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Teaching & Learning PlansPlan 9: The Unit Circle
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 9: The Unit Circle
AimsTo enable students to become familiar with the unit circle•
To use the unit circle to evaluate the trigonometric functions sin, cos and •tan for all angles
Prior Knowledge Students should be able to plot and read coordinates on a Cartesian plane. They should recognise that the circle, as the locus of all points equidistant from a given point, the centre, is uniquely defined by its centre and radius. Students should be familiar with the concept of a function as a one to one or many to one mapping, and with the domain and range of a function. Students should recall the theorem of Pythagoras and be able to calculate sin x, cos x, and tan x from the right-angled triangle, i.e. the trigonometric ratios (SOHCAHTOA) and know that
Learning OutcomesAs a result of studying this topic, students will be able to
associate the coordinates of points on the circumference of the unit circle •with the cos and sin of the angle made by the radius containing these points, with the positive direction of the x-axis
use the reference angle to calculate the sin, cos and tan of any angle • θ, where
find values of sin, cos and tan of negative angles and of angles >360° from •the unit circle
solve equations of the type •
Teaching & Learning Plan 9: The Unit Circle
© Project Maths Development Team 2009 www.projectmaths.ie 2
Relationship to Leaving Certificate Syllabus
Sub-topics Ordinary Level Higher Level2.2 Coordinate
Geometry Recognise that x2+y2= r2 represents the relationship between the x and y co-ordinates of points on a circle centre (0,0) and radius r.
2.3 Trigonometry Define sin x and cos x for all values of x.Define tan x.
Solve trigonometric equations such as a sin nθ=0 and cos nθ=½, giving all solutions.
Resources RequiredCompasses, protractors, clear rulers, pencils, formulae and tables booklet, Geogebra, Autograph, Perspex Model of Unit Circle (last 3 desirable but not essential).
Introducing the TopicTrigonometric functions are very important in science and engineering, architecture and even medicine. Surveying is one of the many applications. Bridge builders and road makers all use trigonometry in their daily work.
The mathematics of sine and cosine functions describes how many physical systems behave. As a playground Ferris wheel rotates, the height above a central horizontal line can be modelled by a sine function. The displacement of the prongs of a vibrating tuning fork from the rest position is described by a sine function. The resulting displacement with time of a weight attached to a spring when it is displaced slightly from its rest position is described by a sine function. Tidal movements and sound waves are two other vibrations described by sine functions. Anything that has a regular cycle (like the tides, temperatures, rotation of the earth, etc) can be modelled using a sine or cosine function. The piston engine is the most commonly used engine in the world. Its motion can be described using a sine curve. http://www.intmath.com/Trigonometric-graphs/2_Graphs-sine-cosine-period.php (Scroll to the bottom of the web page to get the piston applet.)
When students have drawn the graph of sin x, it would then be useful to show the applet on the piston as it may not make much sense initially. Alternatively, show it initially and again after Student Activity 7 when it should make more sense.
The fundamental background of trigonometry finds usage in an area which is a passion for many people i.e. music. Sound travels in waves and in developing computer generated music, combinations of sine and cosine functions are used to generate the sounds of different musical instruments. Hence sound engineers and technologists who
Teaching & Learning Plan 9: The Unit Circle
© Project Maths Development Team 2009 www.projectmaths.ie 3
research advances in computer music and hi-tech music composers have to relate to the basic laws of trigonometry.
Techniques in trigonometry are used in navigation particularly satellite systems and astronomy, naval and aviation industries, oceanography, land surveying, and in cartography (creation of maps).
Domestic a.c. is supplied at 230 V, 50 Hz. The period is 0.02 seconds, and the range is approximately [-325 V, 325 V].
Teac
hing
and
Lea
rnin
g Pl
an 9
: The
Uni
t Ci
rcle
© 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
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
Wh
at 2
item
s o
f »
info
rmat
ion
do
we
nee
d t
o
defi
ne
a ci
rcle
?
Cen
tre
and
rad
ius
•D
istr
ibu
te
»St
ud
ent
Act
ivit
y 1.
(U
nit
Cir
cle)
Giv
en a
un
it c
ircl
e, w
hat
»
dis
tin
gu
ish
es t
he
un
it c
ircl
e fr
om
all
oth
er c
ircl
es?
No
te t
hat
th
e ra
diu
s is
»
1 u
nit
; wat
ch o
ut
for
a re
aso
n w
hy
this
mig
ht
be
use
ful.
It h
as a
rad
ius
of
1 an
d a
•
cen
tre
(0, 0
), a
nd
is d
raw
n
on
a C
arte
sian
pla
ne.
Iden
tify
th
e 4
qu
adra
nts
. »
Wh
at d
irec
tio
n d
o w
e m
ove
in g
oin
g f
rom
th
e fi
rst
to t
he
fou
rth
q
uad
ran
t?
An
ticl
ock
wis
e.•
Ho
w w
ou
ld y
ou
»
des
crib
e p
oin
ts o
n t
he
circ
um
fere
nce
of
the
circ
le?
Poin
ts o
n t
he
circ
um
fere
nce
•
can
be
des
crib
ed b
y an
o
rder
ed p
air
( x, y
)
Enco
ura
ge
use
of
corr
ect
•te
rmin
olo
gy.
Rea
d t
he
Car
tesi
an
»co
ord
inat
es o
f p
oin
ts in
ea
ch q
uad
ran
t. N
ote
th
e si
gn
s o
f th
e co
ord
inat
es in
ea
ch q
uad
ran
t, a
nd
fill
in
Stu
den
t A
ctiv
ity
1A.
Stu
den
ts fi
ll in
th
e •
coo
rdin
ates
an
d n
ote
th
e si
gn
s in
th
e d
iffe
ren
t q
uad
ran
ts.
No
te: E
mph
asis
e to
stu
dent
s th
at
it is
impo
rtan
t to
not
e th
e si
gn
of x
and
y in
eac
h qu
adra
nt f
or
futu
re r
efer
ence
in t
he le
sson
.
Teac
hing
and
Lea
rnin
g Pl
an 9
: The
Uni
t Ci
rcle
© 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: T
each
er In
put
Stud
ent A
ctiv
itie
s:
Poss
ible
and
Exp
ecte
d Re
spon
ses
Teac
her’s
Sup
port
and
A
ctio
nsCh
ecki
ng U
nder
stan
ding
An
an
gle
in s
tan
dar
d p
osi
tio
n h
as
»it
s ve
rtex
at
the
ori
gin
, th
e p
osi
tive
d
irec
tio
n o
f th
e x-
axis
as
the
init
ial
ray
and
th
e o
ther
ray
fo
rmin
g
the
ang
le w
hic
h in
ters
ects
th
e ci
rcu
mfe
ren
ce o
f th
e ci
rcle
is t
he
term
inal
ray
.
