Periodic motion is the regular, repetitive motion of a body which continually retraces its path at regular intervals.
Period T of a periodic motion is the time to make one complete cycle.
Frequency f of a periodic motion is the number of cycles per unit time.
$T = \frac{1}{f}$
Angular frequency $\omega$ of a periodic motion is the rate of change of angular displacement with respect to time.
$\omega \, = \, 2 \pi f \, = \, \frac{2 \pi}{T}$
Displacement $x$ of an object is the distance of the oscillating particle from its equilibrium position at any instant.
Amplitude $x_{o}$ of a periodic motion is the magnitude of the maximum displacement of the oscillating particle from the equilibrium position.
$x = x_{o} \, sin \left( t \frac{2 \pi}{T} \right)$, used when motion starts from equilibrium position.
$x = x_{o} \, cos \left( t \frac{2 \pi}{T} \right)$, used when motion starts from extreme displacement.
If motion starts at somewhere between the amplitude and equilibrium, use:
$x = x_{o} \, sin \left( t \frac{2 \pi}{T} \right) \, + \, \phi$ OR $x = x_{o} \, cos \left( t \frac{2 \pi}{T} \right) \, + \, \phi$, where $\phi$ is the distance from equilibrium
