Write Trigonometric Functions and Models

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.5 a.5, a.6, A..B; P..B TEKS Write Trigonometric Functions and Models Before You graphed sine and cosine functions. Now You will model data using sine and cosine functions. Why?

One method to write a sine or cosine function that models a sinusoid is to find the values of a, b, h, and k for y 5 a sin b( h) k or y 5 a cos b( h) k where a is the amplitude, } p b k is the

y 5d d FIND PERIOD Because the graph repeats every } p units, the period is } p. Solution STEP STEP STEP Find the maimum value M and minimum value m. From the graph, M 5 5 and m 5.

1 .5 a.5, a.6, A..B; P..B TEKS Write Trigonometric Functions and Models Before You graphed sine and cosine functions. Now You will model data using sine and cosine functions. Why? So you can model the number of bicyclists, as in E. 6. Key Vocabulary sinusoid Graphs of sine and cosine functions are called sinusoids. One method to write a sine or cosine function that models a sinusoid is to find the values of a, b, h, and k for y 5 a sin b( h) k or y 5 a cos b( h) k where a is the amplitude, } p b k is the vertical shift. is the period (b > 0), h is the horizontal shift, and E XAMPLE TAKS Solve a REASONING: multi-step problem Multi-Step Problem Write a function for the sinusoid shown below. y 5d d FIND PERIOD Because the graph repeats every } p units, the period is } p. Solution STEP STEP STEP Find the maimum value M and minimum value m. From the graph, M 5 5 and m 5. Identify the vertical shift, k. The value of k is the mean of the maimum and minimum values. The vertical shift is k 5} M m 5} 5 () 5 } 5. So, k 5. Decide whether the graph should be modeled by a sine or cosine function. Because the graph crosses the midline y 5 on the y-ais, the graph is a sine curve with no horizontal shift. So, h 5 0. STEP Find the amplitude and period. The period is p } 5 p } b. So, b 5. The amplitude is a 5 M m } 5 5 () } 5 6 } 5. The graph is not a reflection, so a > 0. Therefore, a 5. c The function is y 5 sin..5 Write Trigonometric Functions and Models 9

Write a model for the height h (in feet) of a rope as a function of the time t (in seconds) if the rope is at its lowest point when t 5 0. 75 in. above ground in.

2 E XAMPLE Model circular motion JUMP ROPE At a Double Dutch competition, two people swing jump ropes as shown in the diagram below. The highest point of the middle of each rope is 75 inches above the ground, and the lowest point is inches. The rope makes revolutions per second. Write a model for the height h (in feet) of a rope as a function of the time t (in seconds) if the rope is at its lowest point when t in. above ground in. above ground Solution STEP STEP Find the maimum and minimum values of the function. A rope s maimum height is 75 inches, so M A rope s minimum height is inches, so m 5. Identify the vertical shift. The vertical shift for the model is: k 5 M m } 5 75 } 5 7 } 5 9 STEP STEP Decide whether the height should be modeled by a sine or cosine function. When t 5 0, the height is at its minimum. So, use a cosine function whose graph is a reflection in the -ais with no horizontal shift (h 5 0). Find the amplitude and period. The amplitude is a 5 M m } 5 75 } 5 6. Because the graph is a reflection, a < 0. So, a 56. Because a rope is rotating at a rate of revolutions per second, one revolution is completed in 0.5 second. So, the period is } p 5 0.5, and b 5. b c A model for the height of a rope is h 56 cos t 9. GUIDED PRACTICE for Eamples and Write a function for the sinusoid.. y (0, ). y d d d. WHAT IF? Describe how the model in Eample would change if the lowest point of a rope is 5 inches above the ground and the highest point is 70 inches above the ground. 9 Chapter Trigonometric Graphs, Identities, and Equations

SINUSOIDAL REGRESSION Another way to model sinusoids is to use a graphing calculator that has a sinusoidal regression feature.

