Proof the $\sin$ is a solution for the [[../02 Areas/Math/Differential Equations|Differential Equation]] for [[../02 Areas/Physics/Simple Harmonic Motion|Simple Harmonic Motion]]: $\begin{align} \vec x(t) &= \vec A \sin (\omega t + \phi) + \vec B \\ \deriv{\vec x}{t} &= \vec A \cos (\omega t + \phi)\omega \\ \deriv{^2\vec x}{t^2} &= -\vec A \sin (\omega t + \phi)\omega^2 \\ \end{align} $