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Teaching & Learning PlansPlan 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.
© Project Maths Development Team 2009 www.projectmaths.ie 1
Teaching & Learning Plan 10: Trigonometric Functions
AimsTo 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 OutcomesAs 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 • , from the graph, for the domain plotted
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
Teaching & Learning Plan 10: Trigonometric Functions
© Project Maths Development Team 2009 www.projectmaths.ie 2
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 Level2.3 Trigonometry Graph the trigonometric functions sine, cosine, tangent.
Graph trigonometric functions of type asin nθ, acos nθ for a, n ∈ N.
Resources RequiredCompasses, 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.
Teac
hing
& L
earn
ing
Plan
10:
Tri
gono
met
ric
Func
tion
s
© P
roje
ct M
aths
Dev
elop
men
t Tea
m 2
009
w
ww
.pro
ject
mat
hs.ie
KE
Y:
» n
ext
step
•
stud
ent
answ
er/r
espo
nse
3
Less
on
In
tera
ctio
nSt
uden
t Le
arni
ng T
asks
: Te
ache
r In
put
Stud
ent A
ctiv
itie
s: P
ossi
ble
and
Expe
cted
Res
pons
esTe
ache
r’s S
uppo
rt a
nd
Act
ions
Chec
king
Und
erst
andi
ng
We
hav
e se
en o
n t
he
un
it
»ci
rcle
th
at s
in, c
os
and
tan
va
ry a
s th
e an
gle
var
ies
fro
m 0
° to
360
°. T
he
sin
fu
nct
ion
is r
epre
sen
ted
by
the
y co
ord
inat
e o
n t
he
un
it c
ircl
e.
Sho
w
»h
ttp
://w
ww
.p
roje
ctm
ath
s.ie
/sys
tem
/fi
les/
Pres
enta
tio
n%
20Lo
nd
on
%20
eye%
20si
ne.
pp
tx a
nd
Per
spex
Mo
del
of
un
it c
ircl
e.
Do
es t
his
pre
sen
tati
on
»
hel
p s
tud
ents
in t
he
task
of
plo
ttin
g t
he
gra
ph
y=
sin
x?
Usi
ng
»
Stu
den
t A
ctiv
ity
1A,
pro
ject
th
e va
lues
of
the
y co
ord
inat
e, w
hic
h is
sin
x
on
to t
he
Car
tesi
an p
lan
e to
g
et a
gra
ph
of
y= s
in x
0
≤ x
≤ 57
0°
Stu
den
ts p
lot
the
po
ints
»
and
join
th
em in
a s
mo
oth
cu
rve.
Dis
trib
ute
»
Stu
den
t A
ctiv
ity
1. Tell
stu
den
ts t
hey
will
nee
d
»a
clea
r ru
ler
and
pen
cil.
Des
crib
e th
e g
rap
h
»St
ud
ent
Act
ivit
y 1B
Q 1
.It
co
nsi
sts
of
rep
eati
ng
•
pat
tern
s.
It h
as a
max
of
•1
and
a m
in
of
-1.
The
gra
ph
is “
rou
nd
ed”
– •
no
sh
arp
po
ints
.
Ask
ind
ivid
ual
stu
den
ts t
o
»d
escr
ibe
the
gra
ph
in t
hei
r o
wn
wo
rds.
Teac
hing
& L
earn
ing
Plan
10:
Tri
gono
met
ric
Func
tion
s
© P
roje
ct M
aths
Dev
elop
men
t Tea
m 2
009
w
ww
.pro
ject
mat
hs.ie
KE
Y:
» n
ext
step
•
stud
ent
answ
er/r
espo
nse
4
Stud
ent
Lear
ning
Tas
ks:
Teac
her
Inpu
tSt
uden
t Act
ivit
ies:
Pos
sibl
e an
d Ex
pect
ed R
espo
nses
Teac
her’s
Sup
port
and
A
ctio
nsCh
ecki
ng U
nder
stan
ding
Sho
w a
n a
nim
atio
n o
f th
is
»ex
erci
se f
rom
Au
tog
rap
h.
(File
, New
Ext
ras
Pag
e,
Trig
on
om
etry
, sin
)
See
also
(sc
roll
to e
nd
of
»p
age)
pis
ton
an
imat
ion
: h
ttp
://w
ww
.intm
ath
.co
m/
Trig
on
om
etri
c-g
rap
hs/
2_G
rap
hs-
sin
e-co
sin
e-p
erio
d.
ph
p
»G
eoG
ebra
inte
ract
ive
web
pag
e f(
x) =
asin
x. S
et
slid
er w
ith
a=
1
Do
es t
he
anim
atio
n
»su
mm
aris
e th
e ex
erci
se f
or
stu
den
ts a
nd
giv
e th
em
incr
ease
d u
nd
erst
and
ing
of
the
rela
tio
nsh
ip b
etw
een
si
n x
an
d t
he
y-co
ord
inat
e o
n t
he
un
it c
ircl
e?
