# Precalculus Project Sine and Cosine Modeling of Periodic Behavior

1. Include your name, title of your project indicating what data you will work with
2. Find periodic data online (possible examples are below)
1. Average monthly temperatures for a city
2. Average monthly ocean temperatures at a certain port
3. Time of sunrise or sunset in a city monthly on a certain day
4. Number of daylight or nighttime hours on a certain day recorded every month

-          If you realized all of these data sets have 12 values, since we look at values in each month

-          You have to choose data sets with at least 12 values

-          Extra  credit (1pt): choosing data sets with at least 24 values, such as every day in a year (365 values) or twice a month (24 values) will gain extra credits

1. Use either sine or cosine curve to model the data

-          show plots of your data in a graph, could be hand drawn, or you can utilize online graphing calculator (explained later)

-          follow steps 1-5, show your work, explain how you calculated values: ‘a’, ‘b’, ‘c’, ‘d’ based on your data and how you found the final equation

-          print a picture of your final curve showing how closely it followed your data

o   use https://www.desmos.com/calculator (easy to use graphing calculator online, where you can input your table and graph, then print it) or any other graphing calculator online

§  input your data: in left corner there is ‘+’ sign, click and add table, you can input your data, x being month and y being observed temperature in previous case (table will go into left side box#1)

§  you can zoom in and out or drag picture right and left to make sure all of your data points are shown (zoom is in right corner)

§  use keypad in the bottom of the page and input your curve using x and y variables on left side into box #2 (if data points are in box #1)

·         or just use your keyboard to input function, in previous case it was: y = 13.5 cos 30(x - 1) + 71.5

§  graph of curve and data plot should be in one graph (checking your work: make sure that you curve will approximate your data points)

§  printed picture will be about .5 page (right click mouse, print)

1. Pick a random value and indicate how closely your model approximated the actual value of your original data. Include how you calculated the approximated value using the equation and how much was the difference between approximation and actual value (step 6)
2. Possible extra credit (1 pt): find which out of your 12 actual data was approximated closest by your periodic function, indicate calculations and difference of each approximated and actual value
3. Be neat if you write by hand or type your project

Good luck with the project!

Prof. Benko

In this project you have to find periodic data to create a sine or cosine model.  The set of data that we will use as an example is the average monthly temperatures for Santiago, Chile.

 Month Average Monthly High Temperatures for Santiago, Chile January 85 February 85 March 81 April 72 May 65 June 58 July 58 August 61 September 65 October 72 November 77 December 83 Below is the data graphed in a scatter plot: This data is in the shape of a cosine curve.  This is due to the fact that the Southern Hemisphere has their coldest months in the middle of the year May – September.  Using the steps to find a cosine model, we will find a model that best fits this data.

Step 1:  Find the amplitude:

To find the amplitude one must take the maximum temperature – minimum temperature and divide it by 2.  This is because the amplitude is the distance from the primary axis to the maximum or minimum point of the data.

Maximum Temperature:  85                            Minimum Temperature:  58 , so the value of ‘a’ in the model: y = a cos b(x - c) + d is 13.5.

Step 2:  Find the vertical translation:

To find the vertical translation one must take the maximum temperature + the minimum temperature and divide it by 2.  This is because the vertical shift moves the primary axis up however many units from the x – axis. , so the value of ‘d’ in the model is 71.5.

So far, we have y = 13.5 cos b(x -h) + 71.5, now I must find the values of ‘b’ and ‘c’.

Step 3:  Find the period, hence finding the value of ‘b’:

To fine the period, we just look at how long it takes to complete one cycle.  For this data, one cycle is completed in 1 year or twelve months.  Therefore, the period is 12.  So to find the value of ‘b’ we must solve the equation: . ; therefore 12b = 360, and b = 30

So, our model is y = 13.5 cos 30(x - c) + 71.5.

Step 4: Find the horizontal translation. (Value of ‘c’)

To do this, we must graph both the data and the model on the same window.  Then we must find out how far to move the graph left or right by choosing the minimum values of both the data and the model.  Minimum (7, 58)                                                       Minimum (6, 58)

Therefore, the graph must be shifted 1 unit to the right.  This is the horizontal shift, so c=1.

Step 5: Write the cosine equation

So our final model is y = 13.5 cos 30(x - 1) + 71.5, and the graph should look like: As you can see, this is a very good model to fit the data.  Most points fall on the model, and it would be a great model to use to predict future temperature.

Step 6: Take a random value and find how closely the model approximates the actual temperatures from the table

So, according to the model let’s figure out the average temperature in the month of February (February is the second month, so x=2):

Y = 13.5 cos (30( 2 – 1)) + 71.5.  The average temperature according to my model is 83.2, and according to the data the average temperature for the data is 85. The model is closed to the actual value, the difference is 1.8F.

Field of study: