The Trigonometry of Temperatures

PRE CALCULUS B

Directions: In this portfolio, you will use your knowledge of the period, amplitude, vertical shift, horizontal shift, domain, and range of a trigonometric function to write sine and cosine functions that model average monthly temperatures in three different cities.

Part A

Step 1

Use your favorite search engine or visit your public library to do research to find the latitude of the city in which you live (or the closest major city) and the average monthly temperatures for that city. You will need to cite your source(s). Enter the information in the spaces provided.

City Name: Toledo

Latitude of the City:41.6528° N

Month 1

Jan 2

Feb 3

Mar 4

Apr 5

May 6

Jun

Average temperature 33 f 36 f 47 f 60 f 71 f 81 f

month 7

Jul 8

Aug 9

Sep 10

Oct 11

Nov 12

Dec

Average temperature 85 f 82 f 75 f 63 f 50 f 36 f

Sources: https://www.currentresults.com/Weather/Ohio/Places/toledo-temperatures-by-month-average.php

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Step 2

Plot the 12 values on a graph. The number corresponding to the month is the first coordinate of each point, and average temperature for that month is the second coordinate. The points should create a periodic pattern. Assume that the data is, in fact, periodic and use the graph to determine the following values for both a sine function and a cosine function:

Sine Cosine

Vertical Shift

Horizontal Shift

Amplitude

Period

Domain

Range

Based on the data in your table, write an equation for a sine function and for a cosine function.

Sine Function:

Cosine Function:

Step 3

Choose one of these functions and sketch it on the same graph as the graph with the 12 data points. Write the name of your city as the title for the graph. You will send this graph as an attachment when you submit the portfolio.

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Part B Repeat steps 1–3 for a new city that is north of the city you used in Part A.

Step 1 Choose a city that is north of your city from Part A and find its latitude. Record the average monthly temperatures for that city. You will need to cite your source(s). Enter the information in the spaces provided.

Name of Northern City: Detroit

Latitude of Northern City: 42.3314° N

Month 1

Jan 2

Feb 3

Mar 4

Apr 5

May 6

Jun

Average Temperature 32 f 35 f 46 f 59 f 70 f 79 f

Month 7

Jul 8

Aug 9

Sep 10

Oct 11

Nov 12

Dec

Average Temperature 83 f 81 f 74 f 62 f 49 f 36 f

Sources: https://www.currentresults.com/Weather/Michigan/Places/detroit-temperatures-by-month-average.php

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Step 2 Plot these 12 points on a graph. Use the graph to determine the following values for both a sine function and a cosine function that models the data:

Sine Cosine

Vertical Shift

Horizontal Shift

Amplitude

Period

Domain

Range

Based on the data in your table, write an equation for a sine function and for a cosine function.

Sine Function:

Cosine Function:

Step 3 Choose one of these functions and sketch it on the same graph as the graph with the 12 data points. Make the title of the graph the name of the city with the word North after the name. You will send this graph as an attachment when you submit the portfolio.

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Part C Repeat steps 1–3 for a new city that is south of the city you used in Part A.

Step 1 Choose a city that is south of your city from Part A and find its latitude. Record the average monthly temperatures for that city. You will need to cite your source(s). Enter the information in the spaces provided.

Name of Southern City: Columbus

Latitude of Southern City: 39.9612° N

Month 1

Jan 2

Feb 3

Mar 4

Apr 5

May 6

Jun

Average temperature 37 f 41 f 51 f 64 f 73 f 82 f

Month 7

Jul 8

Aug 9

Sep 10

Oct 11

Nov 12

Dec

Average temperature 85 f 84 f 77 f 65 f 53 f 40 f

Sources

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Step 2 Plot the 12 values on a graph. Use the graph to determine the following values for both a sine function and a cosine function that models the data:

Sine Cosine

Vertical Shift

Horizontal Shift

Amplitude

Period

Domain

Range

Based on the data in your table, write an equation for a sine function and for a cosine function.

Sine Function:

Cosine Function:

Step 3 Choose one of these functions and sketch it on the same graph as the graph with the 12 data points. Make the title of the graph the name of the city with the word South after the name. You will send this graph as an attachment when you submit the portfolio.

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Part D Compare the tables of average temperatures, the equations, and the graphs created for the three cities.

How well do your functions fit the data points?

What might account for the trigonometric functions not being a perfect fit for the points plotted?

Write at least two paragraphs about any conclusions you can draw from the data as shown in each representation. What values were the same in all three tables? What values were different? What might account for these similarities and differences? Be sure to discuss the impact that the latitude of each city has on the trigonometric function that models its monthly temperatures.

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Part E You will upload the following into the Drop Box:

• this completed worksheet

• a graph of the average monthly temperatures in your city and the sine or cosine function used to model the data

• a graph of the average monthly temperatures in a northern city and the sine or cosine function used to model the data

• a graph of the average monthly temperatures in a southern city and the sine or cosine function used to model the data

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