An
y fu
nct
ion
wh
ich
is
»re
pea
tin
g s
uch
th
at f(
x)
= f(
x+c)
, w
her
e c
is a
co
nst
ant
is c
alle
d a
per
iod
ic
fun
ctio
n.
We
refe
r to
th
e le
ng
th o
f »
the
rep
eati
ng
pat
tern
on
th
e x-
axi
s as
th
e p
erio
d
and
th
e m
ax a
nd
min
y
valu
es a
re t
he
ran
ge
wri
tten
as
[min
val
ue, m
ax v
alue
]
Wri
te d
ow
n t
he
per
iod
an
d
»ra
ng
e o
f y=
sinx
Stu
den
t A
ctiv
ity
1B Q
2 a
nd
3.
Peri
od
= 3
60°
•R
ang
e =
•
[-1,
1]
Teac
hing
& L
earn
ing
Plan
10:
Tri
gono
met
ric
Func
tion
s
© P
roje
ct M
aths
Dev
elop
men
t Tea
m 2
009
w
ww
.pro
ject
mat
hs.ie
KE
Y:
» n
ext
step
•
stud
ent
answ
er/r
espo
nse
5
Stud
ent
Lear
ning
Tas
ks:
Teac
her
Inpu
tSt
uden
t Act
ivit
ies:
Pos
sibl
e an
d Ex
pect
ed R
espo
nses
Teac
her’s
Sup
port
and
A
ctio
nsCh
ecki
ng U
nder
stan
ding
Is
»y=
sin
x a
fun
ctio
n?
Stu
den
t A
ctiv
ity
1B Q
4
Stu
den
ts e
xpla
in in
•
Stu
den
t A
ctiv
ity
1B Q
4 th
at f
or
each
val
ue
of
x th
ere
is
on
ly o
ne
valu
e o
f y,
hen
ce
y=si
n x
is a
fu
nct
ion
.
Is t
he
inve
rse
of
»y=
sin
x a
fun
ctio
n?
Stu
den
t A
ctiv
ity
1B Q
5.
Exp
lain
yo
ur
answ
er.
»
Stu
den
ts e
xpla
in t
hat
fo
r »
any
y va
lue
ther
e is
an
in
fin
ite
nu
mb
er o
f x
valu
es,
so t
he
inve
rse
of
y=si
n x
is
no
t a
fun
ctio
n.
They
may
als
o d
raw
a
»h
ori
zon
tal l
ine
thro
ug
h
any
valu
e o
f y
in t
he
ran
ge,
sh
ow
ing
th
at it
has
m
ult
iple
so
luti
on
s.
Rem
ind
stu
den
ts t
hat
a
»fu
nct
ion
is a
set
of
ord
ered
p
airs
, wh
ere
each
x v
alu
e h
as o
ne
un
iqu
e y
valu
e.
Do
stu
den
ts r
emem
ber
»
wh
at a
fu
nct
ion
is?
Can
th
ey e
xpla
in w
hy
the
»in
vers
e o
f y=
sin
x is
no
t a
fun
ctio
n?
Co
mp
lete
»
Stu
den
t A
ctiv
ity
1B Q
6 an
d 7
.C
an s
tud
ents
so
lve
the
»eq
uat
ion
sin
x=0.
5?H
avin
g s
een
th
e co
nn
ecti
on
»
bet
wee
n y
= s
inx
and
m
oti
on
in a
cir
cle,
we
will
n
ow
plo
t th
e g
rap
h o
f y=
sin
x u
sin
g a
tab
le o
f va
lues
, th
en y
= 2
sin
x an
d
y= 3
sin
x, u
sin
g t
he
sam
e ax
es a
nd
sca
le.
Stu
den
ts p
lot
the
»g
rap
hs
on
gra
ph
pap
er
acco
mp
anyi
ng
Stu
den
t A
ctiv
ity
2 u
sin
g d
iffe
ren
t co
lou
rs f
or
each
gra
ph
.
Dis
trib
ute
»
Stu
den
t A
ctiv
ity
2.
Teac
hing
& L
earn
ing
Plan
10:
Tri
gono
met
ric
Func
tion
s
© P
roje
ct M
aths
Dev
elop
men
t Tea
m 2
009
w
ww
.pro
ject
mat
hs.ie
KE
Y:
» n
ext
step
•
stud
ent
answ
er/r
espo
nse
6
Stud
ent
Lear
ning
Tas
ks:
Teac
her
Inpu
tSt
uden
t Act
ivit
ies:
Pos
sibl
e an
d Ex
pect
ed R
espo
nses
Teac
her’s
Sup
port
and
Act
ions
Chec
king
Und
erst
andi
ng
Wh
at is
th
e ef
fect
of
»va
ryin
g a
? St
ud
ent
Act
ivit
y 2.
The
ran
ge
chan
ges
. Th
e •
ran
ge
is [
-a, a
]If
req
uir
ed g
ive
stu
den
ts s
om
e h
elp
»
in m
akin
g t
his
co
ncl
usi
on
.D
o a
ll st
ud
ents
»
un
der
stan
d t
hat
in t
he
equ
atio
n y
=a
sin
x
the
per
iod
rem
ain
s u
nch
ang
ed a
s a
chan
ges
an
d t
hat
on
ly t
he
ran
ge
chan
ges
?