Wh
en t
he
term
inal
ray
is r
ota
ted
in
»an
an
ticl
ock
wis
e d
irec
tio
n f
rom
th
e in
itia
l ray
a p
osi
tive
an
gle
is f
orm
ed.
On
th
e U
nit
Cir
cle
»St
ud
ent
Act
ivit
y 1A
, usi
ng
a r
ule
r an
d p
rotr
acto
r, d
raw
an
an
gle
of
30°
in s
tan
dar
d
po
siti
on
, wit
h t
he
term
inal
ray
lo
ng
er t
hat
th
e ra
diu
s. M
ark
the
po
int
Q w
her
e th
e te
rmin
al r
ay
inte
rsec
ts t
he
circ
um
fere
nce
.
Stu
den
ts d
raw
an
an
gle
»
of
30°
on
th
e u
nit
cir
cle.
Dem
on
stra
te o
n t
he
»b
oar
d if
stu
den
ts h
ave
dif
ficu
lty
wit
h t
he
term
ino
log
y. Im
po
rtan
t to
get
use
d t
o it
as
it
sim
plifi
es d
escr
ipti
on
s la
ter
on
.
Ho
w w
ou
ld y
ou
dra
w a
n a
ng
le o
f »
-30°
?B
y ro
tati
ng
th
e te
rmin
al
•ra
y b
y 30
° in
a c
lock
wis
e d
irec
tio
n.
An
swer
th
e q
ues
tio
ns
on
»
Stu
den
t A
ctiv
ity
2A.
Dis
trib
ute
»
Stu
den
t A
ctiv
ity
2.C
an s
tud
ents
ap
ply
»
trig
on
om
etri
c ra
tio
s?
Teac
hing
and
Lea
rnin
g Pl
an 9
: The
Uni
t Ci
rcle
© 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
A
ctio
nsCh
ecki
ng
Und
erst
andi
ngW
hat
hav
e yo
u d
isco
vere
d
»ab
ou
t th
e x
and
y
coo
rdin
ates
fo
r an
an
gle
of
30°
on
th
e u
nit
cir
cle?
The
•x
and
y c
oo
rdin
ates
rep
rese
nt
cos
30°
and
sin
30°
.
Wal
k ar
ou
nd
an
d c
hec
k »
that
Stu
den
t A
ctiv
ity
2A
is b
ein
g fi
lled
. Wh
en a
ll th
e sh
eets
are
fille
d a
sk
ind
ivid
ual
stu
den
ts f
or
thei
r an
swer
s.
Wh
at is
th
e si
gn
ifica
nce
»
no
w o
f th
e ra
diu
s o
f th
e ci
rcle
bei
ng
1?
The
sin
30°
=o
pp
/hyp
an
d
•co
s 30
° =
ad
j/hyp
hyp
ote
nu
se =
rad
ius,
an
d,
•⏐
r⏐=1
, si
n 3
0°=
op
p =
y a
nd
co
s 30
° =
ad
j = x
Can
yo
u g
ener
alis
e th
is f
or
»an
y an
gle
θ<
90°?
Co
mp
lete
»
Stu
den
t A
ctiv
ity
2B.
Stu
den
ts c
om
ple
te
»St
ud
ent
Act
ivit
y 2B
fo
r an
y an
gle
θ in
th
e fi
rst
qu
adra
nt
and
sh
ow
th
at
sin
θ =
y/1,
co
s θ=
x/1,
tan
θ =
y/x
The
coo
rdin
ates
of
any
»p
oin
t o
n t
he
un
it c
ircl
e m
ay b
e w
ritt
en a
s (x
, y)
the
Car
tesi
an c
oo
rdin
ates
, or
as
(co
s θ,
sin
θ).
Do
stu
den
ts
»u
nd
erst
and
th
at t
he
x an
d y
co
ord
inat
es o
f p
oin
ts o
n t
he
un
it
circ
le a
re e
qu
al
to t
he
cos
and
si
n o
f th
e an
gle
θ
mad
e b
y th
e ra
diu
s co
nta
inin
g t
hes
e p
oin
ts, w
ith
th
e p
osi
tive
dir
ecti
on
o
f th
e x-
axis
?
Teac
hing
and
Lea
rnin
g Pl
an 9
: The
Uni
t Ci
rcle
© 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
Act
ions
Chec
king
Und
erst
andi
ng
Wh
at a
re t
he
sig
ns
»fo
r si
n, c
os
and
tan
of
an a
ng
le in
th
e fi
rst
qu
adra
nt
and
wh
y? F
ill
in S
tud
ent
Act
ivit
y 2C
.
All
po
siti
ve, s
ince
th
e •
x an
d y
co
ord
inat
es a
re
bo
th p
osi
tive
in t
he
firs
t q
uad
ran
t.
Ask
th
e cl
ass
and
th
en a
n in
div
idu
al
»to
exp
lain
. Ask
th
e cl
ass
if t
hey
ag
ree
wit
h t
he
answ
ers
and
if n
ot
to e
xpla
in w
hy
no
t.
Do
stu
den
ts k
no
w t
hat
»
sin
, co
s an
d t
an a
re
po
siti
ve f
or
ang
les
in t
he
firs
t q
uad
ran
t?
Usi
ng
th
e u
nit
cir
cle,
»
ho
w w
ou
ld y
ou
get
th
e co
s 60
° an
d s
in 6
0°?
Ch
eck
usi
ng
a c
alcu
lato
r.
Stu
den
ts m
ake
an a
ng
le
»o
f 60
° in
sta
nd
ard
po
siti
on
an
d r
ead
th
e co
ord
inat
es.
cos
60°=
x a
nd
sin
60°
= y
Dis
trib
ute
»
Ap
pen
dix
A (
Un
it C
ircl
e).
Did
stu
den
ts u
se t
rig
. »
rati
os
firs
t, o
r w
ere
they
n
ow
co
nfi
den
t in
usi
ng
(x
, y)
for
(co
s A,
sin
A)?
Wh
at a
re t
he
sin
, co
s »
and
tan
of
0° a
nd
90°
fr
om
wh
at y
ou
hav
e le
arn
ed?
