($x,y\in C^{\infty}([a,b])$). How can I prove that there is a unique function $\theta:[a,b]\to\mathbb{R},\ \theta\in C^{\infty}([a,b])$ such that:
$$\begin{cases} x(t)\sin\theta(t)=y(t)\cos\theta(t),\ \forall\ t\in [a,b].\\ (x(t_0),y(t_0))=(x_0, y_0)\neq (0,0)\ \text{is given}\\ \theta(t_0)=\theta_0 \ \text{is given too such that}\ x(t_0)\sin\theta(t_0)=y(t_0)\cos\theta(t_0) . \end{cases}$$