If t
his
was
a s
ou
nd
»
wav
e w
hat
eff
ect
do
yo
u t
hin
k it
wo
uld
hav
e o
n t
he
sou
nd
to
ch
ang
e th
e va
lue
of
a?
The
volu
me
wo
uld
incr
ease
•
as a
incr
ease
d.
In t
he
fun
ctio
n
»y=
a si
n bx
we
hav
e se
en t
he
effe
ct o
f va
ryin
g a
, an
d n
ow
we
will
kee
p a
co
nst
ant
at
a=1,
an
d v
ary
b.
Stu
den
ts fi
ll in
th
e ta
ble
»
and
plo
t th
e g
rap
hs
in 3
d
iffe
ren
t co
lou
rs (
Stu
den
t A
ctiv
ity
3).
Dis
trib
ute
»
Stu
den
t A
ctiv
ity
3.
Rem
ind
stu
den
ts t
hat
th
ey a
re
»p
lott
ing
th
e fu
nct
ion
s
f : x
→ si
n x
g
: x →
sin
2x
h : x
→ si
n 3x
Are
stu
den
ts k
eep
ing
»
the
corr
ect
shap
e o
n t
he
curv
es e
ven
wh
en t
hey
h
ave
few
er p
oin
ts t
o p
lot
for
each
cyc
le?
Fill
in
»St
ud
ent
Act
ivit
y 3.
Wh
at is
th
e ef
fect
of
»va
ryin
g b
?
Stu
den
ts fi
ll in
»
Stu
den
t A
ctiv
ity
3.
As
•b
vari
es t
he
per
iod
va
ries
.
Stu
den
ts s
ho
uld
be
able
to
mo
ve
»fr
om
th
e sp
ecifi
c to
th
e g
ener
al
her
e an
d s
ho
uld
be
allo
wed
to
try
it
ou
t o
n t
hei
r o
wn
firs
t.
»U
sin
g G
eoG
ebra
inte
ract
ive
web
pag
e sh
ow
gra
ph
of
y=a
sin
b x
, var
yin
g a
an
d b
usi
ng
th
e sl
ider
s.
Teac
hing
& L
earn
ing
Plan
10:
Tri
gono
met
ric
Func
tion
s
© P
roje
ct M
aths
Dev
elop
men
t Tea
m 2
009
w
ww
.pro
ject
mat
hs.ie
KE
Y:
» n
ext
step
•
stud
ent
answ
er/r
espo
nse
7
Stud
ent
Lear
ning
Tas
ks:
Teac
her
Inpu
tSt
uden
t Act
ivit
ies:
Pos
sibl
e an
d Ex
pect
ed R
espo
nses
Teac
her’s
Sup
port
and
A
ctio
nsCh
ecki
ng U
nder
stan
ding
Ho
w c
an y
ou
cal
cula
te t
he
»p
erio
d g
iven
b?
Wh
en
•b
is 1
, th
e p
erio
d is
2π
rad
= 3
60°.
Wh
en b
=2
the
per
iod
is π
rad
= 1
80°.
In g
ener
al t
he
per
iod
is
•
Can
stu
den
ts n
ow
cal
cula
te
»th
e p
erio
d a
nd
ran
ge
of
any
fun
ctio
n o
f th
e ty
pe
y=a
sin
bx, f
rom
th
e va
lues
o
f a
and
b?
Use
exa
mp
les
of
fun
ctio
ns
»o
f th
is t
ype
to c
hec
k.
If t
his
was
a s
ou
nd
wav
e »
wh
at w
ou
ld b
e th
e ef
fect
o
f va
ryin
g b
?
No
te: S
tude
nts
do n
ot n
eed
to
know
thi
s fo
r th
e sy
llabu
s, b
ut it
is
goo
d fo
r re
al li
fe c
onte
xt.
The
freq
uen
cy a
nd
hen
ce
•th
e p
itch
of
the
no
te w
ou
ld
chan
ge,
incr
easi
ng
as
b in
crea
sed
.
We
saw
ho
w s
in
»x
is t
he
y-co
ord
inat
e o
f a
po
int
on
th
e ci
rcu
mfe
ren
ce o
f th
e
un
it c
ircl
e an
d t
hat
co
s x
is
the
x-co
ord
inat
e. H
ence
by
plo
ttin
g t
he
x co
ord
inat
es
of
po
ints
on
th
e u
nit
cir
cle
on
to a
Car
tesi
an p
lan
e w
e sh
ou
ld g
et t
he
gra
ph
of
y=co
s x.
Sho
w a
n a
nim
atio
n o
f th
is
»ex
erci
se f
rom
Au
tog
rap
h.
(File
, New
Ext
ras
Pag
e,
Trig
on
om
etry
, co
s) a
nd
Pe
rsp
ex m
od
el o
f u
nit
ci
rcle
.
