Question

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

Answer 1

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:

`Acos(theta + phi)`
Given: `theta = theta_{i} + omega t`
The angular displacement is equal to the angular velocity * time.
We can substitute this into our equation:

`Acos(omega * t + phi)`

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

In your example:
`y = Acos(omega * t + pi/3)`

The Amplitude is equal to 1 (same as `1cos(omega*t + phi)`)
The Phase Change (`phi`) is equal to `pi/3`

We know the period (T) is equal to:

`T = (2pi) / omega`
`omega = (2pi) / T`

Hence, substituting you will get
`y = Acos((2pi) / T * t + pi/3)`

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