Fill
in S
tud
ent
Act
ivit
y 2D
.
Stu
den
ts r
ead
th
e »
coo
rdin
ates
on
th
e ci
rcu
mfe
ren
ce f
or
0° a
nd
90
° fr
om t
he U
nit
Circ
le a
nd
fi
ll in
th
e ta
ble
on
Stu
den
t A
ctiv
ity
2D.
Wh
at d
id y
ou
no
tice
»
abo
ut
tan
90°
?It
’s u
nd
efin
ed, d
ue
to
•d
ivis
ion
by
zero
.
Usi
ng
th
e ca
lcu
lato
r, »
fin
d t
he
tan
89°
, tan
89
.999
°, t
an 8
9.99
999°
. W
hat
do
yo
u n
oti
ce?
Tan
incr
ease
s ve
ry r
apid
ly
•as
th
e an
gle
ten
ds
to 9
0°.
Wo
rkin
g in
pai
rs
»w
rite
in s
ho
rt p
oin
ts a
su
mm
ary
of
wh
at y
ou
h
ave
lear
ned
.
Ask
dif
fere
nt
stu
den
ts t
o r
ead
ou
t »
po
ints
mad
e in
th
eir
sum
mar
ies
- co
nce
pt
of
the
un
it c
ircl
e; a
ng
le in
st
and
ard
po
siti
on
; in
itia
l ray
an
d
term
inal
ray
; fo
rmin
g a
rig
ht
ang
led
tr
ian
gle
, in
th
e fi
rst
qu
adra
nt
wit
h
the
ori
gin
as
a ve
rtex
.
Wri
te t
he
po
ints
on
th
e b
oar
d.
»
Dis
trib
ute
»
Stu
den
t A
ctiv
ity
3.
Wer
e st
ud
ents
ab
le t
o
»su
mm
aris
e w
hat
th
ey
had
do
ne?
Teac
hing
and
Lea
rnin
g Pl
an 9
: The
Uni
t Ci
rcle
© 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: 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 U
nder
stan
ding
Ho
w d
o w
e d
efin
e si
n a
nd
co
s »
for
ang
les
bet
wee
n 9
0° a
nd
18
0° a
s w
e d
on
’t f
orm
rig
ht
ang
led
tri
ang
les
usi
ng
th
ese
ang
les?
Use
th
e u
nit
cir
cle
•co
ord
inat
es.
Fill
in
»St
ud
ent
Act
ivit
y 3A
.
Usi
ng
th
e u
nit
cir
cle,
fin
d t
he
»si
n 1
50°,
co
s 15
0° a
nd
tan
15
0°. C
hec
k an
swer
s u
sin
g a
ca
lcu
lato
r
Stu
den
ts r
ead
sin
150
»°
and
co
s 15
0° f
rom
th
e u
nit
ci
rcle
. Co
nfi
rm u
sin
g t
he
calc
ula
tor.
Co
mp
are
the
resu
lts
wit
h s
in 3
0° a
nd
co
s 30
°.
Ch
eck
answ
ers.
Ask
»
ind
ivid
ual
s.D
id e
very
on
e g
et t
he
»sa
me
answ
ers
for
sin
15
0°, c
os
150°
an
d
tan
150
° fr
om
th
e co
ord
inat
es a
s fr
om
th
e ca
lcu
lato
r?
Can
yo
u f
orm
a r
igh
t an
gle
d
»tr
ian
gle
in t
he
seco
nd
qu
adra
nt
wit
h a
n a
ng
le a
t th
e o
rig
in
wit
h t
he
sam
e va
lues
of
sin
an
d
cos
as 1
50°?
Dro
p a
per
pen
dic
ula
r fr
om
•
Q’ t
o t
he
x-ax
is f
orm
ing
an
ac
ute
an
gle
A a
t th
e o
rig
in
in t
he
seco
nd
qu
adra
nt.
(S
tud
ent
Act
ivit
y 3B
)
Fill
in
»St
ud
ent
Act
ivit
y 3B
.St
ud
ents
fill
in
»St
ud
ent
Act
ivit
y 3B
.C
hec
k va
lues
stu
den
ts
»ar
e fi
llin
g in
on
Stu
den
t A
ctiv
ity
3B.
Fin
d t
he
sin
»
A an
d c
os
A o
f th
e an
gle
A (
no
t in
sta
nd
ard
p
osi
tio
n)
usi
ng
tri
g r
atio
s.
Wh
at d
o y
ou
no
tice
?
An
gle
•
A h
as t
he
sam
e si
n
and
co
s as
150
°.
Ho
w d
o y
ou
cal
cula
te t
he
size
»
of
A?A
•=
180
° –
150°
= 3
0°
Teac
hing
and
Lea
rnin
g Pl
an 9
: The
Uni
t Ci
rcle
© 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
A
ctio
nsCh
ecki
ng U
nder
stan
ding
An
gle
»
A is
cal
led
th
e re
fere
nce
an
gle
. Can
yo
u
des
crib
e it
?Fi
ll in
»
Stu
den
t A
ctiv
ity
3C.
It is
th
e ac
ute
an
gle
•
bet
wee
n t
he
term
inal
ray
o
f an
an
gle
in s
tan
dar
d
po
siti
on
an
d t
he
x-ax
is.
Wh
at t
ran
sfo
rmat
ion
»
cou
ld y
ou
use
to
fo
rm
a co
ng
ruen
t tr
ian
gle
to
O
B’Q
’ in
th
e fi
rst
qu
adra
nt?
Axi
al s
ymm
etry
in t
he
•y-
axis
giv
es c
on
gru
ent
tria
ng
le O
BQ
, wit
h a
ng
le A
at
th
e ve
rtex
.
Hen
ce w
hat
is t
he
»re
lati
on
ship
bet
wee
n t
he
rati
o o
f si
des
in t
rian
gle
O
B’Q
’ an
d t
he
rati
o o
f th
e si
des
of
its
imag
e in
th
e y-
axi
s ?
They
are
nu
mer
ical
ly e
qu
al.
•
Wh
at is
th
e re
lati
on
ship
»
bet
wee
n t
he
trig
rat
ios
for
tria
ng
le O
B’Q
’ an
d t
rian
gle
O
BQ
?
They
are
nu
mer
ical
ly e
qu
al.
