Teaching & Learning Plans Plan 10: Trigonometric Functions Leaving Certificate Syllabus The Teaching & Learning Plans are structured as follows: Aims outline what the lesson, or series of lessons, hopes to achieve. Prior Knowledge points to relevant knowledge students may already have and also to knowledge which may be necessary in order to support them in accessing this new topic. Learning Outcomes outline what a student will be able to do, know and understand having completed the topic. Relationship to Syllabus refers to the relevant section of either the Junior and/or Leaving Certificate Syllabus. Resources Required lists the resources which will be needed in the teaching and learning of a particular topic. Introducing the topic (in some plans only) outlines an approach to introducing the topic. Lesson Interaction is set out under four sub-headings: i. Student Learning Tasks – Teacher Input: This section focuses on teacher input and gives details of the key student tasks and teacher questions which move the lesson forward. ii. Student Activities – Possible and Expected Responses: Gives details of possible student reactions and responses and possible misconceptions students may have. iii. Teacher’s Support and Actions: Gives details of teacher actions designed to support and scaffold student learning. iv. Checking Understanding: Suggests questions a teacher might ask to evaluate whether the goals/learning outcomes are being/have been achieved. This evaluation will inform and direct the teaching and learning activities of the next class(es). Student Activities linked to the lesson(s) are provided at the end of each plan. Teaching & Learning Plan 10: Trigonometric Functions Aims • To help students to graph trigonometric functions sin x, cos x and tan x • To recognise the period and range of each function • To enable students to relate sin x and cos x to the unit circle • To reinforce understanding of the terms function, domain, range, period, inverse function, asymptote Prior Knowledge Students should be familiar with the introduction to trigonometry (T&L Plan 8) and the unit circle (T&L Plan 9) Learning Outcomes As a result of studying this topic, students will be able to • plot the graph of y= sin x, where x is an angle in standard position in the unit circle by projecting the y coordinates (sin x) of points on the unit circle onto an x-y Cartesian plane [This will be done for more than one revolution of the circle to emphasise the periodic nature of the functions. As small intervals are used the non linearity will be emphasised, and they will see that these graphs form smooth curves, which are “rounded” at the top and bottom.] • recognise the periodicity of y = sin x, and be able to give the value of the period and the range • explain why y = sin x is a function for all values of x and why its inverse is not a function for all values of x but could be if we restricted the x-values [more detail on this later in inverse functions] • solve equations of the form plotted , from the graph, for the domain • plot the graph of y= cos x by plotting the x coordinates of points on the unit circle against the corresponding angle in standard position, on an x-y Cartesian plane • plot graphs of sin x, cos x and tan x using tables • plot graphs of the form a sin x, a cos x and state their period and range, and explain the effect of changing the value of a • plot graphs of the form sin bx, cos bx, state their period and range and explain the effect of changing the value of b © Project Maths Development Team 2009 www.projectmaths.ie 1 Teaching & Learning Plan 10: Trigonometric Functions • state the period and range of graphs of the form asin bx, acos bx for a, b ∈ N • solve equations of the type y = asin bx, y =acos bx for the domain used in the graph • sketch functions of the form asin bx or a cos bx for a, b ∈ N from a knowledge of the period and range and the shape of the function Relationship to Leaving Certificate Syllabus Sub-topics Higher Level 2.3 Trigonometry Graph the trigonometric functions sine, cosine, tangent. Graph trigonometric functions of type asin nθ, acos nθ for a, n ∈ N. Resources Required Compasses, protractors, clear rulers, pencils, formulae and tables booklet, Geogebra, Autograph, Perspex Model of Unit Circle (last 3 desirable but not essential) Note The CD icon is used throughout this document to indicate that there is a resource on the Student Disc relating to this topic. © Project Maths Development Team 2009 www.projectmaths.ie 2 © Project Maths Development Team 2009 www.projectmaths.ie • The graph is “rounded” – no sharp points. • student answer/response »» Does this presentation help students in the task of plotting the graph y=sin x? Checking Understanding KEY: » next step »» Ask individual students to describe the graph in their own words. »» Describe the graph Student • It consists of repeating Activity 1B Q 1. patterns. • It has a max of 1 and a min of -1. »» Distribute Student Activity 1. »» Tell students they will need a clear ruler and pencil. »» Using Student Activity 1A, »» Students plot the points project the values of the y and join them in a smooth coordinate, which is sin x curve. onto the Cartesian plane to get a graph of y= sin x 0 ≤ x ≤ 570° Teacher’s Support and Actions »» Show http://www. projectmaths.ie/system/ files/Presentation%20 London%20eye%20sine. pptx and Perspex Model of unit circle. Student Activities: Possible and Expected Responses »» We have seen on the unit circle that sin, cos and tan vary as the angle varies from 0° to 360°. The sin function is represented by the y coordinate on the unit circle. Student Learning Tasks: Teacher Input Lesson Interaction Teaching & Learning Plan 10: Trigonometric Functions 3 © Project Maths Development Team 2009 • Period = 360° • Range = [-1,1] Student Activities: Possible and Expected Responses www.projectmaths.ie »» Write down the period and range of y=sinx Student Activity 1B Q 2 and 3. »» We refer to the length of the repeating pattern on the x- axis as the period and the max and min y values are the range written as [min value, max value] »» Any function which is repeating such that f(x) = f(x+c) , where c is a constant is called a periodic function. Student Learning Tasks: Teacher Input »» KEY: » next step GeoGebra interactive webpage f(x) =asinx. Set slider with a=1 • student answer/response »» Does the animation summarise the exercise for students and give them increased understanding of the relationship between sin x and the y-coordinate on the unit circle? »» Show an animation of this exercise from Autograph. (File, New Extras Page, Trigonometry, sin ) »» See also (scroll to end of page) piston animation: http://www.intmath.com/ Trigonometric-graphs/2_ Graphs-sine-cosine-period. php Checking Understanding Teacher’s Support and Actions Teaching & Learning Plan 10: Trigonometric Functions 4 © Project Maths Development Team 2009 Teacher’s Support and Actions »» They may also draw a horizontal line through any value of y in the range, showing that it has multiple solutions. www.projectmaths.ie • student answer/response »» Can students solve the equation sin x=0.5? »» Can they explain why the inverse of y=sin x is not a function? »» Do students remember what a function is? Checking Understanding KEY: » next step »» Distribute Student Activity 2. »» Students explain that for »» Remind students that a any y value there is an function is a set of ordered infinite number of x values, pairs, where each x value so the inverse of y=sin x is has one unique y value. not a function. • Students explain in Student Activity 1B Q4 that for each value of x there is only one value of y, hence y=sin x is a function. Student Activities: Possible and Expected Responses »» Having seen the connection »» Students plot the between y = sinx and graphs on graph paper motion in a circle, we will accompanying Student now plot the graph of Activity 2 using different y=sin x using a table of colours for each graph. values, then y= 2sinx and y= 3sin x, using the same axes and scale. »» Complete Student Activity 1B Q6 and 7. »» Explain your answer. »» Is the inverse of y=sin x a function? Student Activity 1B Q5. Student Activity 1B Q4 Student Learning Tasks: Teacher Input »» Is y=sin x a function? Teaching & Learning Plan 10: Trigonometric Functions 5 • The range changes. The range is [-a, a] »» What is the effect of varying a? Student Activity 2. • As b varies the period varies. »» What is the effect of varying b? www.projectmaths.ie »» Students fill in Student Activity 3. »» Fill in Student Activity 3. © Project Maths Development Team 2009 »» Students fill in the table and plot the graphs in 3 different colours (Student Activity 3). »» In the function y=a sin bx we have seen the effect of varying a, and now we will keep a constant at a=1, and vary b. »» If this was a sound • The volume would increase wave what effect do as a increased. you think it would have on the sound to change the value of a? Student Activities: Possible and Expected Responses Student Learning Tasks: Teacher Input »» • student answer/response »» Are students keeping the correct shape on the curves even when they have fewer points to plot for each cycle? »» Do all students understand that in the equation y=a sin x the period remains unchanged as a changes and that only the range changes? Checking Understanding KEY: » next step Using GeoGebra interactive webpage show graph of y=a sin b x, varying a and b using the sliders. »» Students should be able to move from the specific to the general here and should be allowed to try it out on their own first. »» Remind students that they are plotting the functions f : x → sin x g : x → sin 2x h : x → sin 3x »» Distribute Student Activity 3. »» If required give students some help in making this conclusion. Teacher’s Support and Actions Teaching & Learning Plan 10: Trigonometric