Course Profile Functions and Relations, Grade 11, University Preparation, Catholic and Public

Unit 4: Trigonometric Functions

Time: 19 hours

Unit Description

Students investigate the periodic nature and graphical properties of the primary trigonometric functions. Using technology, students explore the effects of simple transformations on their graphs and equations. Students apply these concepts to model authentic problems.

Activity 4.1: Surf’s Up! Let’s Catch the Sine Wave

Time: 75 minutes

Description

Students investigate the shape of the graphs of y = sin x and y = cos x by plotting the values of these functions as determined from the unit circle.

Strand(s): Trigonometric Functions

Overall Expectations

TFV.02 - demonstrate an understanding of the meaning and application of radian measure;

TFV.03 - determine, through investigation, the relationships between the graphs and the equations of sinusoidal functions.

Specific Expectations

TF2.07 - demonstrate facility in the use of radian measure in solving and graphing equations;

TF3.01 - sketch the graphs of y = sin x and y = cos x, and describe their periodic properties.

Ontario Catholic School Graduate Expectations

CGE3c - a reflective and critical thinker who thinks reflectively and creatively to evaluate situations and solve problems.

Prior Knowledge & Skills

· Students should be proficient in the use of both radian and degree measure.

· The meaning of q and its use in representing angle measures should be introduced to the students prior to this activity.

Planning Notes

· The teacher requires an overhead projector, acetate sheet and markers, an overhead projection tablet, a graphing calculator with motion sensor attachment (such as a CBR), and a piece of string (40 to 100 cm in length) and a mass (large washers will do) to construct a pendulum.

· The teacher should also provide a unit circle, preferably superimposed on graph paper. The students require graph paper.

Teaching/Learning Strategies

This activity is comprised of two sections. The teacher should facilitate a discussion of periodic and cyclical behaviour, and demonstrate sinusoidal motion in the swinging of a pendulum. The students then derive the graphs of y = sin x and y = cos x from the unit circle.

A. Teacher Facilitation - Demonstration

· As an introduction to the graphs of sinusoidal functions, the teacher should first, by directed questioning, acquaint students with the cyclical character of the unit circle, specifically the repetitive nature of the values of y = sin x and y = cos x. The students should then be asked to provide everyday examples that exhibit similar behaviour (i.e., the hands of a clock, rocking chairs, pendulums, Ferris wheels, the tides, etc.).

· As a demonstration, the teacher should set up a simple pendulum in the class and use the motion sensor to measure the position of the bob (mass) as a function of time. For best results, the mass should be pulled back about 30 cm from the rest position and the motion sensor should measure at least 4 full oscillations. Display the graph on the overhead. The graph will be sinusoidal and should give the students an indication of the type of graphs that they will be exploring throughout this unit. The periodic characteristics of the graph (specifically the amplitude, period, domain, and range) should be discussed at this point.

B. Student Activity

Suggestions for teacher facilitation are included throughout this activity in italics.

1. Convert the angles given in the table below into radian measure and enter their values into the table.

2. Starting from the positive x-axis (calling this q = 0°) and rotating counter-clockwise in 15° increments, determine the coordinates of the points on the unit circle for each angle, and enter these coordinates in the table. Teachers may need to demonstrate how to determine co-ordinates.

3. Determine the values of sin q and cos q for each of the given angles to two decimal places. It is assumed that the students have been acquainted with the relationship of sin q and cos q and the unit circle in Unit 3: Trigonometry.

q (in degrees)	0	15	30	45	60	75	90	105	120	135	150	…	360
q (in radians)
coordinates (x, y)
sin q =
cos q =

4. On the same set of axes, plot the graphs of y = sin q and y = cos q.

C. Follow-Up Skills

Using the graphing calculator or a sample of students’ work transferred to an acetate sheet, the teacher should elicit from the students pertinent characteristics of the graphs of y = sin q and y = cos q. These characteristics should include the period, amplitude, roots, symmetry, domain, and range. The similarity of the graphs of y = sin q and y = cos q should also be noted, and in particular the translation that would map one graph onto the other. The teacher should introduce the term phase shift at this point.

Assessment & Evaluation of Student Achievement

Learning skills can be assessed visually as the students are completing their graphs. Their ability to work independently can be assessed using as criteria accomplishing a task independently and self-direction. Time management skills can be used to assess organization. Initiative can be assessed using the students’ self-motivation and responses to prompts by the teacher as criteria.

Activity 4.2: Transformations of Trigonometric Functions

Time: 150 minutes

Description

Students investigate the effects of simple transformations on the graphs of y = sin x and y = cos x through the use of graphing technology.

Strand(s): Trigonometric Functions, Tools for Operating and Communicating with Functions

Overall Expectations

TFV.02 - demonstrate an understanding of the meaning and application of radian measure;

TFV.03 - determine, through investigation, the relationships between the graphs and the equations of sinusoidal functions;

OCV.02 - demonstrate an understanding of inverses and transformations of functions and facility in the use of function notation.

