MATH 30-2

SINUSOIDAL FUNCTIONS Module Six

Module 6 - Assignment Booklet

Student: _____________________________ Date Submitted:

______________________

http://moodle.blackgold.ca

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Lesson 1: Radian Measure1. Estimate the number of radians for a measure of 210. Sketch this angle

in the space provided.

2. Estimate the number of degrees for a measure of 8.5 rad. Sketch this angle in the space provided.

3. Describe the relationship between degrees and radians. Use a sketch or image to help you visually describe the relationship. Then explain how this relationship can be used to estimate the degree measure of an angle given in radians. Include your sketch in the space provided.

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4. Determine the number of degrees through which the minute hand of a clock rotates between 9:00 a.m. and 11:20 a.m. Answer to the nearest degree.

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Lesson 2: Graphs of Sinusoidal Functions1.Given the following graph, answer the questions.

a. What is the maximum value of the graph?

b. What is the minimum value of the graph?

c. What is the amplitude of the graph?

d. Determine the period of the graph.

e. What is the equation of the midline?

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2. Graph a sinusoidal function with the following characteristics:

period 180 maximum 1 midline that has the equation y 3

3. Find a sinusoidal graph on the Internet. Print a copy of the graph that you find and attach it here. Determine the following characteristics for your graph, and explain how you found each.

amplitude maximum midline minimum period

4. The height over time of a person riding a Ferris wheel can be modelled using a sinusoidal function. Sketch a height versus time graph for a person riding a Ferris wheel with the following characteristics:

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The person reaches a maximum of 15 m. The person reaches a minimum of 1 m. It takes 70 s for the Ferris wheel to turn once. At time 0, the person is at height 1 m.

Describe how you graphed the function.

Lesson 3: Equations of Sinusoidal Functions1. Abe graphed y 5 sin 3(x 2) 1, and he asked you to describe the graph without looking

at it. List five characteristics of the graph.

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2. The sinusoidal function that represents electrical current in Australia and the sinusoidal function that represents electrical current in Canada are shown in the following graph.

Choose the correct answer, and explain how you determined your answer. The sinusoidal function representing electrical current in Australia differs from the sinusoidal function representing electrical current in Canada in

A. amplitude and periodB. period and horizontal shiftC. amplitude and vertical shiftD. horizontal shift and vertical shift

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3. Determine an equation for the following sinusoidal function.

4. The distance from the Sun can be approximated by the equation D 2.5 cos 0.0172(n 185) 150, where D represents the distance in millions of kilometres and n represents the number of days into the year.

a. How far will Earth be from the Sun on March 21, the eightieth day of the year?

b. Determine the range of distances that Earth can be from the Sun.

c. What is the period of this cycle? Explain what the period means in the context of this question.

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Lesson 4: Modelling Data with Sinusoidal Functions1. Recall the information in Focus about the lynx and hare populations. A student determined

a sinusoidal regression function of the form y a sin[b(x c)] d to model the change in the lynx and hare populations.

a. Explain why a sinusoidal function can be used to model the data.

b. What flaws do you see in modelling this data using y a sin[b(x c)] d?

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2. Tides at Minas Basin, Nova Scotia, are the highest tides in the world and range from 15 m from minimum tides to maximum tides. The following table is the data for the Minas Basin for April 9, 2003, where time is given in Atlantic Standard Time hours and the tide level is given in metres.

Hour 0 1 2 3 4 5 6 7 8 9 10 11

Height (m) 3.3 3.6 4.7 6.3 8.1 9.6 10.5 10.6 9.8 8.2 6.3 4.6

Hour 12 13 14 15 16 17 18 19 20 21 22 23

Height (m) 3.4 3.1 3.7 5.1 6.8 8.4 9.7 10.2 9.9 8.8 7.1 5.4

a. By using only six of the data points, perform a regression on the data and determine the equation (to three decimal places). How does this equation compare with one performed by someone using a different set of six points?

b. A large trawler needs 4 m of water to float. For approximately how many hours of the day will the boat sit on the floor of the basin?

c. What is the water depth at 14:36? Note: 17:12 is 17 h of an h 17.2 h in decimal form.

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d. Does your regression look reasonable? By inspecting the data, what is the period of the tides?

e. At what time(s) throughout the day will the tide height be 7.5 m?

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Module Six SUMMARYIn this module you investigated the following question: How can a sinusoidal function be used to model and solve cyclical problems?

You explored sinusoidal functions. The first thing you learned about was the radian, which is an angular measure. Next, you explored some angles that were larger than 360°. You were then introduced to the sine and cosine functions and interpreted various other sinusoidal functions. You then used the parameters a, b, c, and d to describe characteristics of a sinusoidal function from an equation. Finally, you used curves of best fit and regression equations to help interpret sinusoidal data.

In the Module 6 Project: Applications of Sinusoidal Functions, you explored how latitude and the time of year affected the number of hours of daylight. You then modelled hours of daylight data using a sinusoidal curve.

Following are some of the key ideas you learned in each lesson.

Lesson 1 A radian is the size of angle produced at the centre of a circle by a radius laid along the circumference of the circle.

Angles larger than 360° are often represented with a spiral.

Lesson 2 The maximum, minimum, midline, amplitude, and period are characteristics commonly used to describe a sinusoidal function.

Lesson 3 Characteristics of the functions y = a sin b(x – c) + d and y = a cos b(x – c) + d

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can be found by interpreting each parameter separately.

Lesson 4 Data that follows a sinusoidal pattern can be modelled using a curve of best fit or a regression equation.

Mathematics 30-2 Learn EveryWare © 2012 Alberta Education

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MODULE 6 –SINUSOIDAL FUNCTIONS SUMMATIVE ASSIGNMENTComplete the following questions from your text book. Show steps completely and clearly, as marks are assigned for mathematical literacy and communication. Always use graph paper, rulers, and pencils as necessary. Attach questions and study notes securely to this booklet before you hand everything in.

Text: Principles of Mathematics 12 - Chapter 8 – SINUSOIDAL FUNCTIONS

Section 8.1: Pages 489 to 490: #1a, 2a, 3ab, 6, 8, 11

Section 8.2: Pages 494 to 496: #2, 4, 5, 6

Section 8.3: Pages 506 to 512: #4, 6, 8, 11

Section 8.4: Pages 528 to 532: #5, 7, 8ab, 9

Section 8.5: Pages 541 to 548: #3, 9

Module 6 is now complete.

Once you have received your corrected work, review your instructor’s comments and prepare for your module six test.