»Sh
ow
, usi
ng
a
Geo
Geb
ra in
tera
ctiv
e w
ebp
age,
wh
ere
sin
is
dra
gg
ed 9
0° t
o t
he
left
g
ivin
g t
he
cosi
ne
fun
ctio
n.
Teac
hing
& L
earn
ing
Plan
10:
Tri
gono
met
ric
Func
tion
s
© P
roje
ct M
aths
Dev
elop
men
t Tea
m 2
009
w
ww
.pro
ject
mat
hs.ie
KE
Y:
» n
ext
step
•
stud
ent
answ
er/r
espo
nse
8
Stud
ent
Lear
ning
Tas
ks:
Teac
her
Inpu
tSt
uden
t Act
ivit
ies:
Pos
sibl
e an
d Ex
pect
ed R
espo
nses
Teac
her’s
Sup
port
and
A
ctio
nsCh
ecki
ng U
nder
stan
ding
Des
crib
e th
e g
rap
h o
f
»y
= c
os x
Its
shap
e is
th
e sa
me
as t
he
•si
ne
gra
ph
. It
is t
he
gra
ph
o
f si
n x
shif
ted
90°
to
th
e le
ft.
It is
per
iod
ic w
ith
a p
erio
d
•o
f 36
0° a
nd
a r
ang
e o
f [-
1,1]
Are
stu
den
ts n
ow
ab
le t
o
»d
escr
ibe
the
gra
ph
of
y
= c
os x
in t
erm
s o
f p
erio
dic
ity
and
ran
ge,
hav
ing
wo
rked
wit
h
y = si
n x?
Hav
ing
see
n t
he
con
nec
tio
n
»b
etw
een
y =
cos
x a
nd
m
oti
on
in a
cir
cle,
we
will
n
ow
plo
t th
e g
rap
h o
f
y =
cos
x u
sin
g a
tab
le o
f va
lues
, th
en y
= 2
cosx
an
d
y =
3co
sx, u
sin
g t
he
sam
e ax
es a
nd
sca
le (
Stu
den
t A
ctiv
ity
4).
Stu
den
ts p
lot
the
gra
ph
s »
usi
ng
dif
fere
nt
colo
urs
fo
r ea
ch g
rap
h.
Dis
trib
ute
»
Stu
den
t A
ctiv
ity
4.
Wh
at is
th
e p
erio
d a
nd
»
ran
ge
of
y =
cos
x,
y =
2co
sx
and
y =
3co
sx?
Stu
den
t A
ctiv
ity
4.
Stu
den
ts fi
ll in
»
Stu
den
t A
ctiv
ity
4.D
o a
ll st
ud
ents
un
der
stan
d t
hat
»
in t
he
equ
atio
n y
= a
cos x
, th
e p
erio
d r
emai
ns
un
chan
ged
as
a ch
ang
es a
nd
th
at o
nly
th
e ra
ng
e ch
ang
es?
Wh
at is
th
e ef
fect
of
»va
ryin
g a
? St
ud
ent
Act
ivit
y 4.
The
ran
ge
chan
ges
. Th
e •
ran
ge
is [
-a,a
]
In t
he
fun
ctio
n
»y=
a co
s bx
we
hav
e se
en t
he
effe
ct o
f va
ryin
g a
, an
d n
ow
we
will
ke
ep a
co
nst
ant
and
var
y b.
Stu
den
ts fi
ll in
th
e ta
ble
•
and
plo
t th
e g
rap
hs
in 3
d
iffe
ren
t co
lou
rs. S
tud
ent
Act
ivit
y 5.
Dis
trib
ute
»
Stu
den
t A
ctiv
ity
5.A
re s
tud
ents
kee
pin
g t
he
»co
rrec
t sh
ape
on
th
e cu
rves
ev
en w
hen
th
ey h
ave
few
er
po
ints
to
plo
t fo
r ea
ch c
ycle
?
Teac
hing
& L
earn
ing
Plan
10:
Tri
gono
met
ric
Func
tion
s
© P
roje
ct M
aths
Dev
elop
men
t Tea
m 2
009
w
ww
.pro
ject
mat
hs.ie
KE
Y:
» n
ext
step
•
stud
ent
answ
er/r
espo
nse
9
Stud
ent
Lear
ning
Tas
ks:
Teac
her
Inpu
tSt
uden
t Act
ivit
ies:
Pos
sibl
e an
d Ex
pect
ed R
espo
nses
Teac
her’s
Sup
port
and
Act
ions
Chec
king
Und
erst
andi
ng
Wh
at is
th
e p
erio
d a
nd
»
ran
ge
of
y =
cos
x
y =
cos
2x
and
y =
cos
3x?
St
ud
ent
Act
ivit
y 5.
Wh
at is
th
e ef
fect
of
vary
ing
»
b?
Stu
den
ts fi
ll in
»
Stu
den
t A
ctiv
ity
5.
As
•b
vari
es t
he
per
iod
va
ries
.