•
Wh
at is
th
e sa
me
and
wh
at
»is
dif
fere
nt
abo
ut
sin
30°
an
d c
os
30°
and
si
n 1
50°
and
co
s 15
0°?
sin
150
•°
= s
in 3
0° a
nd
co
s 15
0° =
– c
os
30°
Are
stu
den
ts a
ble
to
»
calc
ula
te t
he
valu
e o
f
sin
150
° an
d c
os
150°
usi
ng
th
e re
fere
nce
an
gle
an
d
the
app
rop
riat
e si
gn
s?
Follo
w t
his
lin
e o
f »
reas
on
ing
fo
r an
y an
gle
in
th
e se
con
d q
uad
ran
t,
com
ple
te S
tud
ent
Act
ivit
y 3D
.
Stu
den
ts fi
ll in
»
Stu
den
t A
ctiv
ity
3D.
Cir
cula
te a
nd
ch
eck
»St
ud
ent
Act
ivit
y 3D
as
it is
bei
ng
fi
lled
. Wh
en fi
lled
, ask
d
iffe
ren
t st
ud
ents
to
rea
d
ou
t th
e an
swer
s an
d a
sk
clas
s if
th
ey a
gre
e an
d if
no
t to
just
ify
thei
r an
swer
.
Are
stu
den
ts a
ble
to
»
calc
ula
te t
he
sin
, co
s an
d
tan
of
any
ang
le in
th
e se
con
d q
uad
ran
t u
sin
g
the
refe
ren
ce a
ng
le a
nd
co
rrec
t si
gn
s?
Teac
hing
and
Lea
rnin
g Pl
an 9
: The
Uni
t Ci
rcle
© 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:
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
Fill
in
»St
ud
ent
Act
ivit
y 4.
Stu
den
ts fi
ll in
»
Stu
den
t A
ctiv
ity
4.
Stu
den
ts s
ho
uld
avo
id
»er
rors
in c
alcu
lati
ng
th
e re
fere
nce
an
gle
if t
hey
re
fer
to t
he
dra
win
g o
f th
e u
nit
cir
cle
rath
er t
han
just
m
emo
risi
ng
a f
orm
ula
.
Dis
trib
ute
»
Stu
den
t A
ctiv
ity
4.
Are
stu
den
ts a
ble
to
»
calc
ula
te t
he
sin
, co
s an
d
tan
of
any
ang
le in
th
e th
ird
qu
adra
nt
usi
ng
th
e re
fere
nce
an
gle
an
d c
orr
ect
sig
ns?
Fill
in
»St
ud
ent
Act
ivit
y 5.
Stu
den
ts fi
ll in
»
Stu
den
t A
ctiv
ity
5.R
emin
d s
tud
ents
to
alw
ays
»d
raw
a s
mal
l dia
gra
m o
f th
e u
nit
cir
cle
wit
h t
he
sum
mar
y in
form
atio
n
wh
en d
ealin
g w
ith
th
e tr
ig.
fun
ctio
ns.
Are
stu
den
ts a
ble
to
»
calc
ula
te t
he
sin
, co
s an
d
tan
of
any
ang
le in
th
e fo
urt
h q
uad
ran
t u
sin
g t
he
refe
ren
ce a
ng
le a
nd
co
rrec
t si
gn
s?
htt
p://
ww
w.w
ou
. »
edu
/~b
urt
on
l/tri
g.h
tml
htt
p://
ww
w.m
ath
sisf
un
.co
m/g
eom
etry
/un
it-c
ircl
e.h
tml (
Scro
ll to
th
e b
ott
om
o
f th
e w
eb p
age
to s
ee t
he
app
let
on
ref
eren
ce a
ng
le)
Sho
w s
tud
ents
Jav
a »
app
lets
. D
o J
ava
app
lets
co
ntr
ibu
te
»to
un
der
stan
din
g?
Fill
in t
he
sum
mar
y o
n
»St
ud
ent
Act
ivit
y 6A
.St
ud
ents
fill
in
•St
ud
ent
Act
ivit
y 6A
.D
istr
ibu
te
»St
ud
ent
Act
ivit
y 6.
Fin
d t
he
sin
( »
–210
°). W
e o
nly
dea
lt w
ith
po
siti
ve
valu
es f
or
ang
les
bef
ore
. W
hat
is t
he
dif
fere
nce
b
etw
een
po
siti
ve a
nd
n
egat
ive
ang
les?
Posi
tive
an
gle
s re
pre
sen
t •
anti
clo
ckw
ise
rota
tio
ns
and
neg
ativ
e an
gle
s ar
e cl
ock
wis
e ro
tati
on
s.
Teac
hing
and
Lea
rnin
g Pl
an 9
: The
Uni
t Ci
rcle
© 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: T
each
er 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
U
nder
stan
ding
Can
yo
u s
ug
ges
t a
stra
teg
y fo
r »
dea
ling
wit
h e
valu
atin
g t
he
trig
fu
nct
ion
s fo
r n
egat
ive
ang
les?
In w
hic
h q
uad
ran
t is
»
– 21
0°?
Wh
at is
th
e si
gn
of
sin
in t
he
seco
nd
»
qu
adra
nt?
Wh
at p
osi
tive
an
gle
is e
qu
al t
o
»–
210°
?
Wh
at is
th
e re
fere
nce
an
gle
fo
r »
– 15
0°?
Wh
at is
th
e si
n (
»–
210°
) eq
ual
to
?
Dra
w t
he
un
it c
ircl
e.
• Se
con
d q
uad
ran
t •
Posi
tive
•
360
•°
– 21
0° =
150
°
30•
°
∴•
sin
(–
210)
° =
+ s
in 3
0°.
Ask
stu
den
ts t
o c
om
e »
up
wit
h t
he
stra
teg
y th
emse
lves
firs
t.
Hav
ing
giv
en t
hem
»
a fe
w m
inu
tes,
ask
st
ud
ents
fo
r th
e st
eps
and
sh
ow
eac
h s
tep
o
n t
he
bo
ard
.
Can
stu
den
ts c
om
e »
up
wit
h t
he
stra
teg
y b
y th
emse
lves
, giv
en
that
th
ey a
re u
sed
to
dra
win
g t
he
un
it
circ
le?
Can
stu
den
ts c
alcu
late
»
sin
, co
s an
d t
an f
or
neg
ativ
e an
gle
s?
Do
th
e ex
erci
ses
on
» S
tud
ent
Act
ivit
y 6B
on
neg
ativ
e an
gle
s.
