# Lesson 19

Beyond Circles

- Let's use trigonometric functions to model data.

### 19.1: Notice and Wonder: Examining Data

Here is some data that we will study in today’s lesson.

day | amount |
---|---|

1 | 0.99 |

2 | 1.00 |

3 | 0.98 |

4 | 0.93 |

5 | 0.86 |

6 | 0.77 |

7 | 0.67 |

8 | 0.57 |

9 | 0.46 |

10 | 0.37 |

day | amount |
---|---|

11 | 0.28 |

12 | 0.19 |

13 | 0.13 |

14 | 0.07 |

15 | 0.03 |

16 | 0.01 |

17 | 0.00 |

18 | 0.01 |

19 | 0.04 |

20 | 0.09 |

day | amount |
---|---|

21 | 0.16 |

22 | 0.24 |

23 | 0.33 |

24 | 0.43 |

25 | 0.54 |

26 | 0.65 |

27 | 0.76 |

28 | 0.85 |

29 | 0.92 |

30 | 0.98 |

31 | 1.00 |

What do you notice? What do you wonder?

### 19.2: Watching the Evening Sky

The data from the warm-up is the amount of the Moon that is visible from a particular location on Earth at midnight for each day in January 2018. A value of 1 represents a full moon in which all of illuminated portion of the moon's face is visible. A value of 0.25 means one fourth of the illuminated portion of the moon's face is visible.

- What is an appropriate midline for modeling the Moon data? What about the amplitude? Explain your reasoning.
- What is an appropriate period for modeling the Moon data? Explain your reasoning.
- Choose a sine or cosine function to model the data. What is the horizontal translation for your choice of function?
- Propose a function to model the Moon data. Explain the meaning of each parameter in your model and specify units for the input and output of your function.
- Plot the data using graphing technology and check your choice of parameters (midline, amplitude, period, horizontal translation). What changes did you make to your model?
- Use your model to predict when the next two full moons will be in 2018. Are your predictions accurate?
- How much of the Moon do you expect to be visible on your birthday? Explain your reasoning.

### Summary

Sometimes a phenomenon can be periodic even though it is not connected to motion in a circle. For example, here is a graph of the water level in Bridgeport, Connecticut, over a 50 hour period in 2018.

Notice that each day (or each 24 hour period) there are two tides, a small one where the water goes up to a little less than 3 feet and then a bigger one when the tide goes up to a little more than 3 feet. Since there are two tides per day, the period for this graph is about 12 hours. The data begins about 1 hour before the tide is at the 0 value. Since \(\sin(0) = 0\), this would make the sine a good choice for modeling the tide.

Putting together all of our information gives model \(f(h) = 3\sin\left(\frac{2\pi}{12}(h -1)\right)\), where \(h\) measures hours since midnight on September 1.

Notice that:

- The coefficient of 3 is the amplitude, which averages out the bigger and smaller tides.
- \(\frac{2\pi}{12}\) makes the period 12 hours.
- -1 translates the sine graph to the right by 1 hour to so it has a value of 0 at about 1 hour after midnight.