Ho
w c
an y
ou
cal
cula
te t
he
»p
erio
d g
iven
b?
Wh
en
•b
is 1
, th
e p
erio
d is
2π
rad
= 3
60°.
Wh
en b
=2
the
per
iod
is π
rad
= 1
80°.
In g
ener
al t
he
per
iod
is
•
Stu
den
ts m
ay n
eed
so
me
hel
p
»in
mo
vin
g f
rom
th
e sp
ecifi
c to
th
e g
ener
al h
ere;
allo
w t
hem
to
try
it o
ut
on
th
eir
ow
n fi
rst.
Can
stu
den
ts n
ow
»
calc
ula
te t
he
per
iod
an
d
ran
ge
of
any
fun
ctio
n
of
the
typ
e y=
a co
s bx,
fr
om
th
e va
lues
of
a an
d
b?
Use
mo
re e
xam
ple
s o
f »
fun
ctio
ns
of
this
typ
e to
ch
eck
un
der
stan
din
g.
Sket
ch t
he
gra
ph
s o
f th
e »
fun
ctio
ns
on
Stu
den
t A
ctiv
ity
6, w
ith
ou
t ta
ble
s o
f va
lues
, by
firs
t ca
lcu
lati
ng
th
e p
erio
d a
nd
ran
ge,
an
d k
no
win
g t
he
shap
e o
f fu
nct
ion
s o
f th
e ty
pe
y=a
sin
bx a
nd
y=a
cos b
x.
Stu
den
ts m
ake
4 ev
enly
»
spac
ed m
arks
on
th
e
x ax
is a
nd
mar
k th
e ra
ng
e o
n t
he
y ax
is, a
nd
p
lot
the
curv
es.
Dis
trib
ute
»
Stu
den
t A
ctiv
ity
6.
Wal
k ar
ou
nd
, lo
oki
ng
at
the
»g
rap
hs,
ref
erri
ng
stu
den
ts b
ack
to t
he
pre
vio
us
acti
vity
sh
eets
w
hen
th
ey a
re in
do
ub
t.
»Sh
ow
Geo
Geb
ra in
tera
ctiv
e w
ebp
age
y=a
sin
bx a
nd
y=a
cos b
x
Are
stu
den
ts a
ble
to
»
sket
ch g
rap
hs
of
the
form
y=a
sin
bx a
nd
y=
a co
s bx
fro
m t
he
valu
es o
f p
erio
d a
nd
ra
ng
e?
We
will
no
w lo
ok
at t
he
»g
rap
h o
f th
e fu
nct
ion
y=
a ta
n x.
Dis
trib
ute
»
Stu
den
t A
ctiv
ity
7.
Teac
hing
& L
earn
ing
Plan
10:
Tri
gono
met
ric
Func
tion
s
© P
roje
ct M
aths
Dev
elop
men
t Tea
m 2
009
w
ww
.pro
ject
mat
hs.ie
KE
Y:
» n
ext
step
•
stud
ent
answ
er/r
espo
nse
10
Stud
ent
Lear
ning
Tas
ks: T
each
er
Inpu
tSt
uden
t Act
ivit
ies:
Pos
sibl
e an
d Ex
pect
ed R
espo
nses
Teac
her’s
Sup
port
and
A
ctio
nsCh
ecki
ng
Und
erst
andi
ngW
hat
is t
he
rela
tio
nsh
ip
»b
etw
een
sin
, co
s an
d t
an o
f an
an
gle
x?
Wh
at m
easu
rem
ent
in t
he
»d
iag
ram
rep
rese
nts
fo
r in
stan
ce
the
tan
<A
OA
’’? E
xpla
in.
Tan
<A
OA
’’ =
inte
rcep
t th
e •
rad
ius
fro
m O
to
A’’
cuts
off
th
e lin
e x=
1.
Fill
in t
he
tab
le o
n
»St
ud
ent
Act
ivit
y 7A
.St
ud
ents
fill
in
»St
ud
ent
Act
ivit
y 7A
.
Stu
den
ts p
lot
the
po
ints
. »
(Stu
den
t A
ctiv
ity
7B)
Usi
ng
a c
alcu
lato
r, fi
nd
tan
89°
, »
then
th
e ta
n 8
9.99
9°, t
hen
th
e ta
n 8
9,99
999°
.
Wh
at is
th
e ta
n 9
1°, 9
0.00
1°,
»an
d 9
0.00
001°
?
57.2
9, 5
7295
.78,
572
9577
.951
•
-57.
29, -
5729
5.78
,-57
2957
7.95
•
Stu
den
ts w
ho
may
hav
e »
dif
ficu
lty
un
der
stan
din
g
that
tan
x t
end
s to
infi
nit
y ca
n b
e h
elp
ed in
th
is b
y u
sin
g t
he
calc
ula
tor
to
eval
uat
e ta
n o
f an
gle
s ve
ry
clo
se t
o 9
0°.
Do
stu
den
ts s
ee t
hat
»
as x
get
s cl
ose
r to
90
°, t
an x
incr
ease
s ve
ry r
apid
ly?
