Teaching & Learning Plans

Teaching & Learning PlansPlan 8: Introduction to

Trigonometry

Junior 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 8: Introduction to Trigonometry

AimsTo introduce the concept of trigonometry•

To understand the concept of sin, cos and tan•

To apply trigonometry to solve problems•

Prior Knowledge Students should have knowledge of the concept of ratio and should know that division is not commutative. Students should be able to use a calculator to convert fractions, correct to three decimal places, measure the lengths of sides of a triangle to the nearest millimetre and draw to scale. Students should have studied Pythagoras’ Theorem and know the meaning of the term “hypotenuse”. Students may have studied similar triangles in geometry – the fact that corresponding sides are proportional makes trigonometry possible. Students will have understood how to calculate mean. Students should know how to use a measuring tape and protractor.

Learning OutcomesAs a result of studying this topic, students will be able to

correctly identify the hypotenuse in a right angled triangle, and the •opposite and adjacent sides of a given angle

find a pattern linking the ratio of sides of a triangle with the angles and •hence understand the concepts of sine, cosine and tangent ratios of angles

apply trigonometry to solve problems including those involving angles of •elevation and depression



Relationship to Junior Certificate SyllabusSub-topics Ordinary Level Higher Level

(also includes OL)

Trigonometry Solution of right-angled triangle problems of a simple nature involving heights and distances, including the use of the theorem of Pythagoras.

Cosine, sine and tangent of angles (integer values) between 0˚ and 90˚.

Functions of 30˚, 45˚ and 60˚ in surd form derived from suitable triangles.

Solution of right-angled triangles.

Decimal and DMS values of angles.

Resources RequiredCalculators, measuring tapes, graph paper, cm rulers and a sunny day!

Introducing the TopicTrigonometry is simply geometrical constructions where the ratio of the side lengths in triangles is used to determine angular measurement.

Trigon, meaning triangle and metria, meaning measurement

Mathematicians have used trigonometry for centuries to accurately determine distances without having to physically measure them (Clinometer Activity Appendix A). It can also be used to calculate angles that would be very difficult to measure. Trigonometry has uses in such areas as surveying, navigation, drawing and architecture.

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Plan

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stud

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6

Stud

ent

Lear

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Tas

ks:

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her

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tSt

uden

t Act

ivit

ies:

Pos

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Plan

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stud

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7

Stud

ent

Lear

ning

Tas

ks: T

each

er

Inpu

tSt

uden

t Act

ivit

ies:

Pos

sibl

e an

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her’s

Sup

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use

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nam

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or

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sid

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6 •

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6

pai

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th

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firs

t p

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knew

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rst

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nd

on

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f th

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ld

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cle

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nd

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ll st

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ents

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at y

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ate

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th

e o

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e th

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ltip

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ctiv

itie

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4, 5

, 6,

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and

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or

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the

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an

d o

ne

oth

er

giv

en a

ng

le (

giv

en o

n t

he

shee

t). L

abel

th

e si

des

of

the

rig

ht

ang

led

tri

ang

les

as h

yp

(hyp

ote

nu

se),

ad

j (ad

jace

nt)

, an

d o

pp

(o

pp

osi

te).

Mea

sure

ea

ch s

ide

corr

ect

to t

he

nea

rest

m

m a

nd

cal

cula

te t

he

rati

os

op

p/h

yp, a

dj/h

yp, o

pp

/ad

j.

On

e st

ud

ent

is t

o m

easu

re, o

ne

»to

cal

cula

te r

atio

s an

d o

ne

to

do

ub

le c

hec

k ch

ang

ing

ro

les

on

ea

ch t

rian

gle

.

Stu

den

ts w

ork

on

»

Stu

den

t A

ctiv

itie

s 3,

4, 5

, 6,

7 a

nd

8.

Dis

trib

ute

S »

tud

ent

Act

ivit

y 3,

4,

5, 6

,7 a

nd

8 t

o d

iffe

ren

t g

rou

ps

of

3-4

stu

den

ts e

ach

i.e

. on

e St

ud

ent

Act

ivit

y p

er

gro

up

.

Emp

has

ise

that

stu

den

ts

»sh

ou

ld m

easu

re a

ccu

rate

ly t

o

the

nea

rest

mm

.

Are

stu

den

ts m

easu

rin

g

»ac

cura

tely

an

d

calc

ula

tin

g r

atio

s co

rrec

tly

i.e. o

pp

/hyp

an

d n

ot

hyp

/op

p?