Functions 6 © Project Maths Development Team 2009 • The frequency and hence the pitch of the note would change, increasing as b increased. • In general the period is the period is π rad = 180°. Student Activities: Possible and Expected Responses • When b is 1, the period is 2π rad = 360°. When b=2 www.projectmaths.ie »» We saw how sin x is the y-coordinate of a point on the circumference of the unit circle and that cos x is the x-coordinate. Hence by plotting the x coordinates of points on the unit circle onto a Cartesian plane we should get the graph of y=cos x. Note: Students do not need to know this for the syllabus, but it is good for real life context. »» If this was a sound wave what would be the effect of varying b? »» How can you calculate the period given b? Student Learning Tasks: Teacher Input »» • student answer/response »» Use examples of functions of this type to check. »» Can students now calculate the period and range of any function of the type y=a sin bx, from the values of a and b? Checking Understanding KEY: » next step Show, using a GeoGebra interactive webpage, where sin is dragged 90° to the left giving the cosine function. »» Show an animation of this exercise from Autograph. (File, New Extras Page, Trigonometry, cos) and Perspex model of unit circle. Teacher’s Support and Actions Teaching & Learning Plan 10: Trigonometric Functions 7 • Its shape is the same as the sine graph. It is the graph of sin x shifted 90° to the left. »» Describe the graph of y = cos x • The range changes. The range is [-a,a] »» What is the effect of varying a? Student Activity 4. © Project Maths Development Team 2009 www.projectmaths.ie • Students fill in the table »» In the function y=a cos bx and plot the graphs in 3 we have seen the effect of different colours. Student varying a, and now we will Activity 5. keep a constant and vary b. »» Students fill in Student Activity 4. »» What is the period and range of y = cosx, y = 2cosx and y = 3cosx? Student Activity 4. »» Having seen the connection »» Students plot the graphs between y = cos x and using different colours for motion in a circle, we will each graph. now plot the graph of y = cos x using a table of values, then y = 2cosx and y = 3cosx, using the same axes and scale (Student Activity 4). • It is periodic with a period of 360° and a range of [-1,1] Student Activities: Possible and Expected Responses Student Learning Tasks: Teacher Input »» Distribute Student Activity 5. »» Distribute Student Activity 4. Teacher’s Support and Actions Teaching & Learning Plan 10: Trigonometric Functions KEY: » next step • student answer/response »» Are students keeping the correct shape on the curves even when they have fewer points to plot for each cycle? »» Do all students understand that in the equation y = acos x, the period remains unchanged as a changes and that only the range changes? »» Are students now able to describe the graph of y = cos x in terms of periodicity and range, having worked with y = sin x? Checking Understanding 8 »» Students fill in Student Activity 5. »» What is the period and range of y = cos x y = cos 2x and y = cos 3x? Student Activity 5. © Project Maths Development Team 2009 »» We will now look at the graph of the function y=a tan x. »» Students make 4 evenly spaced marks on the x axis and mark the range on the y axis, and plot the curves. • In general the period is Show GeoGebra interactive webpage y=a sin bx and y=a cos bx • student answer/response »» Are students able to sketch graphs of the form y=a sin bx and y=a cos bx from the values of period and range? »» Use more examples of functions of this type to check understanding. »» Can students now calculate the period and range of any function of the type y=a cos bx, from the values of a and b? Checking Understanding KEY: » next step »» Distribute Student Activity 7. »» »» Walk around, looking at the graphs, referring students back to the previous activity sheets when they are in doubt. »» Distribute Student Activity 6. • When b is 1, the period is »» Students may need some help in moving from the specific to 2π rad = 360°. When b=2 the general here; allow them the period is π rad = 180°. to try it out on their own first. www.projectmaths.ie »» Sketch the graphs of the functions on Student Activity 6, without tables of values, by first calculating the period and range, and knowing the shape of functions of the type y=a sin bx and y=a cos bx. »» How can you calculate the period given b? »» What is the effect of varying • As b varies the period b? varies. Student Activities: Possible Teacher’s Support and Actions and Expected Responses Student Learning Tasks: Teacher Input Teaching & Learning Plan 10: Trigonometric Functions 9 © Project Maths Development Team 2009 • Its period is 180°. »» Students complete Student Activity 7C. • It is periodic but it repeats more often than sin x or cos • There are gaps in it. • Its shape is unlike the graphs of