4-10. Modeling with Trigonometric Functions. Vocabulary. Lesson. Mental Math. build an equation that models real-world periodic data.

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1 Chapter 4 Lesson 4-0 Modeling with Trigonometric Functions Vocabular simple harmonic motion BIG IDEA The Graph-Standardization Theorem can be used to build an equation that models real-world periodic data. Man natural phenomena involving periodic behavior can be modeled b a single sine or cosine function. Activit The tables below show the number N of hours of dalight in Jacksonville, FL as a function of the number D of das after December 3st of a certain ear. The data were collected on the fi rst da of each month over a two-ear period. Neither ear was a leap ear. Mental Math Find an equation for f g if f and g have the given equations. a. f() = 2 and g() = 5 b. f() = 5 and g() = 2 c. f() = _ and g() = 4 7 d. f() = 4 7 and g() = _ D N D N Source: The Old Farmer s Almanac, 2009 Step Work with a partner and use a statistics utilit to create a scatterplot of the data. The data should appear periodic. Step 2 Assuming the data fi t a sine function, estimate the following from the graph and interpret its meaning in this situation. a. amplitude b. period c. vertical translation d. phase shift Step 3 Use the information from Step 2 to create a model of the form _ k b = sin ( _ a h ). Graph our model on the same grid as the scatterplot. You ma need to convert our equation to a form that our technolog can graph. How well does our model seem to fi t the data? Step 4 Check our model b using the sine regression function on our calculator. U.S. Census for 2007 ranked Jacksonville, FL, as the 2th largest cit in the United States with a population of 805, Trigonometric Functions

2 Lesson 4-0 Simple Harmonic Motion The motion of a pendulum swinging back and forth in a vacuum and the motion of a weight bobbing on a spring are eamples of harmonic motion. The graphs of these motions appear to be sine waves. Motion that can be described using a sine or cosine function is called simple harmonic motion. Each point on the graph corresponds to a location of the pendulum or weight at a particular time. Eample A pendulum swings back and forth in a vacuum. Its distance from an object is captured using a motion detector for the object. The setup is pictured below at the left; at the right is a graph of the pendulum s distance from the motion detector. Write an equation for the distance from the pendulum to the object as a function of time. Motion Detector Distance Solution We want to fi nd the values of a, b, h, and k in the equation _ k b = sin ( _ a h). First, identif the period P, because a = P_ 2π. The pendulum makes a round-trip ever 2 seconds, so the period is 2 seconds. Thus a = 2_ Net, calculate the amplitude b. The amplitude is half of 2π the difference between the maimum and minimum -values. So.5.0 b = _ = Then fi nd the vertical shift k, the distance from the - 2 ais to the center line of the graph. This distance is.25, so k =.25. Finall, compare the graph to a graph of the sine or cosine function. We pick the sine function. The phase shift is 0 with respect to = sin, so h = 0. Now substitute in the equation to write a formula for the function. _ -.25 = sin _ 0.25 ( or 0.383) = 0.25 sin _ ( 0.383) +.25, where is the distance (in meters) of the pendulum from the object and is time (in seconds). Interpreting a Given Model When ou are given the equation of a sine wave model, ou can interpret the coefficients to describe the situation. Modeling with Trigonometric Functions 277

3 Chapter 4 GUIDED Eample 2 When an oven is set to a particular temperature, the heat level rises and falls, actuall fl uctuating slightl above and below that level as time passes. Assume that when a particular oven is set to 425ºF, the oven temperature t in degrees Fahrenheit m minutes after the burner fi rst shuts off satisfi es t = cos(0.9m). a. What are the maimum and minimum temperatures of the oven at this setting? b. What is the period of this sine wave? What does the period represent? Solution First, transform the equation into the form _ t - k b = cos (_ m - a h) to identif the coeffi cients of the model. _ t = cos _? ( m? ) a. The sum and difference of the amplitude and vertical translation of the sine wave tell us the maimum and minimum values of the model. This function has an amplitude of? and a vertical translation of?. Using these values, the maimum temperature of the oven is? while the minimum temperature is?. b. The period = 2π a =?. For this sine wave, the period represents how long it takes the oven to go from one maimum (or minimum) temperature to the net maimum (or minimum) temperature. Modeling an arbitrar sine wave, especiall one with a nonzero phase shift, can be an involved process. It is easier if ou focus our attention on one graph characteristic at a time. B asking a question like What s the period? or What s the amplitude? ou can deal with one piece of the model at a time instead of tring to figure out everthing at once. Questions COVERING THE IDEAS In and 2, use the sine regression model from Step 4 of the Activit.. What is the calendar da on which the number of hours of dalight is the greatest (the summer solstice) for the first ear? How man hours of dalight are there on that da? 2. How man hours of dalight does our model predict for Januar st for the net ear after the data shown? 3. Bernie has just purchased an oven. The manufacturer claims that less heat escapes from this oven than from the oven in Eample 2. Assume that Bernie s oven takes twice as long to cool from the maimum to the minimum temperature. Write an equation to model Bernie s oven temperature as he cooks a roast at Trigonometric Functions