Teac

hing

& L

earn

ing

Plan

8: I

ntro

duct

ion

to T

rigo

nom

etry

© P

roje

ct M

aths

Dev

elop

men

t Tea

m 2

009

w

ww

.pro

ject

mat

hs.ie

KE

Y:

» n

ext

step

•

stud

ent

answ

er/r

espo

nse

8

Stud

ent

Lear

ning

Tas

ks:

Teac

her

Inpu

tSt

uden

t Act

ivit

ies:

Pos

sibl

e an

d Ex

pect

ed R

espo

nses

Teac

her’s

Sup

port

and

A

ctio

nsCh

ecki

ng U

nder

stan

ding

Wri

te d

ow

n w

hat

yo

u

»h

ave

ob

serv

ed f

rom

yo

ur

answ

ers.

The

rati

os

are

un

chan

ged

•

for

a p

arti

cula

r an

gle

re

gar

dle

ss o

f th

e si

ze o

f th

e tr

ian

gle

.

If a

gro

up

has

dif

ficu

lty

»se

ein

g t

he

pat

tern

or

verb

alis

ing

it y

ou

will

see

th

is a

s yo

u c

ircu

late

an

d

hel

p b

y u

sin

g le

adin

g

qu

esti

on

s.

Are

th

e st

ud

ents

fin

din

g

»th

at t

he

rati

os

are

un

chan

ged

fo

r a

par

ticu

lar

ang

le r

egar

dle

ss o

f th

e si

ze

of

the

tria

ng

le?

Of

the

3 ra

tio

s w

hic

h o

f »

them

can

nev

er b

e b

igg

er

than

1?

Exp

lain

. »

•

Nu

mer

ato

r w

ill a

lway

s •

be

smal

ler

than

th

e d

eno

min

ato

r as

th

e h

ypo

ten

use

is t

he

lon

ges

t si

de.

Ask

th

e cl

ass

wh

en t

hey

»

hav

e a

few

of

the

rati

os

calc

ula

ted

to

an

swer

th

is

qu

esti

on

an

d t

hen

ask

on

e g

rou

p t

o e

xpla

in.

Cir

cula

te c

hec

kin

g t

he

»p

rog

ress

of

the

dif

fere

nt

gro

up

s to

see

th

at

sid

es a

re b

ein

g la

bel

led

co

rrec

tly

and

th

at s

tud

ents

u

nd

erst

and

th

e ta

sk.

Hav

e th

ey b

een

ab

le t

o

»an

swer

th

is q

ues

tio

n b

ased

o

n t

hei

r kn

ow

led

ge

of

rig

ht

ang

led

tri

ang

les?

Is it

po

ssib

le f

or

any

of

the

»ra

tio

s to

be

big

ger

th

an 1

?

If s

o, w

hic

h o

ne

or

wh

ich

»

on

es a

nd

wh

y?

Yes

•

•

- T

he

op

po

site

is

big

ger

th

an t

he

adja

cen

t w

hen

it is

op

po

site

th

e b

igg

er o

f th

e 2

com

plim

enta

ry a

ng

les.

If s

tud

ents

can

no

t an

swer

»

this

qu

esti

on

, giv

e an

ex

amp

le u

sin

g n

um

ber

s

wh

en f

ract

ion

s g

ive

answ

ers

gre

ater

th

an 1

or

less

th

an 1

.

Do

stu

den

ts u

nd

erst

and

»

that

th

e o

nly

on

e o

f th

e th

ree

rati

os

wh

ich

can

be

big

ger

th

an 1

is

?

Teac

hing

& L

earn

ing

Plan

8: I

ntro

duct

ion

to T

rigo

nom

etry

© P

roje

ct M

aths

Dev

elop

men

t Tea

m 2

009

w

ww

.pro

ject

mat

hs.ie

KE

Y:

» n

ext

step

•

stud

ent

answ

er/r

espo

nse

9

Stud

ent

Lear

ning

Tas

ks:

Teac

her

Inpu

tSt

uden

t Act

ivit

ies:

Pos

sibl

e an

d Ex

pect

ed R

espo

nses

Teac

her’s

Sup

port

and

Act

ions

Chec

king

Und

erst

andi

ng

We

hav

e b

een

stu

dyi

ng

»

ho

w t

he

sid

es a

nd

an

gle

s o

f tr

ian

gle

s ar

e re

late

d

to e

ach

oth

er. T

his

is

calle

d T

RIG

ON

OM

ETR

Y –

Tr

igo

n m

ean

ing

tri

ang

le

and

met

ria

mea

nin

g

mea

sure

men

t.

