I



I. LESSON PLAN

Lesson Title Applying Trigonometric Functions to Everyday Life Situations

Lesson Summary Using “The Farmers Almanac”, students will discover how the length of day over a year can be modeled by a sine curve.

Key Words Period, Amplitude, Phase Shift, Tangent Lines, Local Extrema, Vertical Shift

Background knowledge

The assumptions of this inquiry lesson are:

• Students will have already successfully completed graphs of trig functions including horizontal and vertical translations as well as stretches and shrinks (when given the actual function).

• Students will already be proficient at entering data into a graphing calculator and performing regression.

Standards Addressed

Alignment to the Ohio Academic Content Standard (OACS): K – 12 Mathematics

Find one indicator that is addressed well by this lesson. List the standard, grade level, number of the indicator and then write out the indicator. Note: Give an indicator, not a benchmark.

Standard: Patterns, Functions, and Algebra

Grade Level: 12; Indicator #: 3

The indicator says: The indicator says: Describe and compare the characteristics of transcendental and periodic functions; e.g., general shape, number of roots, domain and range, asymptotic behavior, extrema, local and global behavior.

Name a process standard that was used in this lesson and describe how it was used.

Apply mathematical modeling to workplace and consumer situations, including problem formulation, identification of a mathematical model, interpretation of solution within the model, and validation to original problem situation.

List one OACS grade level indicators that students must have been mastered in order to be able to do this activity successfully:

Standard: Patterns, Functions, and Algebra

Grade Level: 11; Indicator #: 4

The indicator says: Identify the maximum and minimum points of polynomial, rational and trigonometric functions graphically and with technology.

Learning Objectives

• To make it clear to the students that trigonometry exists outside the classroom.

• To become more proficient at creating trig functions to fit a set of data

Materials

• Graph paper

• Graphing calculator

Suggested procedures

• Once students have grasped the basic concepts of sinusoidal graphs, this activity can be used to reinforce and expand the students’ understanding of the graphing process.

Assessments

Students will complete the extension exercises successfully to demonstrate proficiency.

II. INQUIRY BASED ACTIVITY

Title Applying Trigonometric Functions to Everyday Life Situations

Investigation’s Goals Based on the equation of a sinusoidal function, students will be able to identify Period, Amplitude, Phase Shift, Vertical Shift, Tangent Lines, and Local Extrema

Investigation

This investigation will explore the relationship between the day of the year and the amount of daylight using data from the “Old Farmer’s Almanac, 2008”

Create a scatter plot of the following data using the day of the year as the independent variable and the amount of daylight as the dependent variable.

|Day of Month – 2008 |Day of Year |Amount of Daylight (min) |

|January 6th |6 |553 |

|January 20th |20 |574 |

|February 3rd |34 |604 |

|February 17th |48 |640 |

|March 2nd |62 |678 |

|March 16th |76 |719 |

|March 30th |90 |759 |

|April13th |104 |798 |

|April 27th |118 |835 |

|May 11th |132 |869 |

|May 25th |146 |895 |

|June 8th |160 |912 |

|June 22nd |174 |917 |

|July 6th |188 |909 |

|July 20th |202 |890 |

|August 3rd |216 |861 |

|August 17th |230 |828 |

|August 31st |244 |790 |

|September 14th |258 |751 |

|September 28th |272 |711 |

|October 12th |286 |673 |

|October 26th |300 |634 |

|November 9th |314 |600 |

|November 23rd |328 |570 |

|December 7th |342 |551 |

|December 21st |356 |544 |

Regressions

1. Based on the shape of the graph, what type of regression will be appropriate? (Hint: You might want to anticipate what the data might look like for the next year or two.) Explain your decision by providing two aspects of the graph that are unique to this type of equation.

2. Write your regression equation. How well does it fit the data? Explain.

3. Find the Domain and Range of the regression curve.

In this activity, we will find the period, amplitude and the phase shift of a sinusoidal curve. Before we can accomplish this task we must expand our view of the sinusoidal regression because we can only see the curve for one year. Let’s expand the viewing window by raising the x-max to 3 years or 1100 days. Now, we can see that the curve repeats for each calendar year.

