# Trigonometric functions ```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.
»» Does this presentation
help students in the task of
plotting the graph
y=sin x?
Checking Understanding
KEY: » next step
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.
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.
Teacher Input
»»
KEY: » next step
GeoGebra interactive
webpage f(x) =asinx. Set
slider with a=1
»» 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 )
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
»» 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.
»» Is the inverse of y=sin x a
function? Student Activity
1B Q5.
Student Activity 1B Q4
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
Teacher Input
»»
»» 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?
Teacher Input
»»
»» 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
Teacher Input
»» Distribute Student
Activity 5.
»» Distribute Student
Activity 4.
Teacher’s Support and
Actions
Teaching & Learning Plan 10: Trigonometric Functions
KEY: » next step
»» 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
»» 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
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
»» 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).
Teacher Input
»» Describe how tan x varies
as x approaches 90°?
»»
y=tan x
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
trigonometric functions.
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
KEY: » next step
»» 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
```