Can
yo
u s
ug
ges
t a
stra
teg
y fo
r fi
nd
ing
»
trig
fu
nct
ion
s o
f an
gle
s g
reat
er t
han
36
0°?
For
exam
ple
co
s 91
0°?
Do
th
e ex
erci
se in
»
Stu
den
t A
ctiv
ity
6C
on
an
gle
s g
reat
er t
han
360
°.
Dra
w t
he
un
it c
ircl
e.
•
Each
ro
tati
on
of
360
•°
is
equ
ival
ent
to a
n a
ng
le o
f 0°
.
910
•°=
2(3
60°)
+ 1
90°
=
190°
, wh
ich
is in
th
e th
ird
q
uad
ran
t w
her
e co
s is
n
egat
ive
and
th
e re
fere
nce
an
gle
190
•°
– 18
0°=
10°
.
cos
910
•°
= –
co
s 10
°.
Ask
stu
den
ts t
o c
om
e »
up
wit
h t
he
stra
teg
y th
emse
lves
firs
t.
Giv
en t
he
stu
den
ts
»a
few
min
ute
s th
en
asks
stu
den
ts f
or
the
step
s an
d s
ho
w e
ach
st
ep o
n t
he
bo
ard
.
Can
stu
den
ts c
alcu
late
»
sin
, co
s an
d t
an f
or
ang
les
gre
ater
th
an
360°
?
Teac
hing
and
Lea
rnin
g Pl
an 9
: The
Uni
t Ci
rcle
© 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 U
nder
stan
ding
In w
hic
h q
uad
ran
ts c
an
»si
n θ
be
po
siti
ve?
In w
hic
h q
uad
ran
ts c
an
»si
n θ
be
neg
ativ
e?
In w
hic
h q
uad
ran
ts c
an
»co
s θ
be
po
siti
ve?
In w
hic
h q
uad
ran
ts c
an
»co
s θ
be
neg
ativ
e?
The
pre
vio
us
acti
viti
es
»co
nce
ntr
ated
on
fin
din
g
the
refe
ren
ce a
ng
le A
, fo
r a
giv
en a
ng
le θ
in e
ach
of
the
4 q
uad
ran
ts. W
e w
ill
no
w t
ry t
o fi
nd
θ, k
no
win
g
the
valu
e o
f A.
Firs
t an
d s
eco
nd
•
Thir
d a
nd
fo
urt
h
• Fi
rst
and
fo
urt
h
• Se
con
d a
nd
th
ird
•
Ref
er s
tud
ents
bac
k to
th
e »
sum
mar
y d
iag
ram
, Stu
den
t A
ctiv
ity
6A.
Ask
ind
ivid
ual
stu
den
ts.
»
Are
stu
den
ts c
lear
•
on
th
e si
gn
s o
f th
e tr
igo
no
met
ric
fun
ctio
ns
in t
he
dif
fere
nt
qu
adra
nts
?
Giv
en a
ref
eren
ce a
ng
le
»A,
ho
w m
any
valu
es o
f θ
cou
ld t
his
ref
eren
ce a
ng
le
hav
e?
4•
Dis
trib
ute
»
Stu
den
t A
ctiv
ity
7.
Ask
an
ind
ivid
ual
stu
den
t »
and
ask
th
e w
ho
le c
lass
if
they
ag
ree.
Ref
er t
o t
he
dia
gra
ms
on
»
Stu
den
t A
ctiv
ity
7A.
Stu
den
ts fi
ll in
th
e fo
rmu
lae
•in
Stu
den
t A
ctiv
ity
7A.
Wal
kin
g a
rou
nd
. Ch
eck
»ea
ch s
tud
ent’
s w
ork
an
d
then
ask
ind
ivid
ual
s to
cal
l o
ut
his
/her
an
swer
s fo
r cl
ass
agre
emen
t/d
isag
reem
ent
wit
h ju
stifi
cati
on
.
Teac
hing
and
Lea
rnin
g Pl
an 9
: The
Uni
t Ci
rcle
© 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
13
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
an
d A
ctio
nsCh
ecki
ng
Und
erst
andi
ngW
e ar
e n
ow
go
ing
to
so
lve
the
»eq
uat
ion
in S
tud
ent
Act
ivit
y 7B
u
sin
g t
he
sum
mar
y o
f th
e u
nit
ci
rcle
an
d f
orm
ula
e o
n S
tud
ent
Act
ivit
y 7A
.
Stu
den
ts w
ork
in p
airs
, dis
cuss
ing
•
the
pro
ble
m.
Wh
at a
re w
e tr
yin
g t
o fi
nd
ou
t »
her
e?
We
are
tryi
ng
to
fin
d a
ll th
e •
ang
les
θ w
her
e
Teac
her
ref
ers
»st
ud
ents
to
th
eir
sum
mar
y o
n
Stu
den
t A
ctiv
ity
6A.
Hav
e st
ud
ents
»
bee
n a
ble
to
wri
te
θ in
ter
ms
of
A in
St
ud
ent
Act
ivit
y 7B
?
Ho
w d
o y
ou
kn
ow
wh
ich
qu
adra
nts
»
the
ang
le θ
co
uld
bel
on
g t
o?
Sin
is p
osi
tive
, th
eref
ore
•
θ co
uld
b
e in
th
e fi
rst
or
the
seco
nd
q
uad
ran
t.
Ref
er s
tud
ents
bac
k »
to S
tud
ent
Act
ivit
y 6A
.
Ho
w w
ill y
ou
fin
d t
he
refe
ren
ce
»an
gle
A?
The
refe
ren
ce a
ng
le is
th
e ac
ute
•
ang
le w
ho
se s
in is
i
.e. 4
5°
Wh
en s
tud
ents
»
get
a p
rob
lem
, are
th
ey fi
rst
clar
ifyi
ng
w
hat
th
e p
rob
lem
is
aski
ng
?
Co
uld
th
e re
fere
nce
an
gle
hav
e a
»n
egat
ive
trig
on
om
etri
c fu
nct
ion
as
soci
ated
wit
h it
? Ex
pla
in.
Ref
eren
ce a
ng
le c
ann
ot
hav
e a
•n
egat
ive
trig
fu
nct
ion
ass
oci
ated
w
ith
it a
s it
is a
lway
s an
acu
te
ang
le.
Are
stu
den
ts u
sin
g
»th
e u
nit
cir
cle
effe
ctiv
ely
to s
olv
e th
is p
rob
lem
?