Des
crib
e th
e g
rap
h o
f
»y
= ta
n x?
Its
shap
e is
un
like
the
gra
ph
s •
of
y =
sin
x an
d y
= c
os x
.
Ther
e ar
e g
aps
in it
. •
It is
per
iod
ic b
ut
it r
epea
ts
•m
ore
oft
en t
han
sin
x o
r co
s x
Stu
den
ts c
om
ple
te
»St
ud
ent
Act
ivit
y 7C
.
Enco
ura
ge
stu
den
ts t
o
»d
escr
ibe
the
gra
ph
in t
hei
r o
wn
wo
rds,
bu
t b
ecau
se
of
fam
iliar
ity
wit
h s
in a
nd
co
s g
rap
hs
they
sh
ou
ld b
e u
sin
g t
he
wo
rds
per
iod
an
d
ran
ge.
Wh
at is
th
e p
erio
d
»y
= ta
n x?
(St
ud
ent
Act
ivit
y 7C
)It
s p
erio
d is
180
°.•
Teac
hing
& L
earn
ing
Plan
10:
Tri
gono
met
ric
Func
tion
s
© P
roje
ct M
aths
Dev
elop
men
t Tea
m 2
009
w
ww
.pro
ject
mat
hs.ie
KE
Y:
» n
ext
step
•
stud
ent
answ
er/r
espo
nse
11
Stud
ent
Lear
ning
Tas
ks:
Teac
her
Inpu
tSt
uden
t Act
ivit
ies:
Pos
sibl
e an
d Ex
pect
ed R
espo
nses
Teac
her’s
Sup
port
and
A
ctio
nsCh
ecki
ng U
nder
stan
ding
Des
crib
e h
ow
tan
»
x va
ries
as
x a
pp
roac
hes
90°
?
As
the
ang
le a
pp
roac
hes
•
90°,
tan
x in
crea
ses
very
ra
pid
ly a
nd
ten
ds
to +
∞ a
s it
ap
pro
ach
es 9
0° f
rom
th
e le
ft,
and
to
- ∞
as
it a
pp
roac
hes
90
° fr
om
th
e ri
gh
t.
»Sh
ow
th
e G
eoG
ebra
in
tera
ctiv
e w
ebp
age
on
y=
tan
x
Do
stu
den
ts a
pp
reci
ate
»th
e p
hra
se “
ten
ds
to
infi
nit
y” a
s x
ten
ds
to
90°?
Wh
at is
th
e ra
ng
e o
f »
y =
tan
x (S
tud
ent
Act
ivit
y 7C
).
[-∞
,+∞
]•
Are
stu
dent
s ab
le t
o pl
ot t
he
grap
h of
y =
tan
x an
d lis
t th
e pe
riod
and
rang
e?
Wh
at is
th
e ta
n 9
0°?
»U
nd
efin
ed•
The
gra
ph
of
»y
= ta
n x
app
roac
hes
a li
ne
as x
ap
pro
ach
es f
or
exam
ple
, 90
° an
d 2
70°.
Th
ese
lines
ar
e ca
lled
ASY
MPT
OTE
S –
a st
raig
ht
line
to w
hic
h t
he
gra
ph
bec
om
es c
lose
r an
d
clo
ser
bu
t d
oes
n’t
to
uch
.
Dra
w in
th
e as
ymp
tote
s o
n
»th
e g
rap
h.
Stu
den
ts d
raw
in li
nes
»
y =
–90
° y
= 9
0°
y =
270
°
Teac
hing
& L
earn
ing
Plan
10:
Tri
gono
met
ric
Func
tion
s
© P
roje
ct M
aths
Dev
elop
men
t Tea
m 2
009
w
ww
.pro
ject
mat
hs.ie
KE
Y:
» n
ext
step
•
stud
ent
answ
er/r
espo
nse
12
Stud
ent
Lear
ning
Tas
ks:
Teac
her
Inpu
tSt
uden
t Act
ivit
ies:
Pos
sibl
e an
d Ex
pect
ed R
espo
nses
Teac
her’s
Sup
port
and
A
ctio
nsCh
ecki
ng
Und
erst
andi
ngR
eflec
tio
nW
rite
do
wn
3 t
hin
gs
you
»
lear
ned
ab
ou
t to
day
tr
igo
no
met
ric
fun
ctio
ns.
Wri
te d
ow
n a
nyt
hin
g y
ou
»
fou
nd
dif
ficu
lt.
Wri
te d
ow
n a
ny
qu
esti
on
s »
you
may
hav
e.
Dra
w t
he
gra
ph
of
sin
1.
x,
a
sin
x, s
in b
x an
d a
sin
bx.
Fin
d t
he
per
iod
an
d r
ang
e o
f si
n
2.
x,
a si
n x
, sin
bx
and
a s
in b
x.
Co
nce
pt
of
vary
ing
3.
a
and
b in
y
= a
sin
bx
Dra
w t
he
gra
ph
of
cos
4.
x,
a co
s x,
co
s bx
an
d a
co
s bx
.