y = sin x and y = cos x. x • -57.29, -57295.78,-5729577.95 • 57.29, 57295.78, 5729577.951 »» Students plot the points. (Student Activity 7B) »» Students fill in Student Activity 7A. www.projectmaths.ie »» What is the period y = tan x? (Student Activity 7C) »» Describe the graph of y = tan x? »» What is the tan 91°, 90.001°, and 90.00001°? »» Using a calculator, find tan 89°, then the tan 89.999°, then the tan 89,99999°. »» Fill in the table on Student Activity 7A. »» What measurement in the • Tan <AOA’’ = intercept the diagram represents for instance radius from O to A’’ cuts off the tan <AOA’’? Explain. the line x=1. »» What is the relationship between sin, cos and tan of an angle x? KEY: » next step »» Encourage students to describe the graph in their own words, but because of familiarity with sin and cos graphs they should be using the words period and range. »» Students who may have difficulty understanding that tan x tends to infinity can be helped in this by using the calculator to evaluate tan of angles very close to 90°. Student Learning Tasks: Teacher Student Activities: Possible and Teacher’s Support and Input Expected Responses Actions Teaching & Learning Plan 10: Trigonometric Functions • student answer/response »» Do students see that as x gets closer to 90°, tan x increases very rapidly? Checking Understanding 10 © Project Maths Development Team 2009 www.projectmaths.ie »» Draw in the asymptotes on the graph. »» Students draw in lines y = –90° y = 90° y = 270° • Undefined »» What is the tan 90°? »» The graph of y = tan x approaches a line as x approaches for example, 90° and 270°. These lines are called ASYMPTOTES – a straight line to which the graph becomes closer and closer but doesn’t touch. • [-∞,+∞] • As the angle approaches 90°, tan x increases very rapidly and tends to + ∞ as it approaches 90° from the left, and to - ∞ as it approaches 90° from the right. Student Activities: Possible and Expected Responses »» What is the range of y = tan x (Student Activity 7C). Student Learning Tasks: Teacher Input »» Describe how tan x varies as x approaches 90°? »» y=tan x • student answer/response Are students able to plot the graph of y = tan x and list the period and range? »» Do students appreciate the phrase “tends to infinity” as x tends to 90°? Checking Understanding KEY: » next step Show the GeoGebra interactive webpage on Teacher’s Support and Actions Teaching & Learning Plan 10: Trigonometric Functions 11 © Project Maths Development Team 2009 »» Write down any questions you may have. »» Write down anything you found difficult. Reflection »» Write down 3 things you learned about today trigonometric functions. Student Learning Tasks: Teacher Input www.projectmaths.ie 10.Draw asymptotes on the graph y = tan x. 9. Concept of asymptotes 8. Find the period and range of tan x. 7. Draw the graph of y = tan x. 6. Concept of varying a and b in y = a cos bx 5. Find the period and range of cos x, a cos x, cos bx and a cos bx. 4. Draw the graph of cos x, a cos x, cos bx and a cos bx. 3. Concept of varying a and b in y = a sin bx 2. Find the period and range of sin x, a sin x, sin bx and a sin bx. Student Activities: Possible and Expected Responses 1. Draw the graph of sin x, a sin x, sin bx and a sin bx. »» Have all students learned and understood these items? »» Circulate and take note particularly of any questions students have and help them to answer them. KEY: » next step • student answer/response »» Are they using the terminology with understanding and communicating with each other using these terms? Checking Understanding Teacher’s Support and Actions Teaching & Learning Plan 10: Trigonometric Functions 12 © Project Maths Development Team 2009 www.projectmaths.ie 13 7.How many solutions has the equation in Q6?_ ______________________________________________________________________________ 6.Using the graph solve for x the equation sin x = 0.5_________________________________________________________________________ 5.Is the inverse of y = sin x a function? Explain._ ______________________________________________________________________________ 4.Is y = sin x a function? Explain._____________________________________________________________________________________________ 3.What is the range of y = sin x?_____________________________________________________________________________________________ 2.What is the period of y = sin x?