4 Lesson 4-0 In 4 and 5, a pendulum s motion is captured and graphed below a. To the nearest 0. second, identif the time(s) at which the pendulum is furthest from the motion detector. b. To the nearest 0. second, identif the time(s) at which the pendulum is closest to the motion detector. 5. a. For each attribute below, describe its meaning in this situation, and find its value. i. amplitude ii. period iii. vertical shift b. Write a formula for the pendulum s motion in terms of time.. Write a formula describing the graph below A model for an earthquake tremor is f(t) =. sin(88.5t). Find the period and amplitude of the tremor. APPLYING THE MATHEMATICS 8. A pendulum swings for 00 seconds; its motion is captured and graphed below. Considering onl period, vertical shift, and amplitude, which properties seem to be changing over time? Which seem to be constant? Wh might this be so? Modeling with Trigonometric Functions 279

5 Chapter 4 9. Rose Tate s height above ground level was tracked as she traveled on a Ferris wheel. The graph is shown below. Distance (feet) Time (minutes) a. What is the radius of the Ferris wheel? b. How high was Rose above the ground at her lowest point? c. How high was she when she boarded the Ferris wheel? d. How long does it take the Ferris wheel to make one revolution? e. Write an equation that approimates the graph. 0. The average adult normall breathes in and out ever 5 seconds. The amount taken in and epelled from the lungs in a single breath is called the person s tidal volume. In adults, the tidal volume is about fluid ounces. The maimum amount usuall held in the lungs is 48 ounces. a. Sketch a sine wave graph to model 3 ccles of this situation. Assume that the function starts when the air is epelled from the lungs. b. Find a sine equation to model this situation.. Average monthl temperatures in degrees Fahrenheit for Chicago for the ears are given below. month Jan Feb Mar Apr Ma Jun Jul Aug Sep Oct Nov Dec avg. temp (ºF) Source: Statistical Abstract of the United States, 2009 a. Draw a scatterplot of these data. Plot = the number of the month on the horizontal ais, and = the temperature on the vertical ais. b. Carefull sketch a sine curve to fit the data and estimate its amplitude to the nearest hundredth. c. What is the period of this function? d. Multiple Choice Which of these four models is best for these data? A _ a = cos π ( _ - 49 ) B _ π a = cos ( _ ) - 49 C _ π a = cos ( _ ) D _ -a = cos π ( _ ) e. Find another equation equivalent to our answer in Part d that also describes these data. 280 Trigonometric Functions

6 Lesson Listed below are the hours of dalight in International Falls, Minnesota, for ten das of the ear. Date / 2/28 3/2 4/27 5/ /2 8/4 9/23 0/25 2/2 Hours Source: The Old Farmer s Almanac 2009 a. Draw a scatterplot of the data indicating dates as das of the ear. (/ would be, 2/28 would be 59, and so on.) Assume this ear was not a leap ear. b. Using paper and pencil, fit a sine wave to the scatterplot. c. Determine an equation of a cosine function that models the data. d. Using the model in Part c, estimate the hours of dalight on Jul 3. e. Estimate what das International Falls had at least 0 hours of dalight. REVIEW 3. Consider the function with equation = cos _ ( + π 4 ). a. Sketch a graph of the function. b. Give the amplitude, period, and phase shift of its graph. c. The graph is the image of the graph of = cos under the composite of which two transformations? (Lesson 4-9) 4. Given cos θ = m, find each value. (Lesson 4-3) a. cos( θ) b. cos(π θ) c. sin(θ π) In 5 and, refer to the unit circle at the right. (Lesson 4-3) 5. a. If c = d, find the eact value of c. b. What is the value of θ?. If θ 2 = _ 2π, find e and f. 3 In 7 and 8, let f() = 2 2 and g() = 7. a. Calculate g f g( 2). b. Calculate f g f( 2). 8. a. Give the domain of f g f. b. Give the domain of g f g. _ 3 -. (Lesson 3-7) O (e, f) = (cos θ 2, sin θ 2 ) θ 2 (c, d) = (cos θ, sin θ ) θ (, 0) EXPLORATION 9. The French phsicist Léon Foucault used a large pendulum to demonstrate the fact that Earth rotates on its ais. The original Foucault Pendulum hangs in the Panthéon in Paris, France; copies are ehibited in science museums around the world. How does the pendulum s motion show Earth s rotation? Research Foucault to find the answer and to find out about his other inventions. Modeling with Trigonometric Functions 28