Fin
d t
he
per
iod
an
d r
ang
e o
f co
s 5.
x,
a
cos
x, c
os
bx a
nd
a c
os
bx.
Co
nce
pt
of
vary
ing
6.
a
and
b in
y
= a
co
s bx
Dra
w t
he
gra
ph
of
7.
y
= t
an x
.
Fin
d t
he
per
iod
an
d r
ang
e o
f ta
n
8.
x.
Co
nce
pt
of
asym
pto
tes
9.
Dra
w a
sym
pto
tes
on
th
e g
rap
h
10. y
= t
an x
.
Cir
cula
te a
nd
tak
e »
no
te p
arti
cula
rly
of
any
qu
esti
on
s st
ud
ents
h
ave
and
hel
p t
hem
to
an
swer
th
em.
Hav
e al
l stu
den
ts
»le
arn
ed a
nd
u
nd
erst
oo
d t
hes
e it
ems?
Are
th
ey u
sin
g t
he
»te
rmin
olo
gy
wit
h
un
der
stan
din
g a
nd
co
mm
un
icat
ing
wit
h
each
oth
er u
sin
g t
hes
e te
rms?
Teac
hing
& L
earn
ing
Plan
10:
Tri
gono
met
ric
Func
tion
s
© P
roje
ct M
aths
Dev
elop
men
t Tea
m 2
009
w
ww
.pro
ject
mat
hs.ie
13
Stud
ent A
ctiv
ity
1Stu
den
t A
ctiv
ity 1
AG
raph
of y
= sin
x C
ompl
ete
the
proj
ectio
n of
sin
val
ues
from
the
unit
circ
le o
nto
the
Car
tesia
n pl
ane
on th
e rig
ht a
nd th
en jo
in th
e po
ints
with
a s
moo
th c
urve
.
Stu
den
t A
ctiv
ity 1
B1.
Des
crib
e th
e g
rap
h o
f y
= s
in x
. __
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
_
2. W
hat
is t
he
per
iod
of
y =
sin
x?
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
___
3. W
hat
is t
he
ran
ge
of
y =
sin
x?
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
4. I
s y
= s
in x
a f
un
ctio
n?
Exp
lain
. ___
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
_
5. I
s th
e in
vers
e o
f y
= s
in x
a f
un
ctio
n?
Exp
lain
. __
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
6. U
sin
g t
he
gra
ph
so
lve
for
x th
e eq
uat
ion
sin
x =
0.5
___
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
_
7. H
ow
man
y so
luti
on
s h
as t
he
equ
atio
n in
Q6?
___
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
___
Teac
hing
& L
earn
ing
Plan
10:
Tri
gono
met
ric
Func
tion
s
© P
roje
ct M
aths
Dev
elop
men
t Tea
m 2
009
w
ww
.pro
ject
mat
hs.ie
14
Stud
ent A
ctiv
ity
2
Usin
g a
calc
ulat
or, o
r the
uni
t circ
le, fi
ll in
the
tabl
e fo
r the
follo
win
g gr
aphs
and
plo
t all
of th
em u
sing
the
sam
e ax
es.
y = si
n x,
y = 2
sin x,
y =
3sin
x U
se d
iffer
ent c
olou
rs fo
r eac
h gr
aph.
x/º
-90
-60
-30
030
6090
120
150
180
210
240
270
300
330
360
sin
x2s
in x
3sin
x
Peri
odRa
nge
y =
sin
xy
= 2
sin
xy
= 3
sin
xy
= a
sin
x
In
th
e fu
nct
ion
, wh
at is
th
e ef
fect
on
th
e g
rap
h o
f va
ryin
g a
in a
sin
x?
