The differential equation for the angle as a function of time $t$ is

$\frac{{\ⅆ}^2\mathrm{\θ}}{\ⅆ t^2}\+\frac{g}{L}\sin \left(\mathrm{\theta }\right)\=0$.

It is not possible to write a formula for the solution to this equation in terms of an elementary function. Instead, we use an approximation which is fairly accurate if the angle $\mathrm{\theta }$ is sufficiently small (i.e. when $\sin \left(\mathrm{\θ}\right)\approx \mathrm{\θ}$ ):

  $\frac{{\ⅆ}^2\mathrm{\θ}}{\ⅆ t^2}\+\frac{g}{L}\mathrm{\θ}\=0$.

Solving this differential equation allows us to find formula for the angle of the pendulum at a given time $t$ :

$\mathrm{\θ}\left(t\right)\={\mathrm{\θ}}_0\cdot \cos \left(\sqrt{\frac{g}{L}}t\right)$

and the angular speed at a given time $t$ :

$\frac{\ⅆ\mathrm{\θ}}{\ⅆ t}\={\mathrm{\θ}}_0\cdot \sqrt{\frac{g}{L}}\sin \left(\sqrt{\frac{g}{L}}t\right)$ 

where ${\mathrm{\theta }}_0$ is the initial angle of the pendulum. Note that the angular frequency of the pendulum is a constant${\mathrm{\ω}}_0\=\sqrt{\frac{g}{L}}$.