Specific Expectations

TF2.07 - demonstrate facility in the use of radian measure in solving and graphing equations;

TF3.02 - determine, through investigation, using graphing calculators or graphing software, the effect of simple transformations (e.g., translations, reflections, stretches) on the graphs and equations of y = sin x and y = cos x;

TF3.03 - determine the amplitude, period, phase shift, domain, and range of sinusoidal functions whose equations are given in the form y = a sin(kx + d) + c or y = a cos(kx + d) + c;

TF3.04 - sketch the graphs of simple sinusoidal functions [e.g., y = a sin x, y = cos kx, y = sin(x + d),
y = a cos kx + c];

TF3.05 - write the equation of a sinusoidal function, given its graph and given its properties;

OC2.06 - represent transformations (e.g., translations, reflections, stretches) of the functions defined by
f(x) = x, f(x) = x², f(x) = , f(x) = sin x, and f(x) = cos x, using function notation;

OC2.07 - describe, by interpreting function notation, the relationship between the graph of a function and its image under one or more transformations;

OC2.08 - state the domain and range of transformations of the functions defined by f(x) = x, f(x) = x²,
f(x) = , f(x) = sin x, and f(x) = cos x.

Ontario Catholic School Graduate Expectations

CGE3c - a reflective and critical thinker who thinks reflectively and creatively to evaluate situations and solve problems;

CGE5a - works effectively as an interdependent team member;

CGE5e - respects the rights, responsibilities and contributions of self and others;

CGE5f - exercises Christian leadership in the achievement of individual and group goals.

Prior Knowledge & Skills

· Students should be proficient in the use of radian measure.

· The graphs and characteristics of y = sin x and y = cos x should be familiar to all students.

· Students should be proficient in the use of graphing calculators to plot functions. Students should also be able to graph manually.

· The meaning of q and its use in representing angle measures should be introduced to the students prior to this activity.

Planning Notes

· The teacher requires an overhead projector, transparencies, and markers. (An overhead projection tablet could be used as well.) In addition, the teacher must prepare a student worksheet for this activity.

· The students require graphing calculators (or dynamic graphing software such as Zap-A-Graph or Geometer’s Sketchpad), and graph paper.

· The first part of the activity is a discovery exercise, after which the students summarize their results. The second part of the activity includes a similar investigation involving more complex transformations.

· This activity may also be done without the use of technology, if required.

Part 1: Transformations: More than Meets the Eye

Time: 75 minutes

Teaching/Learning Strategies

A. Teacher Facilitation

· Students are placed into groups of two or three. Each group is assigned a set of four particular examples of sinusoidal functions of the following types:

Group 1 y = a sin x Group 2 y = a cos x

Group 3 y = sin kx Group 4 y = cos kx

Group 5 y = sin(x + s) Group 6 y = cos(x + s)

Group 7 y = sin x + c Group 8 y = cos x + c

· This activity is described for Group 1 only. Groups 2 to 8 answer similar questions using a variety of values for the given parameters. In particular, it is recommended that both positive and negative values be used, and that both whole numbers and fractions are included. For example, Group 3 might be assigned the functions y = sin 3x, y = sin(-x), y = sin , and y = sin(-4x).

· As trigonometric functions are traditionally considered to be functions of radian measure, it should be noted that the values given for the constant s (Groups 5 and 6) should be multiples of p.

· Students are asked to describe the graphs of the assigned functions. The teacher should expect these descriptions to include at least the maximum and minimum points, the amplitude, the roots, the period, symmetry, the phase shift, and the domain and range.

B. Student Activity

Suggestions for teacher facilitation are included throughout this activity in italics.

1. a) Plot a graph of y = sin x on the graphing calculator, using a window large enough to display two complete cycles, one on either side of the y- axis. Reproduce this sketch on the graph paper.

b) Graph the following curves on the calculator, stopping after each to reproduce the plot on the graph paper, using the same set of axes as in question 1a: y = 2 sin x, y = –sin x, y = sin x, and y = sin x.

c) In what ways are the graphs similar? In what ways are they different?

2. a) Clear the screen, then plot a graph of y = cos x on the graphing calculator, using a window large enough to display two complete cycles, one on either side of the y- axis. Reproduce this sketch on the graph paper using a new set of axes.

b) Use the calculator to graph each of the functions y = -3 cos x, y = 5 cos x, y = cos x, and
y = cos x. After graphing each equation, reproduce the plot on the graph paper, using the same set of axes as in question 2a.

c) In what ways are the graphs similar? In what ways are they different?

d) Compare and contrast these graphs to those sketched in question 1.

3. Without plotting, compare and contrast the graphs of the following equations:

a) y = sin x and y = 12 sin x b) y = sin x and y = sin x c) y = cos x and y = -3 cos x

After each group has completed their respective investigation, they present their results to the rest of the class, drawing attention to the role of the constants in each transformation, and thus making connections between the function and its graph.

