How did we get $\phi=A \sin (\omega t + \delta)$ for the skateboard example?

This came from the solution to the differential equation that came from the Newton's second law in the $\hat{\phi}$ direction, which was $\ddot{\phi} = -\frac{g}{R} \phi$. The solution to this equation is a sinusoid, and the most general sinusoidal solution to this DE can be written $\phi=A \sin (\omega t + \delta)$, where $\omega = \sqrt{g/R}$ (try plugging it in!). $\delta$ and $A$ are constants to be determined by initial conditions.