# How do you find the amplitude, period, vertical and phase shift and graph y=cos(theta+pi/3)?

May 15, 2018

Simple Harmonic Motion/Oscillations (SHM) are graphed with cosine or sine functions, and the amplitude; period and phase shift can easily be found in the following way:

The graph for displacement can be described with:

$A \cos \left(\theta + \phi\right)$
Given: $\theta = {\theta}_{i} + \omega t$
The angular displacement is equal to the angular velocity * time.
We can substitute this into our equation:

$A \cos \left(\omega \cdot t + \phi\right)$

Where:
A == Amplitude
$\omega$ == Angular Velocity
$t$ == time
$\phi$ == phase change

$y = A \cos \left(\omega \cdot t + \frac{\pi}{3}\right)$

The Amplitude is equal to 1 (same as $1 \cos \left(\omega \cdot t + \phi\right)$)
The Phase Change ($\phi$) is equal to $\frac{\pi}{3}$

We know the period (T) is equal to:

$T = \frac{2 \pi}{\omega}$
$\omega = \frac{2 \pi}{T}$

Hence, substituting you will get
$y = A \cos \left(\frac{2 \pi}{T} \cdot t + \frac{\pi}{3}\right)$

To find the period we will have to substitute a value for time or find a time for displacement = 0