Stu

den

ts w

rite

th

is h

ead

ing

»

and

th

e ra

tio

s in

to t

hei

r co

pie

s.

Wri

te t

he

wo

rd

»TR

IGO

NO

MET

RY

on

th

e b

oar

d.

Tell

the

stu

den

ts t

hat

th

e »

rati

os

they

hav

e in

vest

igat

ed

hav

e sp

ecia

l nam

es a

nd

wri

te

them

on

th

e b

oar

d.

»

Go

bac

k to

»

Stu

den

t A

ctiv

itie

s 3

- 8

and

fill

in

the

nam

e o

f ea

ch r

atio

, fo

r ex

amp

le f

or

the

shee

t w

ith

an

gle

s o

f 30

° fi

ll in

op

p/h

yp

= s

in 3

0° e

tc.

Bes

ide

each

rat

io s

tud

ents

»

fill

in t

he

app

rop

riat

e n

ame

plu

s th

e an

gle

it r

efer

s to

.

Tell

stu

den

ts t

hat

sin

is t

he

»sh

ort

ver

sio

n o

f si

ne,

co

s is

sh

ort

fo

r co

sin

e an

d t

hat

tan

is

th

e sh

ort

ened

ver

sio

n o

f th

e w

ord

‘tan

gen

t’.

On

th

e m

aste

r ta

ble

, »

Stu

den

t A

ctiv

ity

9 fi

ll in

th

e m

ean

val

ue

you

hav

e ca

lcu

late

d f

or

sin

, co

s an

d

tan

of

the

ang

le f

rom

yo

ur

ow

n S

tud

ent

Act

ivit

y 3,

4,

5, 6

, 7, o

r 8.

Tel

l th

e re

st

of

the

clas

s th

e va

lues

yo

u

hav

e ca

lcu

late

d.

Each

gro

up

giv

es t

he

»an

swer

s fo

r ea

ch r

atio

fo

r th

e an

gle

th

ey h

ave

wo

rked

on

.

Han

d o

ut

»St

ud

ent

Act

ivit

y 9.

Dra

w a

mas

ter

tab

le o

n t

he

»b

oar

d a

nd

fill

in t

he

answ

ers

as t

hey

are

giv

en t

ellin

g

stu

den

ts t

hat

th

ey w

ill b

e ab

le

to c

hec

k th

ose

an

swer

s in

th

e n

ext

step

.

Hav

e al

l stu

den

ts

»u

nd

erst

oo

d t

he

task

s so

far

, co

mp

lete

d t

hem

fo

r al

l th

e tr

ian

gle

s, a

nd

fi

lled

ou

t th

e m

aste

r ta

ble

?

Teac

hing

& L

earn

ing

Plan

8: I

ntro

duct

ion

to T

rigo

nom

etry

© 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

Act

ions

Chec

king

U

nder

stan

ding

Wh

at u

nit

s ar

e an

gle

s »

mea

sure

d in

?D

egre

es•

Ther

e ar

e o

ther

un

its

for

»m

easu

rin

g a

ng

les

such

as

rad

ian

s, w

hic

h y

ou

will

m

eet

late

r o

n s

o y

ou

mu

st

be

sure

yo

ur

calc

ula

tor

is in

d

egre

e m

od

e if

yo

u a

re u

sin

g

deg

rees

.

Sho

w s

tud

ents

ho

w t

o c

hec

k »

if t

hei

r ca

lcu

lato

r is

in d

egre

e m

od

e an

d, i

f n

ot,

ho

w t

o p

ut

it in

to t

his

mo

de.

Usi

ng

th

e ca

lcu

lato

r ch

eck

»th

e va

lues

of

sin

, co

s an

d

tan

of

the

ang

les,

wh

ich

yo

u

hav

e ca

lcu

late

d t

hro

ug

h

mea

sure

men

t. C

hec

k th

e m

easu

rem

ents

of

the

rest

of

the

clas

s al

so.

Stu

den

ts c

hec

k th

e va

lues

»

in t

he

mas

ter

tab

le w

ith

th

e va

lues

go

t u

sin

g t

he

calc

ula

tor

and

fill

in t

o

“ch

eck”

co

lum

ns

on

th

e m

aste

r ta

ble

Stu

den

t A

ctiv

ity

9.