Kn
ow
ing
th
e re
fere
nce
an
gle
, ho
w
»d
o y
ou
fin
d t
he
ang
le θ
?U
se t
he
form
ula
e fo
r ea
ch
•q
uad
ran
t.
Firs
t q
uad
ran
t:
θ
= A
= 4
5°,
Seco
nd
qu
adra
nt:
180
° –
θ =
A
∴
θ =
180
° –
A =
135
°
Solv
e th
e ab
ove
eq
uat
ion
i.e.
»
, f
or
all v
alu
es o
f θ.
U
se w
hat
yo
u le
arn
ed a
bo
ut
trig
fu
nct
ion
s o
f an
gle
s >
360
°.
∴
•θ
= 45
° +
n360
° or
θ
= 1
35° +
n36
0°, n
∈ Z
Are
stu
den
ts u
sin
g
»w
hat
th
ey h
ave
lear
ned
pre
vio
usl
y ab
ou
t an
gle
s g
reat
er
than
360
°?
Teac
hing
and
Lea
rnin
g Pl
an 9
: The
Uni
t Ci
rcle
© 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
14
Stud
ent
Lear
ning
Ta
sks:
Tea
cher
In
put
Stud
ent A
ctiv
itie
s: P
ossi
ble
and
Expe
cted
Res
pons
esTe
ache
r’s S
uppo
rt
and
Act
ions
Chec
king
U
nder
stan
ding
Refl
ecti
on
Wri
te d
ow
n
»3
thin
gs
you
le
arn
ed a
bo
ut
trig
on
om
etry
to
day
.
Wri
te d
ow
n
»an
yth
ing
yo
u
fou
nd
dif
ficu
lt.
Wri
te d
ow
n a
ny
»q
ues
tio
ns
you
m
ay h
ave.
Co
nce
pt
of
the
Un
it C
ircl
e 1.
An
gle
in s
tan
dar
d p
osi
tio
n
2.
Init
ial r
ay a
nd
ter
min
al r
ay
3.
Form
ing
a r
igh
t an
gle
d t
rian
gle
in t
he
firs
t q
uad
ran
t 4.
w
ith
th
e o
rig
in a
s a
vert
ex
Usi
ng
tri
g r
atio
s, a
nd
dis
cove
rin
g t
hat
th
e 5.
x
and
y
coo
rdin
ates
rep
rese
nt
the
cos
and
sin
of
an a
ng
le
cos
150
6.
° an
d s
in 1
50°
can
be
go
t fr
om
th
e x
and
y
coo
rdin
ates
of
wh
ere
the
term
inal
ray
of
the
ang
le
mee
ts t
he
circ
um
fere
nce
of
the
circ
le.
A r
igh
t an
gle
d t
rian
gle
wit
h t
he
term
inal
ray
as
the
7.
hyp
ote
nu
se in
th
e se
con
d q
uad
ran
t g
ives
an
gle
A a
t th
e o
rig
in in
th
e ri
gh
t an
gle
d t
rian
gle
, wh
ich
has
th
e sa
me
cos
and
sin
as
150°
.
150
8.
° in
th
e se
con
d q
uad
ran
t h
as a
ref
eren
ce a
ng
le
in t
he
firs
t q
uad
ran
t w
ith
th
e sa
me
mag
nit
ud
e o
f si
n
and
co
s b
ut
wit
h a
dif
fere
nt
sig
n f
or
cos.
The
sam
e lin
e o
f re
aso
nin
g f
or
ang
les
in t
he
thir
d a
nd
9.
fo
urt
h q
uad
ran
ts.
Solv
ing
tri
g e
qu
atio
ns
for
exam
ple
10
.
.
Cir
cula
te a
nd
tak
e »
no
te p
arti
cula
rly
of
any
qu
esti
on
s st
ud
ents
hav
e an
d h
elp
th
em t
o
answ
er t
hem
.
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
w
ith
eac
h o
ther
u
sin
g t
hes
e te
rms?
Teaching and Learning Plan 9: The Unit Circle
© Project Maths Development Team 2009 www.projectmaths.ie 15
Student Activity 1
Student Activity 1A
The unit circle – A circle whose centre is at (0,0) and whose radius is 1
Any point on the circumference of the circle can be described by an ordered pair (x,y).
The coordinates of B are (0.6, 0.8)
What are the coordinates of C, D, and E?
C = _________________, D = _________________, E = _________________.
In which quadrant are both x and y positive? _______________________________________
In which quadrant is x negative and y positive?_____________________________________
In which quadrant is x positive and y negative?_____________________________________
In which quadrant is x negative and y negative? ____________________________________
Angles in standard position - the vertex is at the origin, with the initial ray as the
positive direction of the x-axis, and the other ray forming the angle is the terminal ray.
Student Activity 1B
Angles in the first quadrant
Draw an angle of 30° in standard position on the unit circle on Student Activity Sheet
1A. Mark the initial ray and the terminal ray. Label the point where the terminal ray
meets the circumference as θ.
The coordinates of θ are _________________________________________________________ .
Teaching and Learning Plan 9: The Unit Circle
© Project Maths Development Team 2009 www.projectmaths.ie 16
Student Activity 2B
Application to any angle in the first quadrant 0°<θ<90°
sin θ =________ cos θ =________ tan θ =________
Student Activity 2
Trigonometric function Sign in the first quadrant
sin
cos
tan
A/° 0° 90°
Coordinates on
the unit circle
cos Asin Atan A
Student Activity 2A
Drop a perpendicular from Q to the x-axis to construct a right angled triangle, centred at (0, 0).
What is the length of the hypotenuse? _________________________________________________
What is the length of the opposite? ____________________________________________________
What is the length of the adjacent? ____________________________________________________
Using trigonometric ratios, (not a calculator), calculate the sin 30°, cos 30°and the tan 30°.
sin 30° =________ cos 30° =________ tan 30° =________
Compare these with the values of the x and y coordinates of Q. What do you notice about the
x and y coordinates of Q and the trigonometric functions sin 30°,cos 30° and tan 30°?
Check the answers using a calculator.
sin 30° =________ cos 30° =________ tan 30° =________
Student Activity 2C
What signs will sin, cos and tan
have in the first quadrant ? Why
have sin, cos and tan got these
signs in the first quadrant?
_________________________________________________________________________________
Student Activity 2D
Mark angles of 0° and 90° degrees in
standard position on the unit circle, and
from what you have just learned, without
using a calculator, write down the sin, cos,
and tan of 0° and 90°.
