**Unformatted text preview: **750 Applications of Trigonometry Example 11.1.2. According to the U.S. Naval Observatory website, the number of hours H of
daylight that Fairbanks, Alaska received on the 21st day of the nth month of 2009 is given below.
Here t = 1 represents January 21, 2009, t = 2 represents February 21, 2009, and so on.
Month
Number
Hours of
Daylight 1 2 3 4 5 6 7 8 9 10 11 12 5.8 9.3 12.4 15.9 19.4 21.8 19.4 15.6 12.4 9.1 5.6 3.3 1. Find a sinusoid which models these data and use a graphing utility to graph your answer
along with the data.
2. Compare your answer to part 1 to one obtained using the regression feature of a calculator.
Solution.
1. To get a feel for the data, we plot it below.
H
22
20
18
16
14
12
10
8
6
4
2
1 2 3 4 5 6 7 8 9 10 11 12 t The data certainly appear sinusoidal,5 but when it comes down to it, ﬁtting a sinusoid to
data manually is not an exact science. We do our best to ﬁnd the constants A, ω , φ and B
so that the function H (t) = A sin(ωt + φ) + B closely matches the data. We ﬁrst go after
the vertical shift B whose value determines the baseline. In a typical sinusoid, the value of B
is the average of the maximum and minimum values. So here we take B = 3.3+21.8 = 12.55.
2
Next is the amplitude A which is the displacement from the baseline to the maximum (and
minimum) values. We ﬁnd A = 21.8 − 12...

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