Emp

has

ise

that

sin

, co

s an

d

»ta

n a

re f

un

ctio

ns

of

ang

les.

Cir

cula

te t

o s

ee t

hat

th

e »

valu

es o

f si

n, c

os

and

tan

of

ang

les

calc

ula

ted

th

rou

gh

m

easu

rem

ents

ag

ree

wit

h

tho

se f

ou

nd

on

th

e ca

lcu

lato

r.

Usi

ng

th

e an

swer

s o

n

»St

ud

ent

Act

ivit

y 9,

an

swer

th

e q

ues

tio

ns

on

Stu

den

t A

ctiv

ity

10.

Stu

den

ts s

ee p

atte

rns

in t

he

»an

swer

s o

n t

he

mas

ter

tab

le.

Dis

trib

ute

»

Stu

den

t A

ctiv

ity

10.

Cir

cula

te t

o e

nsu

re t

hat

»

stu

den

ts a

re a

ble

to

see

th

e p

atte

rns

and

are

usi

ng

co

rrec

t te

rmin

olo

gy.

If y

ou

kn

ew t

he

rati

os

ho

w

»w

ou

ld y

ou

fin

d o

ut

the

ang

le?

Giv

en t

hat

th

e si

n o

f an

an

gle

is 0

.5 h

ow

do

yo

u

fin

d t

he

ang

le?

Stu

den

ts m

ay b

e fa

mili

ar w

ith

•

the

SHIF

T o

r 2n

d f

un

ctio

n

bu

tto

n o

n t

he

calc

ula

tor

and

hen

ce s

ug

ges

t u

sin

g t

his

b

utt

on

.

Wri

te o

n t

he

bo

ard

- G

iven

a

»tr

ig r

atio

, fo

r ex

amp

le s

in A

=

0, t

he

ang

le A

= s

in-1(0

.5).

Em

ph

asis

e th

at s

in-1x,

co

s-1 x

an

d t

an-1 x

rep

rese

nt

ang

les

wh

ere

x is

a r

atio

of

sid

es in

a

rig

ht

ang

led

tri

ang

le.

Teac

hing

& L

earn

ing

Plan

8: I

ntro

duct

ion

to T

rigo

nom

etry

© P

roje

ct M

aths

Dev

elop

men

t Tea

m 2

009

w

ww

.pro

ject

mat

hs.ie

KE

Y:

» n

ext

step

•

stud

ent

answ

er/r

espo

nse

11

Stud

ent

Lear

ning

Tas

ks:

Teac

her

Inpu

tSt

uden

t Act

ivit

ies:

Pos

sibl

e an

d Ex

pect

ed R

espo

nses

Teac

her’s

Sup

port

and

A

ctio

nsCh

ecki

ng U

nder

stan

ding

Giv

en

»

Stu

den

ts u

se t

hei

r ca

lcu

lato

rs t

o

»ev

alu

ate

thes

e an

gle

s.

Refl

ecti

on

List

wh

at y

ou

hav

e le

arn

ed

»to

day

.

Trig

on

om

etry

is a

bo

ut

the

stu

dy

of

•th

e re

lati

on

ship

bet

wee

n a

ng

les

and

ra

tio

of

sid

es in

tri

ang

les.

The

sid

es in

a r

igh

t an

gle

d t

rian

gle

•

are

lab

elle

d h

ypo

ten

use

, an

d t

hen

ad

jace

nt

and

op

po

site

dep

end

ing

o

n w

hic

h o

f th

e tw

o c

om

ple

men

tary

an

gle

s is

of

inte

rest

.

The

thre

e ra

tio

s o

f si

des

in a

rig

ht

•an

gle

d t

rian

gle

are

:

Wri

te d

ow

n a

nyt

hin

g y

ou

»

fou

nd

dif

ficu

lt.

Wri

te d

ow

n a

ny

qu

esti

on

s »

you

may

hav

e.

Cir

cula

te a

nd

tak

e »

no

te p

arti

cula

rly

of

any

qu

esti

on

s st

ud

ents

h

ave

and

hel

p t

hem

to

an

swer

th

em.

Are

stu

den

ts u

sin

g

»th

e te

rmin

olo

gy

wit

h

un

der

stan

din

g?