Attributes of a Sinusoidal Graph Questions:

Period:

Now, let’s discuss the period of a curve. The period is the measure of one length of a trigonometric cycle. To find the period of our sine regression, we need to find two days that have the same length in time.

4. The easiest values to locate on a periodic graph are the maximums and minimums. Use your technology to find the first two maximums or the first two minimums.

5. Subtract the two consecutive x-values to find the period. What is the period? How do you know that your answer is correct?

6. If you found the maximum values, find the minimum values or vice versa.

7. Write the days that give you the shortest and longest days. Using the Old Farmer’s Almanac tables, change them from days of the year to days of each month. What do you notice about each day? Why are these days familiar?

8. Why does the regression equation not give us the exact dates?

9. In general, what is the period of the standard function [pic]

10. The period of the regression equation is different than [pic]. Right now, it cannot be seen in the equation[pic], but it’s there. To find it we must find the ratio of the period of sin(x) and the period of the regression equation. Where is this ratio in the regression equation?

11. So, describe how you could work backwards using the equation [pic] to find the period of any sine equation?

Amplitude:

Next, we want to investigate the amplitude of a sinusoidal curve. Generally, amplitude is defined as height of the curve above a horizontal axis. Usually the x-axis is the horizontal axis. Here, the regression curve certainly exceeds y = 0. So, we will find the amplitude by saying that it is half of the difference between the maximum and minimum of the graph.

12. Using the values of the local maximums and minimums that you found above, write equations of the horizontal lines that bound the curve.

13. Use these two lines to find the height of the curve and divide by 2. What is the amplitude of the graph?

14. Now, look at the regression equation [pic]. Do you see the amplitude in the equation? What coefficient a, b, c, or d is it?

Phase Shift:

Finally, let’s consider the phase shift. The phase shift is the horizontal shift left or right. So, we need to recall where the function f(x) = sin(x) crosses the y-axis.

15. Using your knowledge of the graph of [pic], does it cross the y-axis nearer a minimum, a maximum, or halfway between each? Explain?

16. We need to find this same point on our regression. To do this, change your window to see the minimum value just left of the y-axis. Find the point that was discussed in the previous question.

17. How far right of the y-axis is this point? How would you describe the phase shift?

18. Do you see this value in the equation?

19. This value is not in the equation because it is affected by the period/horizontal stretch. It would usually be the value subtracted from x. However, the coefficient of x changes this value. So, we need to factor [pic]into [pic]. What is the result? What do you notice about the value subtracted from x?

20. What specific day of the year does this phase shift represent? What is the significance of this date?

Vertical Shift:

Finally, let’s consider the vertical shift. The vertical shift is the shift up or down. So, we need to recall where the function f(x) = sin(x) crosses the y-axis.

21. Using your knowledge of the graph of [pic], does it cross the x-axis nearer a minimum, a maximum, or halfway between each? Explain?

22. We need to find this same point on our regression. To do this, change your window to see the minimum value just left of the y-axis. Find the point that was discussed in the previous question. (This is the same answer to #2 to find the Phase Shift.)

23. How far above of the x-axis is this point? How would you describe the vertical shift?

24. Where in the equation [pic]do you see this value? a, b, c, or d?

Generalize your results:

Using a sine equation, [pic], explain how to find the following:

25. Period –

26. Amplitude –

27. Phase Shift –

28. Vertical Shift-

Numerical Analysis:

Tangent Lines and Rates of Change.

In this portion of the lab, you will explore how rates of change, as observed in a table, can be used to find interesting points on the graph of a function.

29. As an appendix to this activity, there are several charts featuring the day of the month, the day of the year, and the length of daylight in minutes. Please refer to the table showing the entire year. What general trend do you observe in the amount of daylight from the beginning of the year to the end?

30. Estimate which month and day of the year that each of the following occurs. [Hint: The answers for the questions below may not occur on a specific table value date, but rather in-between values]:

a. Maximum amount of daylight.

b. Minimum amount of daylight.

c. Greatest increase in the amount of daylight.

d. Greatest decrease in the amount of daylight.

