The trigonometric functions exhibit periodic behavior. Periodic behavior is common in our everyday lives; here are a few examples:

- Number of people online in a game with respect to time
- See my personal project DarzaCharts, a tool which tracks the live playercount of a multiplayer game

- Height of tide with respect to time
- Vibrations of a stringed instrument (guitars, violins)

This section will explore how to model some of these behaviors using the trigonometric functions.

If the equation describing the displacement $y$ of an object at time $t$ is \[y = a \sin \omega t \qquad \text{ or } y = a \cos \omega t\]
then the object is in **simple harmonic motion**. In this case,

amplitude | $\lvert a \rvert$ | maximum displacement of the object |

period | $\frac{2\pi}{\omega}$ | time required to complete one cycle |

frequency | $\frac{\omega}{2\pi}$ | number of cycles per unit of time (usually in Hertz, denoted Hz) |

A mass is suspended by a spring in the air. External force is introduced to the spring, and the displacement of the mass modeled by the function \[y = 10 \sin 4 \pi t\]
where $y$ is measured in inches and $t$ in seconds.

- Find the amplitude, period, and frequency of the motion of the mass with the proper units.
- Sketch one period of the displacement.
- How high above the resting point is the mass at $t = \frac{1}{8}$ seconds?

A trombone player plays the note E and holds the note. Suppose the E causes a variation in air pressure from the normal air pressure with respect to the equation \[V(t) = 0.2 \sin 80\pi t\]
where $V(t)$ is measured in pounds per square inch and $t$ is measured in seconds.

- Find the amplitude, period, and frequency of $V(t)$ with the proper units.
- If the player increases the loudness of the note, what changes in $V(t)$?
- If the player plays a lower note instead, what changes in $V(t)$?

A mass is suspended from a spring; it is at rest. The spring is compressed a distance of $4$ cm upwards and then released. It is observed that the mass returns to the compressed position after $\frac{1}{3}$ seconds.

- Find a function that models the displacement of the mass.
- Graph one period of the displacement of the mass over time.

Any sine curve can be expressed in the following equivalent forms

$y = A \sin (kt - b)$ | where the phase is $b$ |

$y = A \sin k\left(t - \frac{b}{k}\right)$ | where the horizontal shift is $\frac{b}{k}$ |

Given two functions \[y_1 = A \sin (kt - b) \qquad y_2 = A\sin(kt - c)\] the **phase difference** is $b - c$. If this is a factor of $2\pi$, then the functions are called **in phase.**

If two functions have the same period and have the same phase, the two graphs will lie on top of each other.

Three objects are in simple harmonic motion given by the following equations \[y_1 = 10 \sin \left(3t - \frac{\pi}{6}\right) \qquad y_2 = 10 \sin \left(3t - \frac{\pi}{2}\right) \qquad y_3 = 10 \sin \left(3t + \frac{23\pi}{6}\right)\]

- Find the phase difference between $y_1$ and $y_2$. Are they in phase?
- Find the phase difference between $y_1$ and $y_3$. Are they in phase?