teaching & learning plans - project maths

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 •



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



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


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 _________________________________________________________ .



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°.



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? ________________________

_________________________________________________________________________________

_________________________________________________________________________________



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 ________________________________________________________________________


Write down the relationship between tan θ in the second quadrant and the tan A in the

first quadrant____________________________________________________________________


0.5

A A

θ

-0.5

(-1,0) (1,0)

(0,1)

(0,-1)

P’(-x,y) P(x,y)

|r|=1

O



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?

_________________________________________________________________________________

_________________________________________________________________________________

_________________________________________________________________________________

_________________________________________________________________________________

_________________________________________________________________________________


_________________________________________________________________________________

_________________________________________________________________________________

_________________________________________________________________________________

_________________________________________________________________________________

_________________________________________________________________________________


_________________________________________________________________________________

_________________________________________________________________________________

_________________________________________________________________________________

_________________________________________________________________________________

_________________________________________________________________________________

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



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)



Student Activity 4



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. ____________________

_________________________________________________________________________________

_________________________________________________________________________________

_________________________________________________________________________________




A/º 270º

Coordinates on the unit circle

cos Asin Atan A

Student Activity 4



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



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)



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


_________________________________________________________________________________

Write down the relationship between tan θ and the tan A in the fourth quadrant and


_________________________________________________________________________________

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. _______________________________________________________________________

_________________________________________________________________________________




A/º 360ºCoordinates on the unit circlecos Asin Atan A

Student Activity 5



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



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



Appendix A

The Unit Circle

teaching & learning plans - project maths

Documents