____________________________________________________________________________________________ 1.Describe the graph of y = sin x._ ___________________________________________________________________________________________ Student Activity 1B Graph of y= sin x Complete the projection of sin values from the unit circle onto the Cartesian plane on the right and then join the points with a smooth curve. Student Activity 1A Student Activity 1 Teaching & Learning Plan 10: Trigonometric Functions Period -90 Range -60 -30 0 30 60 90 120 150 180 210 240 270 300 330 360 © Project Maths Development Team 2009 www.projectmaths.ie 14 _________________________________________________________________________________________________________________________ _________________________________________________________________________________________________________________________ _________________________________________________________________________________________________________________________ _________________________________________________________________________________________________________________________ In the function, what is the effect on the graph of varying a in asin x?_ _______________________________________________________ y = asin x y = 3sin x y = 2sin x y = sin x sin x 2sin x 3sin x x/º Using a calculator, or the unit circle, fill in the table for the following graphs and plot all of them using the same axes. y = sin x, y = 2sin x, y = 3sin x Use different colours for each graph. Student Activity 2 Teaching & Learning Plan 10: Trigonometric Functions Range 30 45 60 75 90 105 120 135 150 165 180 195 210 225 240 255 270 285 300 315 330 345 360 www.projectmaths.ie 15 _________________________________________________________________________________________________________________________ _________________________________________________________________________________________________________________________ In the graph of y = sin bx, what is the effect on the graph of varying b in sin bx?_______________________________________________ y = sin bx y = sin 3x Period 15 © Project Maths Development Team 2009 0 y = sin 2x y = sin x sin x 2x sin 2x 3x sin 3x x/º Fill in the table first, and using the same axes but different colours for each graph, draw the graphs of: y = sin x, y = sin 2x = sin (2 x x), y = sin 3x = sin (3 x x) Graphs of the form y = sin bx Student Activity 3 Teaching & Learning Plan 10: Trigonometric Functions -30 0 30 60 90 120 150 180 210 240 270 300 330 360 © Project Maths Development Team 2009 www.projectmaths.ie 16 _________________________________________________________________________________________________________________________ _________________________________________________________________________________________________________________________ _________________________________________________________________________________________________________________________ In the function y = acos x , what is the effect on the graph of varying a in acos x?______________________________________________ y = 3cos x -60 Period Range -90 y = 2cos x y = cos x cos x 2cos x 3cos x x/º Using a table, find the coordinates for the following graphs and plot all of them using the same axes: y = cos x, y = 2cos x, y = 3cos x Student Activity 4 Teaching & Learning Plan 10: Trigonometric Functions 0 Period 15 45 Range 30 60 75 90 105 120 135 150 165 180 195 210 225 240 255 270 285 300 315 330 345 360 © Project Maths Development Team 2009 www.projectmaths.ie 17 _________________________________________________________________________________________________________________________ _________________________________________________________________________________________________________________________ In the graph of y = acos bx, what is the effect on the graph of varying b in cosbx?_____________________________________________ y = cos bx y = cos 3x y = cos 2x y = cos x cos x 2x cos 2x 3x cos 3x x/º By filling in a table first, and using the same axes but different colours for each graph, draw the graphs of Student Activity 5 Teaching & Learning Plan 10: Trigonometric Functions 18 Period =_________________ Range =_________________ y = 2sin 3x Sketch each of the following graphs: (0°≤ x ≤ 360°) www.projectmaths.ie Period =_________________ Range =_________________ y = cos4 x Sketch each of the following graphs: (0°≤ x ≤ 360°) © Project Maths Development Team 2009 Period =_________________ Range =_________________ y = 4sin x Sketch each of the following graphs: (0°≤ x ≤ 360°) Student Activity 6 Teaching & Learning Plan 10: Trigonometric Functions © Project Maths Development Team 2009 www.projectmaths.ie |∠AOA’’| 60.00° |∠AOA’’’| 75.00° Student Activity 7A |∠AOA’’’’| 80.00° |∠AOA’’’’’| 85.00° Angle θ/º tan θ 0 30 the table using the trigonometric ratios. 60 75 80 19 85 Using the diagram of the unit circle, read the approximate value of the tan of the angles in y=tan x |∠AOA’| 30.00° Student Activity 7 Teaching & Learning Plan 10: Trigonometric Functions 0 www.projectmaths.ie -90 -75 -60 -45 -30 © Project Maths Development Team 2009 tan x x/º 30 60 y = tan x 45 90 Period Range 20 105 120 135 150 180 210 225 240 255 270 285 300 330 360 Student Activity 7C 75 By filling in a table first, and using the same axes but different colours for each graph, draw the graphs of Student Activity 7B Student Activity 7 Teaching & Learning Plan 10: Trigonometric Functions

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