Graphing Trig Functions

An object whose displacement or movement can be described using a sinusoidal function is said to follow simple harmonic motion. The displacement \(d\) of the object can be modeled as a function over time \(t\) using one of the formulas below.

\[\begin{align*} d &= a\sin(\omega t) \\ d &= a\cos(\omega t) \end{align*}\]

The \(a\)-value is the amplitude, or maximum displacement of the object from rest. The \(\omega\)-value, "omega", is related to the period of the object's motion by the formula \(\omega = \frac{2\pi}{\text{period}}\), or by \(\text{period} = \frac{2\pi}{\omega}\). The frequency of the object's motion is the reciprocal of the period, so \(f = \frac{1}{\text{period}} = \frac{\omega}{2\pi}\), or \(\omega = 2\pi f\). We use sine if the object's initial position is at rest or at equilibrium, and we use cosine if its initial position is at its maximum displacement.