C. Follow-up Skills

Time: 30 minutes

The teacher should assign additional questions to the students in class to reinforce the new learning. The teacher could facilitate a class discussion to confirm responses, and use the opportunity to provide remediation as necessary. The given questions should include a variety of constants, as outlined above.

Part 2: Give Me A Sine

Time: 75 minutes

Teaching/Learning Strategies

A. Teacher Facilitation

· The second part of the activity proceeds in much the same manner as the first. The following sinusoidal functions will be assigned to each group.

Group 1 y = a sin kx Group 2 y = a cos kx

Group 3 y = a sin(x + s) Group 4 y = a cos(x + s)

Group 5 y = sin kx + c Group 6 y = cos kx + c

Group 7 y = sin k(x + s) Group 8 y = cos k(x + s)

B. Student Activity

Once again, each group is asked to present their results to the rest of the class, drawing attention to the effect of each numerical value in the given functions.

C. Follow-up Skills

Time: 45 minutes

To prepare students for subsequent activities, the teacher should further explore some of the concepts learned in this activity:

· Functions of the form y = sin(x + s) and y = cos(x + s) have, until now, included only multiples of p for the value of s. The teacher should have students explore the effect of other real values of s, as they play a most important role in the modelling of authentic problems later in this unit.

· A connection must be established between the functions y = sin k(x + s) (studied in this activity) and y = sin(kx + d) (an alternate form), and between the functions y = cos k(x + s) and y = cos(kx + d). Specifically, it is imperative that students recognize the roles of k, s, and d in determining the period and the phase shift of a given function. Both of these forms are examined in subsequent activities, so students should be comfortable working with them both.

· In this activity, the students are asked to compare and contrast the equations and graphs of simple transformations of y = sin x and y = cos x. The teacher should provide additional questions to reinforce these skills. In addition, students should be asked to sketch a variety of given sinusoidal functions, and to determine the equation of various sinusoidal functions given their graphs. These skills are of great importance to achieve success in the rest of this unit.

· Suitable textbook questions should be assigned to consolidate these concepts.

Assessment & Evaluation of Student Achievement

The students’ oral reports may be assessed using an appropriate group presentation rubric, with emphasis paid to the assessment of Communication skills, particularly the use of mathematical symbols and conventions. Other criteria may include the computation and construction of graphs (Knowledge and Communication), and reasoning skills (Inquiry and Communication). After completing the recommended textbook exercise, a short pencil-and-paper task could be used to assess the core knowledge and concepts. Learning skills, specifically initiative, could be assessed as the teacher facilitates the classroom discussion to conclude the first part of the activity. Teamwork skills can be assessed visually as the groups complete their investigation. The students’ work habits and organization can be assessed during their presentations.

Accommodation

The teacher should ensure that those students with difficulties in understanding concepts be placed into groups in which they can receive support from other students.

Activity 4.3: Don’t Go Off on a Tangent

Time: 75 minutes

Description

Students identify the defining characteristics of the function y = tan x, and establish connections with the functions y = sin x and y = cos x.

Strand(s): Trigonometric Functions

Overall Expectations

TFV.03 - determine, through investigation, the relationships between the graphs and the equations of sinusoidal functions.

Specific Expectations

TF3.01 - sketch y = sin x and y = cos x, and describe their periodic properties;

TF3.06 - sketch the graph of y = tan x; identify the period, domain, and range of the function; and explain the occurrence of asymptotes.

Ontario Catholic School Graduate Expectations

CGE3c - a reflective and creative thinker who thinks reflectively and creatively to evaluate situations and solve problems;

CGE4b - a self-directed, responsible, life long learner who demonstrates flexibility and adaptability;

CGE4f - a self-directed, responsible, life long learner who applies effective communication, decision-making, problem-solving, time and resource management skills.

Prior Knowledge & Skills

· Students should possess a comprehensive knowledge of the division of rational numbers, and specifically the concept of division by 0.

· The primary trigonometric ratios should be familiar to all students.

· Students should understand the concept of an asymptote, as defined in Unit 1: Algebraic Manipulation of Functions and Unit 2: Function Notation, Inverses, and Transformations.

· Students must be able to use the radian and degree modes of a scientific calculator.

Planning Notes

· Students must have graph paper and a scientific calculator for use in the graphing of the primary trigonometric functions.

· The teacher has to prepare the tables to be filled out by the students in this activity.

Teaching/Learning Strategies

A. Teacher Facilitation

· Students use characteristics of the graphs of y = sin x and y = cos x and the relationship of the primary trigonometric ratios to determine the graph of y = tan x, after which they make observations about its properties.

· The teacher may pair or group students for this activity, or they may have the students work independently.

· As the class is completing this investigation, the teacher should be addressing students’ concerns individually or, if necessary, in small groups. In addition, students should be encouraged to seek help from other students.

B. Student Activity

Suggestions for teacher facilitation are included throughout this activity in italics. Some solutions are included to aid in the flow of the activity.

1. Find the missing angle measures in the given table.

2. Using a scientific calculator, determine the values of sin x and cos x for the given angles to the nearest thousandth.

x (in radians)

…