458 CHAPTER 6 Trigonometric Functions Solution (a) Step 1 = − = − = Amplitude largest data value smallest data value 2 15.28 9.07 2 3.105 Step 2 = + = + = Vertical shift largest data value smallest data value 2 15.28 9.07 2 12.175 Step 3 The data repeat every 365 days. Since π ω = = T 2 365, we find ω π = 2 365 So far, we have π φ ( ) = − + y x 3.105sin 2 365 12.175. Step 4 To determine the horizontal shift, use the period = T 365 and divide the interval [ ] 0, 365 into four subintervals of length ÷ = 365 4 91.25: [ ] [ ] [ ] [ ] 0, 91.25 , 91.25, 182.5 , 182.5, 273.75 , 273.75, 365 The sine curve is increasing on the interval [ ] 0, 91.25 and is decreasing on the interval [ ] 91.25, 273.75 , so a local maximum occurs at = x 91.25. Since the maximum occurs on the summer solstice at = x 172, we must shift the graph of the function − = 172 91.25 80.75 units to the right by replacing x by −x 80.75. Doing this yields π ( ) ( ) = − + y x 3.105sin 2 365 80.75 12.175 Next, multiply out to obtain the form ω φ ( ) = − + y A x B sin . π π ( ) = − + y x 3.105sin 2 365 323 730 12.175 (b) To predict the number of hours of daylight on April 1, let = x 91 in the function found in part (a) and obtain π π ( ) = ⋅ − + ≈ y 3.105sin 2 365 91 323 730 12.175 12.72 The prediction is that there will be about = 12.72 hours 12 hours 43 minutes of sunlight on April 1 in Boston. (c) Using a TI-84 Plus CE graphing calculator, the graph of the function found in part (a) is given in Figure 89. (d) According to the Old Farmer’s Almanac, there will be 12 hours 44 minutes of sunlight on April 1 in Boston. Figure 89 0 8 16 400 Now Work PROBLEM 35 Certain graphing utilities (such as the TI-84 Plus CE, GeoGebra, and Desmos) have the capability of finding the sine function of best fit for sinusoidal data.At least four data points are required for this process. Finding the Sine Function of Best Fit Use a graphing utility to find the sine function of best fit for the data in Table 11 (p. 455). Graph this function with the scatter plot of the data. EXAMPLE 5 Solution Enter the data from Table 11 and execute the SINe REGression program. The result using a TI-84 Plus CE is shown in Figure 90 on the next page. The output that the utility provides shows the equation ( ) = + + y a bx c d sin
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