____
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___
___
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____
____
____
____
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____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
_
___
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
_
___
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
_
___
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
_
Teac
hing
& L
earn
ing
Plan
10:
Tri
gono
met
ric
Func
tion
s
© P
roje
ct M
aths
Dev
elop
men
t Tea
m 2
009
w
ww
.pro
ject
mat
hs.ie
15
Stud
ent A
ctiv
ity
3
Fill
in th
e ta
ble
first
, and
usin
g th
e sa
me
axes
but
diff
eren
t col
ours
for e
ach
grap
h, d
raw
the
grap
hs o
f:y =
sin
x, y =
sin
2x =
sin
(2 x
x), y
= si
n 3x
= si
n (3
x x)
G
raph
s of
the
form
y =
sin b
x
x/º
015
3045
6075
9010
512
013
515
016
518
019
521
022
524
025
527
028
530
031
533
034
536
0
sin
x2x si
n 2x
3x sin
3x
Peri
odRa
nge
y =
sin
xy
= si
n 2x
y =
sin
3xy
= si
n bx
In
th
e g
rap
h o
f y
= si
n bx
, wh
at is
th
e ef
fect
on
th
e g
rap
h o
f va
ryin
g b
in si
n bx
? ___
____
____
____
____
____
____
____
____
____
____
___
___
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
_
___
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
_
Teac
hing
& L
earn
ing
Plan
10:
Tri
gono
met
ric
Func
tion
s
© P
roje
ct M
aths
Dev
elop
men
t Tea
m 2
009
w
ww
.pro
ject
mat
hs.ie
16
Stud
ent A
ctiv
ity
4
Usin
g a
tabl
e, fi
nd th
e co
ordi
nate
s fo
r the
follo
win
g gr
aphs
and
plo
t all
of th
em u
sing
the
sam
e ax
es: y
= co
s x, y
= 2
cos x
, y =
3co
s x
x/º
-90
-60
-30
030
6090
120
150
180
210
240
270
300
330
360
cos x
2cos
x3c
os x
Peri
odRa
nge
y =
cos
xy
= 2
cos x
y =
3co
s x
In
th
e fu
nct
ion
y =
acos
x ,
wh
at is
th
e ef
fect
on
th
e g
rap
h o
f va
ryin
g a
in a
cos x
? __
____
____
____
____
____
____
____
____
____
____
___
___
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
_
___
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
_
___
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
_
Teac
hing
& L
earn
ing
Plan
10:
Tri
gono
met
ric
Func
tion
s
© P
roje
ct M
aths
Dev
elop
men
t Tea
m 2
009
w
ww
.pro
ject
mat
hs.ie
17
Stud
ent A
ctiv
ity
5
By fi
lling
in a
tabl
e fir
st, a
nd u
sing
the
sam
e ax
es b
ut d
iffer
ent c
olou
rs fo
r eac
h gr
aph,
dra
w th
e gr
aphs
of
x/º
015
3045
6075
9010
512
013
515
016
518
019
521
022
524
025
527
028
530
031
533
034
536
0
cos x
2x cos 2
x3x co
s 3x
Peri
odRa
nge
y =
cos
xy
= c
os 2
xy
= c
os 3
xy
= c
os b
x
In
th
e g
rap
h o
f y
= ac
os b
x, w
hat
is t
he
effe
ct o
n t
he
gra
ph
of
vary
ing
b in
cos
bx?
____
____
____
____
____
____
____
____
____
____
____
___
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
_
___
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
_
Teac
hing
& L
earn
ing
Plan
10:
Tri
gono
met
ric
Func
tion
s
© P
roje
ct M
aths
Dev
elop
men
t Tea
m 2
009
w
ww
.pro
ject
mat
hs.ie
18
Stud
ent A
ctiv
ity
6
Sket
ch e
ach
of
the
follo
win
g g
rap
hs:
(0°
≤ x
≤ 36
0°)
y
= 4
sin
x
Pe
rio
d =
____
____
____
____
_ R
ang
e =
____
____
____
____
_
Sket
ch e
ach
of
the
follo
win
g g
rap
hs:
(0°
≤ x
≤ 36
0°)
y
= c
os4
x
Pe
rio
d =
____
____
____
____
_ R
ang
e =
____
____
____
____
_
Sket
ch e
ach
of
the
follo
win
g g
rap
hs:
(0°
≤ x
≤ 36
0°)
y
= 2
sin
3x
Per
iod
=__
____
____
____
___
Ran
ge
=__
____
____
____
___
Teac
hing
& L
earn
ing
Plan
10:
Tri
gono
met
ric
Func
tion
s
© P
roje
ct M
aths
Dev
elop
men
t Tea
m 2
009
w
ww
.pro
ject
mat
hs.ie
19
Stud
ent A
ctiv
ity
7
Stu
den
t A
ctiv
ity 7
A
|∠AO
A’|
|∠AO
A’’|
|∠AO
A’’’|
|∠AO
A’’’’
||∠
AOA’
’’’’|
30.0
0°60
.00°
75.0
0°80
.00°
85.0
0°
y=ta
n x
Usi
ng
th
e d
iag
ram
of
the
un
it c
ircl
e, r
ead
th
e ap
pro
xim
ate
valu
e o
f th
e ta
n o
f th
e an
gle
s in
the
tab
le u
sin
g t
he
trig
on
om
etri
c ra
tio
s.
Ang
le θ
/º0
3060
7580
85ta
n θ
Teac
hing
& L
earn
ing
Plan
10:
Tri
gono
met
ric
Func
tion
s
© P
roje
ct M
aths
Dev
elop
men
t Tea
m 2
009
w
ww
.pro
ject
mat
hs.ie
20
Stud
ent A
ctiv
ity
7
Stu
den
t A
ctiv
ity 7
B
By fi
lling
in a
tabl
e fir
st, a
nd u
sing
the
sam
e ax
es b
ut d
iffer
ent c
olou
rs fo
r eac
h gr
aph,
dra
w th
e gr
aphs
of
x/º
-90
-75
-60
-45
-30
030
4560
7590
105
120
135
150
180
210
225
240
255
270
285
300
330
360
tan
x
Stu
den
t A
ctiv
ity 7
C
y =
tan
xPe
riod
Rang
e