E XAMPLE Use sinusoidal regression ENERGY The table below shows the number of kilowatt hours K (in thousands) used each month for a given year by a hangar at the

t 5 6 7 9 0 K 6.9 59 6 70.. 9. 0. 06. 05. 9.9. 69.9 Solution STEP Enter the data in a graphing calculator. STEP Make a scatter plot. L L 6.9 59 6 70. 5. L()= L STEP Perform a sinusoidal regression, because the scatter plot appears sinusoidal.

So, a model for the data is K 5.9 sin (0.5t.69).. GUIDED PRACTICE for Eample.

3 SINUSOIDAL REGRESSION Another way to model sinusoids is to use a graphing calculator that has a sinusoidal regression feature. The advantage of this method is that it uses all of the data points to find the model. E XAMPLE Use sinusoidal regression ENERGY The table below shows the number of kilowatt hours K (in thousands) used each month for a given year by a hangar at the Cape Canaveral Air Station in Florida. The time t is measured in months, with t 5 representing January. Write a trigonometric model that gives K as a function of t. t K Solution STEP Enter the data in a graphing calculator. STEP Make a scatter plot. L L L()= L STEP Perform a sinusoidal regression, because the scatter plot appears sinusoidal. STEP Graph the model and the data in the same viewing window. SinReg y=a*sin(b+c)+d a=.9059 b=.556 c= d=.09 c The model appears to be a good fit. So, a model for the data is K 5.9 sin (0.5t.69).. GUIDED PRACTICE for Eample. METEOROLOGY Use a graphing calculator to write a sine model that gives the average daily temperature T (in degrees Fahrenheit) for Boston, Massachusetts, as a function of the time t (in months), where t 5 represents January. t T Write Trigonometric Functions and Models 9

.5 EXERCISES SKILL PRACTICE. VOCABULARY What is a sinusoid? HOMEWORK KEY 5 WORKED-OUT SOLUTIONS on p. WS for Es. 5, 9, and 5 5 TAKS PRACTICE AND REASONING Es., 9,, 0, and EXAMPLE on p. 9 for Es. 9. WRITING Describe two methods you can use to model a sinusoid.

y d (0, ) d ERROR ANALYSIS Describe and correct the error in finding the amplitude and vertical shift for a sinusoid with a maimum point at (, 0) and a minimum point at (, 6). 7. a 5 M m }.

4 .5 EXERCISES SKILL PRACTICE. VOCABULARY What is a sinusoid? HOMEWORK KEY 5 WORKED-OUT SOLUTIONS on p. WS for Es. 5, 9, and 5 5 TAKS PRACTICE AND REASONING Es., 9,, 0, and EXAMPLE on p. 9 for Es. 9. WRITING Describe two methods you can use to model a sinusoid. WRITING FUNCTIONS Write a function for the sinusoid.. y s d, d. y 5 (0, 5) 5d 5. y (, 6) 6. y d (0, ) d ERROR ANALYSIS Describe and correct the error in finding the amplitude and vertical shift for a sinusoid with a maimum point at (, 0) and a minimum point at (, 6). 7. a 5 M m }. k 5 M m } } 5 } 5 5 WRITING FUNCTIONS Write a function for the sinusoid with maimum at point A and minimum at point B. 9. A(, 6), B(, 6) 0. A(0, ), B(, ). A}, 5, B(0, ). A}, 6, B(0, 6). A }, 9, B(, 5). A(0, 5), B(6, ) 5. A(0, 0), B(, ) 6. A},, B }, 7 7. A }, 0, B(0, ). TAKS REASONING During one cycle, a sinusoid has a minimum at (6, ) and a maimum at (, 60). What is the amplitude of this sinusoid? A B C D 9 9 Chapter Trigonometric Graphs, Identities, and Equations

Eplain which type of function you would choose to model a sinusoid whose y-intercept occurs at the minimum value of the function.

