Sunrise, Sunset; a Sinusoidal Story

Today in precalculus, we looked at some data that follow a sinusoidal pattern and calculated a regression function to model it. Here’s how we did it:

The hours of daylight over a year increase, then decrease in a regular, periodic fashion – just the kind of data that result in a nice sine curve. The US Navy maintains a website where you can enter any location and see the sunrise and sunset times over an entire year:

http://aa.usno.navy.mil/data/docs/RS_OneYear.php

We entered our location (Nashville, TN), and used the sunrise and sunset times for the middle of the months:

Raw Data

The times are given in hours and minutes, so we converted them to decimal hours. We then subtracted sunrise time from sunset time to get the number of daylight hours for each month:

Daylight Hours

Now that we had our data, we entered it into our graphing calculators:

Stat

After setting our window parameters, we plotted the data:

StatPlot

Nice! Next, we calculated the regression function that best fits the data (This might be a good time to discuss pros and cons of different regression functions: quadratic, cubic, quartic, etc.):

We then plotted the regression function with the data points. Nice fit!

Plot

Because we did this in mid-February (x = 2.5), we used our model to estimate how many hours of daylight we should expect to have today: 11.4 hours.

If you’re ready to move beyond graphing calculators (and I certainly am!), then you can do this activity with the Desmos online function grapher. What’s nice about that approach is the ability to set up sliders, and let students fit the regression function to the data manually.

Here are a couple of screenshots:

Desmos Data

Desmos Regression Function

 

Update: Apparently this is the source of the Chinese Yin/Yang symbol.

Math Screencast Library

In my previous post, I mentioned using screencasts as part of the “reverse classroom” teaching method I am beginning to implement. I use screencast-o-matic to create the screencasts. You can access all of my Precalculus ones here. My BC Calculus ones are here. To view them, your browser must have Java installed.

Here’s an example of one I have uploaded to YouTube:

I embed these screencasts into my courses’ Moodle pages, where I can make them visible or invisible as needed. Eventually, I hope to have a substantial library of lectures that I can use year after year.