Teaching and Learning Plan 9: The Unit Circle
© Project Maths Development Team 2009 www.projectmaths.ie 17
Student Activity 3
Student Activity 3A
Angles in the second quadrant 90° < θ < 180°
On the unit circle on Appendix A, mark an angle of 150° in standard position. Read the
x and y coordinates of the point Q’, where the terminal ray intersects the circumference.
(x, y) of point Q’ are _____________________________________________________________
Using what you have learned about the coordinates of points on the circumference of
the unit circle, fill in the following:
cos 150° = ________ sin 150° = ________ tan 150° = ________
Check these values using the calculator.
cos 150° = ________ sin 150° = ________ tan 150° = ________
Compare with
cos 30° = _________ sin 30° = _________ tan 30° = _________
Student Activity 3B
Drop a perpendicular from point
Q’, to the negative direction of the
x- axis, to make a right angled
triangle, with angle A at the origin.
What is the value of A in degrees?
_________________________________
_________________________________
Using the trigonometric ratios on triangle OB’Q’, what is sin A =________ cos A =________
Student Activity 3C
A is called the reference angle. Describe the reference angle? ________________________
_________________________________________________________________________________
_________________________________________________________________________________
Teaching and Learning Plan 9: The Unit Circle
© Project Maths Development Team 2009 www.projectmaths.ie 18
Student Activity 3D
What do you notice about the sin and cos of the reference angle and the sin 150° and
cos 150°? ________________________________________________________________________
What is the image of triangle OB’Q’ by reflection in the y- axis? _____________________
What is the relationship between triangle OB’Q’ and its image in the y- axis? _________
_________________________________________________________________________________
Hence what is the relationship between ratio of sides in triangle OB’Q’ and the ratio of
the sides of its image in the y- axis? _______________________________________________
Therefore 150° in the second quadrant has a reference angle of ___ in the first quadrant
sin 150° =________ sin 30° =________ cos 150° =________ cos 30° =________
Application to any angle in the second quadrant
sin θ = _____________
cos θ = _____________
tan θ = _____________
sin A = opp/hyp = ____ (A in the 2nd quadrant)
cos A = adj/hyp = ____ (A in the 2nd quadrant)
sin A = ____ (A in the 1st quadrant)
cos A = ____ (A in the 1st quadrant)
Express A in terms of θ and 180º _________________________ Equation (i)
Write down the relationship between sin θ in the second quadrant and sin A in the first
quadrant ________________________________________________________________________
Rewrite the answer using equation (i) above _______________________________________
Write down the relationship between cos θ and the second quadrant cos A in the first
quadrant ________________________________________________________________________
Rewrite the answer using equation (i) above _______________________________________
Write down the relationship between tan θ in the second quadrant and the tan A in the
first quadrant____________________________________________________________________
Rewrite the answer using equation (i) above _______________________________________
0.5
A A
θ
-0.5
(-1,0) (1,0)
(0,1)
(0,-1)
P’(-x,y) P(x,y)
|r|=1
O
Teaching and Learning Plan 9: The Unit Circle
© Project Maths Development Team 2009 www.projectmaths.ie 19
Fill in the signs for cos and sin and tan of an angle in the second quadrant.
Trigonometric function Sign in the second quadrant
sin
cos
tan
Using the reference angle, how would you calculate the sin 130º, cos 130º, tan 130º?
_________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
Using the reference angle, how would you calculate the sin 110º, cos 110º, tan 110º?
_________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
Using the reference angle, how would you calculate the sin 170º, cos 170º, tan 170º?
_________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
Mark an angle of 180º degrees in standard position on the unit circle, and using
coordinates, not a calculator, write down the sin, cos, and tan of 180º. Check the
answers using a calculator.
A/º 180ºCoordinates on the unit circlecos Asin Atan A
Student Activity 3D
Teaching and Learning Plan 9: The Unit Circle
© Project Maths Development Team 2009 www.projectmaths.ie 20
Angles in the third quadrant 180° < θ < 270°
On the unit circle on Appendix A, mark an angle of 210º in standard position. Read the x
and y coordinates of the point Q’’, where the terminal ray intersects the circumference.
(x, y) of point Q’’ are _____ Hence: cos 210° =______ sin 210° =______ tan 210° =______
Check these values using the calculator. cos 210° =_____ sin 210° =_____ tan 210° =_____
Compare with cos 30° =________ sin 30° =________ tan 30° =________
Is there a relationship between trig functions of angles in the first and third quadrants?
_________________________________________________________________________________
Explanation of the relationship
Drop a perpendicular from point Q’’, to the negative
direction of the x- axis, to make a right angled
triangle OB’Q’’. What is the value of A in degrees? _
Using the trigonometric ratios on triangle OB’Q’’,
what is sin A =________ cos A =________
A is called the Reference angle. Describe the
reference angle? ________________________________
What do you notice about the sin and cos of the
reference angle and sin 210º and
cos 210°? ________________________________________________________________________
What is the image of triangle OB’Q’’ by S0? ________________________________________
What is the relationship between triangle OB’Q’’ and its image by S0?_______________
Hence what is the relationship between ratio of sides in triangle OB’Q’’ and the ratio of
the sides of its image by S0? ______________________________________________________
Therefore 210º in the third quadrant has a reference angle of_____in the first quadrant.
sin 210º = _______ sin 30º= _______, cos 210º = _______ cos 30°= _______
Application to any angle in the third quadrant.
sin θ = _____________
cos θ = _____________
tan θ = _____________
sin A = opp/hyp = ____ (A in the 3rd quadrant)
cos A = adj/hyp = ____ (A in the 3rd quadrant)
sin A = ____ (A in the 1st quadrant)
cos A = ____ (A in the 1st quadrant)
Student Activity 4
Teaching and Learning Plan 9: The Unit Circle
© Project Maths Development Team 2009 www.projectmaths.ie 21
Express A in terms of θ and 180º. _______________________________________equation (i)
Write down the relationship between sin θ and the sin A in the third quadrant and then
rewrite the answer using equation (i) above. _______________________________________
_________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
Write down the relationship between cos θ and the cos A in the third quadrant and
then rewrite the answer using equation (i) above. _________________________________
_________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
Write down the relationship between tan θ and the tan A in the third quadrant and
then rewrite the answer using equation (i) above. __________________________________
_________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
Fill in the signs for cos and sin and tan of an angle in the third quadrant.
