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Trigonometry Lab Hours of Daylight

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    Two important events on the earth take place everyday that is Sunrise and Sunset. The time duration from the Sunrise and Sunset is day length (or length of days or length of daytime. However, it can be noticed that over the course of a year, the length of the day or the number of hours of daylight hours does not remain the same in any place.

    It varies from a certain minimum to maximum and than again minimum. The length of day at any place on a given day depends on its latitude. In each hemisphere, the higher the latitude, the shorter the day in winter. The length of day increases from winter to summer and than again decreases from summer to winter.

    On the Equator, the day length remains approximately of 12 hours.In this lab report, the number of daylight hours for a year for Boston, Massachusetts and Fairbanks, Alaska will be modeled using cosine function and differences in model will be discussed.

    The maximum daylight hour is equal to 15.3 hours and minimum daylight hour is equal to 9.1 hours.

    Therefore, the amplitude of the cosine function (model) will beHours. The scatterplot with day of year t, as the independent variable and number of hours of daylight, H as the dependent variable.F

    For day 0, the value of daylight hours is 9.1 hours; therefore, the phase shift of the cosine function is equal to 180o radians.Since, number of hours of daylight at the end of the year and at start of the year is same; therefore, the period of the cosine function will be 2 or 360o.

    The scatterplot along with graph of the cosine function  for Boston with day of year t, as the independent variable and number of hours of daylight, H as the dependent variable. It can be seen that both scatterplot and graph of the cosine function shows same cycle for the number of hours of daylight. Therefore, modeled function can be used for calculating number of hours of daylight for any given day for Boston, Massachusetts (42N).  The maximum daylight hour is equal to 20.7 hours and a minimum daylight hour is equal to 3.7 hours.

    The scatterplot for Fairbanks, Alaska with day of year t, as the independent variable and number of hours of daylight, H as the dependent variable. For day 0, the value of daylight hours is 3.7 hours; therefore, the phase shift of the cosine function will be equal to 180o radians. In addition, since, number of hours of daylight at the end of the year and at start of the year is same; therefore, the period of the cosine function will be 2 or 360o.

    The scatterplot along with graph of the cosine function  for Fairbanks with day of year t, as the independent variable and number of hours of daylight, H as the dependent variable. It can be seen that both scatterplot and graph of the cosine function shows approximately same cycle for the number of hours of daylight. Therefore, modeled function can be used for calculating number of hours of daylight for any given day for Fairbanks, Alaska (64N).Difference in Equation of Boston and Fairbanks and Other CitiesThe equation of the Boston and Fairbanks areBoston (42N):Fairbanks (64N):The only difference in above equation for Boston and Fairbanks is in amplitude of the cosine function.

    For Boston, the amplitude is 3.1 hours and for Fairbanks, the amplitude is 8.5 hours. Therefore, the amplitude of daylight hours increases as the latitude of the place increases.

    The only parameter that will change if the data is given for Tromso, Norway, which has latitude of 69 N and is a bit more than 200 miles inside the Arctic Circle, is Amplitude. As the latitude of Tromso is higher as compared to Fairbanks, therefore, its amplitude will be also higher (greater than 8.5 hours). The approximate cosine function will be given byTromso (69N): There are five important lines of latitude.

    Arctic Circle — 66° 33 39 N2.      Tropic of Cancer — 23° 26 21 N3.      Equator—0°4.      Tropic of Capricorn — 23° 26 21.

    Antarctic Circle — 66° 33 39 SArctic Circle in the Northern Hemisphere has latitude of 66° 33 39 N (or 66.56083°) and it is region north of equator. The zone just to the south of Arctic Circle is called the Northern Temperature Zone and equivalent polar circle in the Southern Hemisphere is called the Antarctic Circle. The Arctic Circle has the polar day that is 24-hour sunlit day also called as the “Midnight Sun” and polar night that is 24-hour sunless night.

    Both events happen at least once per year.The amplitude of the cosine function will be close to zero for the city Singapore that has latitude 1N.  The approximate cosine function will be given bySingapore (1N):The curve of the city Wellington, New Zealand (42S) will be mirror image of the city Boston, Massachusetts (42N) . The approximate cosine function for the city Wellington, New Zealand which is approximately as far south of the equator as Boston is north of the equator (latitude  42S). It can be seen that the days are lengthening the most rapidly for days 60 to 90 (orange shaded).

    Therefore, the slope of the line through these two points will beHours per daySlope (Minutes per day): Minutes per daySlope (minutes per week):  Minutes per weekThe equation of the line will be given byWhere H represents Daylight hours, and D represent Day of the year.

    It can be seen that the days are lengthening the most rapidly for days 30 to 40 (orange shaded). Therefore, the slope of the line through these two points will beHours per day Slope (Minutes per day): Minutes per daySlope (minutes per week):  Minutes per weekThe equation of the line will be given byWhere H represents Daylight hours, and D represent Day of the year.

    The slope of the line when the days are lengthening most rapidly for Boston and Fairbanks are 0. 05 hours per day (or 3 minutes per day or 21 minutes per week), and 0.25 hours per day (or 15 minutes per day or 105 minutes per week). Therefore, the days at Fairbanks are lengthening more sharply as compared to Boston.

    References

    1. Latitude and Longitude Coordinates For 7525 Cities Around The World. Accessed on March 16, 2009 from http://www.timegenie.com/latitude_and_longitude/#south_pacific
    2. Day Length for Various Latitudes. Accessed on March 16, 2009 from http://www.orchidculture.com/COD/daylength.html
    3. File:Day length.jpeg, retrieved on March 16, 2009 from http://en.wikipedia.org/wiki/File:Day_length.jpeg

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