31. Now examine the table featuring the values for the month of June.

a. Using information from day 1 and day 3 of June, find the rate of change in the daylight per day.

i. Subtract the daylight for day 3 from day 1.

ii. Divide this value by 2 (since the difference in the days is 2).

iii. What are the units of this quotient? _________________

b. Repeat steps i) and ii) for the following days: Days 18 and 20.

c. Repeat the steps for the following days: Days 28 and 30.

d. Explain why the value for a) is positive, the value for b) is 0, and the value for c) is negative, and what it means for each to be positive, negative or zero.

e. What you found in part b) above is known as the “summer solstice,” and marks the “longest day of the year,” or the day on which the amount of daylight reaches a maximum (the “winter solstice,” or “shortest day of the year,” happens exactly six months later in December.) For 2008, the summer solstice falls on June 21. How close was this to your guess from part 30 a) from the previous page?

f. What else changes or “begins” on the summer solstice? On the winter solstice?

32. Now examine the table featuring the values for the month of March.

a. Using information from day 1 and day 3 of March, find the rate of change in the daylight per day.

i. Subtract the daylight for day 3 from day 1.

ii. Divide this value by 2 (since the difference in the days is 2).

Be sure to include proper units.

b. Repeat steps i) and ii) for the following days: Days 19 and 21.

c. Repeat the steps for the following days: Days 28 and 30.

d. Which of the above has the highest value?

e. What your found in b) above is known as the “spring equinox,” and marks the day on which the rate of change of daylight reaches a positive maximum (the “Autumnal Equinox,” or “shortest day of the year,” happens exactly six months later in September.) For 2008, the spring equinox falls on March 20. How close was this to your guess from part 30 c) from two pages previous?

f. What else changes or “begins” on the spring equinox? On the autumnal equinox?

33. One major concept that you will see in a future calculus course is that of a tangent line to a curve. The “tangent line” problem goes back to the time of Sir Isaac Newton, when he and other mathematicians were trying to understand how a tangent line function could be defined at any point on a curve.

a. Consult the table for the month of June. Using the data from June 19, create an ordered pair featuring (day of the year, amount of daylight)

b. Using [pic], where m is the value of the rate of change of daylight on June 19, (which you calculated in 31 b)), create a tangent line equation.

c. What kind of line is this tangent line (how would you describe its orientation)?

d. On what other day of the year would this kind of line appear on a graph of amount of daylight vs. day of the year? Why?

e. Now consult the table for the month of March. Using the data from March 20, repeat steps a) and b) from above.

f. On what other day of the year would this kind of line, featuring the same slope, except negative, appear on a graph of amount of daylight vs. day of the year? Why?

Extension:

While pushing your nephew on a swing in the local park, you study how the swing goes back and forth and begin to think of your work with trigonometry in math class. Because you know that the chains that hold the swing are fastened at the center of a circle so that the motion of the swing is circular, you surmise that the distance of the swing from the center point of its motion is actually a cosine function. You decide that your nephew is not going to tire of the swing any time soon, so you take out your trusty tape measure and stopwatch and record the following times and distances:

|Time in seconds |0 |1 |2 |3 |4 |5 |

|Distance in feet |4 |0 |-4 |0 |4 |0 |

34. On graph paper, plot the points putting time on the x-axis and distance on the y-axis.

35. Since the swing takes four seconds to complete a cycle, the period of our function is four. Calculate the B value using the fact that [pic].

36. Calculate the amplitude of the function by finding the distance from the smallest and largest distance and dividing by 2.

37. Since the highest distance occurs when time = 0, is there any need to consider a phase shift of this cosine function?