5 9. TAKS REASONING What is an equation of the graph shown at the right? 5 y (6, 5) A y 5 cos } p 6 B y 55 cos } p 0 6 C y 5 sin } p 6 D y 55 sin } p 0 6 (0, 5) 0. WRITING Any sinusoid can be modeled by both a sine function and a cosine function. Therefore, you can choose the type of function that is more convenient. Eplain which type of function you would choose to model a sinusoid whose y-intercept occurs at the minimum value of the function.. REASONING Model the sinusoid in Eample on page 9 with a cosine function of the form y 5 a cos b( h) k. Use identities to show that the model you found is equivalent to the sine model in Eample.. CHALLENGE Write a sine function for the sinusoid with a minimum at } p, and a maimum at } p,. PROBLEM SOLVING EXAMPLE on p. 9 for Es.. CIRCUITS A circuit has an alternating voltage of 00 volts that peaks every 0.5 second. Use the graph shown at the right to write a sinusoidal model for the voltage V as a function of the time t (in seconds). 00 V 00d 00d t. CLIMATOLOGY The graph below shows the average daily temperature of Houston, Teas. Write a sinusoidal model for the average daily temperature T (in degrees Fahrenheit) as a function of time t (in months). Daily Temperature in Houston Temperature (F) T 0 0 (0, 5) (6, ) t Months since January EXAMPLE on p. 9 for E CIRCULAR MOTION One of the largest sewing machines in the world has a flywheel (which turns as the machine sews) that is 5 feet in diameter. Write a model for the height h (in feet) of the handle at the edge of the flywheel as a function of the time t (in seconds). Assume that the wheel makes a complete turn every seconds and the handle is at its minimum height of feet above the ground when t Write Trigonometric Functions and Models 95

The time t is measured in months, with t 5 representing October 00. Use a graphing calculator to write a sinusoidal model that gives R as a function of t. t 5 6 7 9 0 R 5 0 6 9 5 9 7 7.

The time t is measured in years, with t 5 representing the first year. t 5 6 7 9 0 N 0..7.6. 0 7.5 6.7 7.. a.

6 EXAMPLE on p. 9 for Es BICYCLISTS The table below shows the number of adult residents R (in millions) in the United States who rode a bicycle during the months of October 00 through September 00. The time t is measured in months, with t 5 representing October 00. Use a graphing calculator to write a sinusoidal model that gives R as a function of t. t R MULTI-STEP PROBLEM The table below shows the number of employees N (in thousands) at a sporting goods company each year for eleven years. The time t is measured in years, with t 5 representing the first year. t N a. Model Use a graphing calculator to write a sinusoidal model that gives N as a function of t. b. Calculate Predict the number of employees in the twelfth year.. TAKS REASONING The low tide at Eastport, Maine, is.5 feet and occurs at midnight. After 6 hours, Eastport is at high tide, which is 6.5 feet. a. Model Write a sinusoidal model that gives the tide depth d (in feet) as a function of the time t (in hours). Let t 5 0 represent midnight. b. Calculate Find all the times when low and high tides occur in a hour period. c. Reasoning Eplain how the graph of the function you wrote in part (a) is related to a graph that shows the tide depth d at Eastport t hours after :00 A.M. 9. CHALLENGE The table below shows the average monthly sea temperatures T (in degrees Celsius) for Santa Barbara, California. The time t is measured in months, with t 5 representing January. t T a. Use a graphing calculator to write a sine model that gives T as a function of t. b. Find a cosine model for the data WORKED-OUT SOLUTIONS on p. WS 5 TAKS PRACTICE AND REASONING

MIXED REVIEW FOR TAKS TAKS PRACTICE at classzone.com REVIEW TAKS Preparation p. ; TAKS Workbook 0. TAKS PRACTICE The top, front, and side views of a solid built with cubes are shown below.

