628 CHAPTER 6 The Circular Functions and Their Graphs A Trigonometric Model Sine and cosine functions may be used to model many real-life phenomena that repeat their values in a cyclical, or periodic, manner. Average temperature in a certain geographic location is one such example. −45 45 25 0 Figure 30 6.3 Exercises CONCEPT PREVIEW Fill in the blank(s) to correctly complete each sentence. 1. The amplitude of the graphs of the sine and cosine functions is , and the period of each is . 2. For the x-values 0 to p 2 , the graph of the sine function and that of the cosine function . (rises / falls) (rises / falls) EXAMPLE 7 Interpreting a Sine Function Model The average temperature (in °F) at Mould Bay, Canada, can be approximated by the function ƒ1x2 = 34 sin c p 6 1x - 4.32d , where x is the month and x = 1 corresponds to January, x = 2 to February, and so on. (a) To observe the graph over a two-year interval, graph ƒ in the window 30, 254 by 3-45, 454. (b) According to this model, what is the average temperature during the month of May? (c) What would be an approximation for the average annual temperature at Mould Bay? SOLUTION (a) The graph of ƒ1x2 = 34 sin C p 6 1x - 4.32D is shown in Figure 30. Its amplitude is 34, and the period is 2p p 6 = 2p , p 6 = 2p # 6 p = 12. Simplify the complex fraction. Function ƒ has a period of 12 months, or 1 year, which agrees with the changing of the seasons. (b) May is the fifth month, so the average temperature during May is ƒ152 = 34 sin c p 6 15 - 4.32d ≈12°F. Let x = 5 in the given function. See the display at the bottom of the screen in Figure 30. (c) From the graph, it appears that the average annual temperature is about 0°F because the graph is centered vertically about the line y = 0. S Now Try Exercise 57.
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