38. Write the equation of the function.

39. List all of the definitions and properties you learned in this activity.

APPENDIX

March’s Data:

|Day of March |Day of Year|Amount of |Change of time |

| | |Daylight (min) |between Consecutive |

| | | |Days (min) |

| 1 |61 |676 |NA |

|2 |62 |678 | |

|3 |63 |681 | |

|4 |64 |685 | |

|5 |65 |687 | |

|6 |66 |690 | |

|7 |67 |693 | |

|8 |68 |695 | |

|9 |69 |699 | |

|10 |70 |702 | |

|11 |71 |704 | |

|12 |72 |707 | |

|13 |73 |710 | |

|14 |74 |712 | |

|15 |75 |716 | |

|16 |76 |719 | |

|17 |77 |722 | |

|18 |78 |724 | |

|19 |79 |727 | |

|20 |80 |730 | |

|21 |81 |733 | |

|22 |82 |736 | |

|23 |83 |739 | |

|24 |84 |742 | |

|25 |85 |745 | |

|26 |86 |747 | |

|27 |87 |750 | |

|28 |88 |753 | |

|29 |89 |757 | |

|30 |90 |759 | |

|31 |91 |762 | |

June’s Data:

|Day of June |Day of Year|Length of |Change of time |

| | |Day (min) |between Consecutive |

| | | |Days (min) |

|1 |153 |905 |NA |

|2 |154 |906 | |

|3 |155 |908 | |

|4 |156 |908 | |

|5 |157 |910 | |

|6 |158 |911 | |

|7 |159 |911 | |

|8 |160 |912 | |

|9 |161 |913 | |

|10 |162 |914 | |

|11 |163 |915 | |

|12 |164 |915 | |

|13 |165 |916 | |

|14 |166 |916 | |

|15 |167 |916 | |

|16 |168 |917 | |

|17 |169 |917 | |

|18 |170 |917 | |

|19 |171 |917 | |

|20 |172 |917 | |

|21 |173 |917 | |

|22 |174 |917 | |

|23 |175 |917 | |

|24 |176 |916 | |

|25 |177 |916 | |

|26 |178 |916 | |

|27 |179 |916 | |

|28 |180 |915 | |

|29 |181 |914 | |

|30 |182 |914 | |

September’s Data:

|Day of September |Day of Year |Amount of |Change of time |

| | |Daylight (min) |between Consecutive |

| | | |Days (min) |

|1 |245 |788 |NA |

|2 |246 |785 | |

|3 |247 |782 | |

|4 |248 |780 | |

|5 |249 |777 | |

|6 |250 |774 | |

|7 |251 |771 | |

|8 |252 |769 | |

|9 |253 |765 | |

|10 |254 |762 | |

|11 |255 |759 | |

|12 |256 |757 | |

|13 |257 |754 | |

|14 |258 |751 | |

|15 |259 |748 | |

|16 |260 |746 | |

|17 |261 |743 | |

|18 |262 |740 | |

|19 |263 |737 | |

|20 |264 |735 | |

|21 |265 |732 | |

|22 |266 |729 | |

|23 |267 |726 | |

|24 |268 |723 | |

|25 |269 |720 | |

|26 |270 |717 | |

|27 |271 |714 | |

|28 |272 |711 | |

|29 |273 |709 | |

|30 |274 |706 | |

| | | | |

December’s Data:

|Day of December |Day of Year |Amount of |Change of time |

| | |Daylight (min) |between Consecutive |

| | | |Days (min) |

|1 |336 |558 |NA |

|2 |337 |556 | |

|3 |338 |555 | |

|4 |339 |554 | |

|5 |340 |553 | |

|6 |341 |552 | |

|7 |342 |551 | |

|8 |343 |551 | |

|9 |344 |550 | |

|10 |345 |549 | |

|11 |346 |548 | |

|12 |347 |547 | |

|13 |348 |546 | |

|14 |349 |546 | |

|15 |350 |546 | |

|16 |351 |555 | |

|17 |352 |555 | |

|18 |353 |555 | |

|19 |354 |555 | |

|20 |355 |555 | |

|21 |356 |554 | |

|22 |357 |554 | |

|23 |358 |554 | |

|24 |359 |555 | |

|25 |360 |555 | |

|26 |361 |555 | |

|27 |362 |556 | |

|28 |363 |557 | |

|29 |364 |557 | |

|30 |365 |557 | |

|31 |366 |558 | |

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