Trigonometric function Sign in the third quadrant
sin
cos
tan
Using the reference angle, calculate the sin, cos and tan of 220º. ____________________
_________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
Mark an angle of 270º degrees in standard position on the unit circle, and using
coordinates, not a calculator, write down the sin, cos, and tan of 270º. Check the
answers using a calculator.
A/º 270º
Coordinates on the unit circle
cos Asin Atan A
Student Activity 4
Teaching and Learning Plan 9: The Unit Circle
© Project Maths Development Team 2009 www.projectmaths.ie 22
Angles in the fourth quadrant 270° < θ < 360°
On the unit circle on Appendix A, mark an angle of 330º in standard position. Read the x
and y coordinates of the point Q’’’, where the terminal ray intersects the circumference.
(x, y) of point Q’’’ are _______________
Hence: cos 330° =_______ sin 330° =_______ tan 330° =_______
Using the calculator: cos 330° =_______ sin 330° =_______ tan 330° =_______
Compare with: cos 30° =_______ sin 30° =_______ tan 30° =_______
Is there a relationship between trig functions of angles in the first and fourth
quadrants? ______________________________________________________________________
Explanation of the relationship
Drop a perpendicular from
point Q’’’, to the negative
direction of the x- axis, to
make a right angled triangle
OBQ’’’. What is the value of A in
degrees?_____________
Using the trigonometric ratios
on triangle OBQ’’’,
sin A =___________
cos A =___________
A is called the reference angle. Describe the reference angle? ________________________
_________________________________________________________________________________
What do you notice about the sin and cos of the reference angle and that of
330º?____________
What is the image of triangle OBQ’’’ by SX?________________________
What is the relationship between triangle OBQ’’’ and its image by SX?_______________
Hence what is the relationship between ratio of sides in triangle OBQ’’’ and the ratio of
the sides of its image in the y- axis?_________________________________________
Therefore 330º in the fourth quadrant has a reference angle of _____ in the first
quadrant
sin 330º = _______ sin 30º= _______ cos 330º = _______ cos 30º= _______
Student Activity 5
Teaching and Learning Plan 9: The Unit Circle
© Project Maths Development Team 2009 www.projectmaths.ie 23
Application to any angle in the fourth quadrant.
sin θ = _____________
cos θ = _____________
tan θ = _____________
sin A = opp/hyp = ____ (A in the 4th quadrant)
cos A = adj/hyp = ____ (A in the 4th quadrant)
sin A = ____ (A in the 1st quadrant)
cos A = ____ (A in the 1st quadrant)
Express A in terms of θ and 360º._______________________________________equation (i)
Write down the relationship between sin θ and the sin A in the fourth quadrant and
then rewrite the answer using equation (i) above __________________________________
_________________________________________________________________________________
Write down the relationship between cos θ and the cos A in the fourth quadrant and
then rewrite the answer using equation (i) above. __________________________________
_________________________________________________________________________________
Write down the relationship between tan θ and the tan A in the fourth quadrant and
then rewrite the answer using equation (i) above. __________________________________
_________________________________________________________________________________
Fill in the signs for cos and sin and tan of an angle in the fourth quadrant.
Trigonometric function Sign in the fourth quadrant
sin
cos
tan
Using the reference angle, calculate the sin, cos and tan of 315º, writing the answers in
surd form. _______________________________________________________________________
_________________________________________________________________________________
Mark an angle of 360º degrees in standard position on the unit circle, and using
coordinates, not a calculator, write down the sin, cos, and tan of 360º. Check the
answers using a calculator.
A/º 360ºCoordinates on the unit circlecos Asin Atan A
Student Activity 5
Teaching and Learning Plan 9: The Unit Circle
© Project Maths Development Team 2009 www.projectmaths.ie 24
Student Activity 6A
Summary on finding trig functions of all angles
Fill in on the unit circle, in each quadrant, the first letter of the trigonometric function
which is positive in each quadrant.
Mark in an angle θ and its reference angle A for each
quadrant. Use a different unit circle for each situation.
Fill in also in each quadrant, the formula for the reference
angle given an angle θ in that quadrant.
Student Activity 6
0.5
-0.5
(-1,0) (1,0)
(0,1)
(0,-1)
Firstquadrant
Fourthquadrant
Thirdquadrant
Secondquadrant
G
Student Activity 6B
Negative angles
Find, without using the calculator. Show steps.
sin (–120º) ______________________________________________________________________
cos (–120º) ______________________________________________________________________
tan (–120º) ______________________________________________________________________
Student Activity 6C
Angles greater than 360º
Find, without using the calculator. Show steps.
sin 450º _______________________ sin 1250º _________________________________________
cos 450º _______________________ cos 1250º _________________________________________
tan 450º _______________________ tan 1250º _________________________________________
0.5
-0.5
(-1,0) (1,0)
(0,1)
(0,-1)
Firstquadrant
Fourthquadrant
Thirdquadrant
Secondquadrant
G
0.5
-0.5
(-1,0) (1,0)
(0,1)
(0,-1)
Firstquadrant
Fourthquadrant
Thirdquadrant
Secondquadrant
G
0.5
-0.5
(-1,0) (1,0)
(0,1)
(0,-1)
Firstquadrant
Fourthquadrant
Thirdquadrant
Secondquadrant
G
Teaching and Learning Plan 9: The Unit Circle
© Project Maths Development Team 2009 www.projectmaths.ie 25
Student Activity 7A
If we know A, how do we calculate θ? Always draw a unit circle.
Quadrant in which terminal ray of θ lies
Ist 2nd 3rd 4th
Formula for A (reference angle)
A= A= A= A=
Formula for θ in terms of A
θ= θ= θ= θ=
Student Activity 7
Student Activity 7B
Solve the equation cos θ =
where (use Tables)
What are we trying to find? ______________________________________________________
_________________________________________________________________________________
In what quadrants could θ be located and why? ___________________________________
_________________________________________________________________________________
As the reference angle is acute what sign will the trig functions of reference angles
have? ___________________________________________________________________________
_________________________________________________________________________________
What is the value of the reference angle (the acute angle with this value of cos)?
_________________________________________________________________________________
_________________________________________________________________________________
Knowing the reference angle, what are the possible values of θ? ____________________
_________________________________________________________________________________
_________________________________________________________________________________
Solve the equation sin θ = – 3
2
Teaching and Learning Plan 9: The Unit Circle
© Project Maths Development Team 2009 www.projectmaths.ie 26
Appendix A
The Unit Circle