7 MIXED REVIEW FOR TAKS TAKS PRACTICE at classzone.com REVIEW TAKS Preparation p. ; TAKS Workbook 0. TAKS PRACTICE The top, front, and side views of a solid built with cubes are shown below. How many cubes are needed to construct this solid? TAKS Obj. 7 Top view Front view Side view A 7 B C 9 D 0 REVIEW TAKS Preparation p. 70; TAKS Workbook. TAKS PRACTICE What is the area of the blue figure shown at the right? TAKS Obj. F.5 cm G.0 cm H.5 cm J.0 cm 5 cm 0 cm cm 9 cm QUIZ for Lessons..5 Simplify the epression. (p. 9). sin sec. sin u ( cot u). tan p } u cot u csc u. cos u sin u tan u 5. tan p } sec } csc 6. sin () cos () } csc } sec Find the general solution of the equation. (p. 9) 7. cos cos () 5. Ï } cos sin cos sin sin 5 Write a function for the sinusoid. (p. 9) 0. y (0.5, ). y (, ) (0, 6) (.5, 5). DAILY TEMPERATURES The table below shows the average daily temperature D (in degrees Fahrenheit) in Detroit, Michigan. The time t is measured in months, with t 5 representing January. Use a graphing calculator to write a sinusoidal model that gives D as a function of t. (p. 9) t D EXTRA PRACTICE for Lesson.5, p. 0 ONLINE QUIZ at classzone.com 97

calculator a.5, a.6, A..B; P..B classzone.com Keystrokes Q UESTION How is music related to trigonometry?

The sound of a pure note can be represented using a sine (or cosine) wave.

E XPLORE Analyze the sound of a musical instrument Play a note on a musical instrument. Write a sine function to describe the note.

Use the CBL and the CBL microphone to collect the sound data and store it in a graphing calculator.

STEP Find characteristics of graph Use the graph of the sound data to calculate the note s amplitude and frequency (the number of

DRAW CONCLUSIONS Use your observations to complete these eercises. Choose a note to play and have a classmate also choose a note.

8 ACTIVITY TEXAS CBL Use after Lesson.5.5 Collect and Model Trigonometric Data MATERIALS musical instrument CBL microphone TEKS Calculator Based Laboratory (CBL) graphing calculator a.5, a.6, A..B; P..B classzone.com Keystrokes Q UESTION How is music related to trigonometry? Sound is a variation in pressure transmitted through air, water, or other matter. Sound travels as a wave. The sound of a pure note can be represented using a sine (or cosine) wave. More complicated sounds can be modeled by the sum of several sine waves. E XPLORE Analyze the sound of a musical instrument Play a note on a musical instrument. Write a sine function to describe the note. STEP Play note Play a pure note on a musical instrument. Use the CBL and the CBL microphone to collect the sound data and store it in a graphing calculator. STEP Graph function Use the graphing calculator to graph the pressure of the sound as a function of time. STEP Find characteristics of graph Use the graph of the sound data to calculate the note s amplitude and frequency (the number of cycles in one second). STEP Write function Write a sine function for the note. DRAW CONCLUSIONS Use your observations to complete these eercises. Choose a note to play and have a classmate also choose a note. Find two sine functions y 5 f() and y 5 g() that model the two notes. Then play the notes simultaneously and use the CBL and a graphing calculator to graph the resulting sound wave. Compare this graph with the graph of y 5 f() g(). What do you notice?. The pitch of a sound wave is determined by the wave s frequency. The greater the frequency, the higher the pitch. Which of the notes in Eercise had a higher pitch?. When you change the volume of a note, what happens to the graph of the sound wave?. Compare the sine waves for different instruments playing the same note. 9 Chapter Trigonometric Graphs, Identities, and Equations

Ready To Go On? Skills Intervention 14-1 Graphs of Sine and Cosine

14A Ready To Go On? Skills Intervention 14-1 Graphs of Sine and Cosine Find these vocabulary words in Lesson 14-1 and the Multilingual Glossary. Vocabulary periodic function cycle period amplitude frequency