Student Activity 1

Me and my shadow

Safety warning: Never look directly at the sun

Name: _________________________________________ Class: _____________________________

Date: _________________________________________ Time: ______________________________

When the sun is high, your shadow is short. When the sun is low, your shadow is long.

Student Activity 1AShow the angle of elevation of the sun on the above diagram. Call it A. •Describe the angle of elevation of the sun in terms of the two arms of the •angle._________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________Measure the height of one of the students in your group and the length of their shadow.•Height of the student: __________cm. Length of the shadow _____________________cm.•Draw a rough sketch of a right-angled triangle to model the situation and write in the measurements.•

Student Activity 1B

Measure the length of the shadow of some tall object e.g. flagpole or goalpost. Length

of the shadow of a tall object which you cannot physically measure e.g. goalpost

______________________________cm



Student Activity 1C

Back in class – Measuring the angle of elevation of the sunDecide what scale to use.•

Draw an accurate diagram on graph paper.•

Diagram 1Measure the angle of elevation of the sun from Diagram 1 above using a protractor.•

Angle of elevation of the sun at _________ (time) on _______ (date) was ________.•

Check your answer with other students in the class.•

If you were to measure the angle of elevation of the sun at 10 a. m and another class measured the •angle at 11 a.m. what would be the difference in the measurements?_______________________________



Student Activity 1D

Knowing the angle of elevation of the sun, measure the height of a tall object using the length of its

shadow as previously measured. Decide what scale to use. Scale: ______________________•

Draw an accurate diagram on graph paper using the length of the shadow, the •angle of elevation of the sun and forming right-angled triangle (ASA).

Diagram 2Measure the height of the goalpost from Diagram 2 above and using the scale factor convert to its •actual height.

Check the answer with other students in the class.•

Conclusion for part 2: The height of the goalpost is ___________________cm approximately.Would you expect the same answer if you took the measurements at different times of the day? Explain your answer.________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________



Student Activity 1E

Using Similar TrianglesUsing graph paper draw the above 2 diagrams overlapping, with the angles of •elevation of the sun superimposed as shown by example in the diagram on the right. Label the diagram on the graph paper as in the diagram on the right.

What do you notice about the 2 vertical lines in the triangles? ______________________________________•_______________________________________________________________________________________________

Measure the heights of the 2 vertical lines • |ED| and |CB|. ____

Measure the 2 horizontal lines • |AB| and |AD|. _____

What do you notice about the two ratios? ________________________________________________________•________________________________________________________________________________________________________________________________________________________________________________________________

Knowing • |AB| and |CB| and the distance |AD| how could you find |ED| without knowing the angle of elevation |∠EAD| of the sun?



Student Activity 1E

Ratios in Similar TrianglesDraw 3 different right angled triangles with the arms of the 90 degree angle being vertical and •horizontal line segments, using the same angle of elevation which you calculated for the sun. Call the triangles T1, T2, T3.

Measure the length of the vertical and horizontal line segments in these triangles.

Vertical T1 Horizontal T1 Vertical T2 Horizontal T2 Vertical T3 Horizontal T3

What do you notice about the ratios of any 2 vertical line segments of any 2 of these triangles and the ratio of the corresponding horizontal line segments? ______________________________________________ ________________________________________________________________________________________________ ________________________________________________________________________________________________ ________________________________________________________________________________________________



Student Activity 2

Labelling Sides in Right Angled Triangles



Student Activity 3

Calculating ratios for similar right angled triangles with angles of 30°Measure and label the 90° and the 30° angles in the following triangles. What is the measure of the •third angle?

Label the hypotenuse as “hyp”. With respect to the 30° angle, label the other sides as “adj” for •adjacent and “opp” for

Complete the table below. •

Marked Angle Size=30°

|opp|/mm |hyp|/mm |adj|/mm

(for angle=30°) (for angle=30°) (for angle=30°)

fraction decimal fraction decimal fraction decimal

T1T2T3T4T5Mean Value (correct to 2 decimal places)



Student Activity 4

Calculating ratios for similar right angled triangles with angles of 40°Measure and label the 90° and the 40° angles in the following triangles. What is the measure of the •third angle?

Label the hypotenuse as “hyp”. With respect to the 40° angle, label the other sides as “adj” for •adjacent and “opp” for opposite.

Complete the table below.•








Student Activity 5

Calculating ratios for similar right angled triangles with angles of 45°Measure and label the 90° and the 45° angles in the following triangles. What types of right angled •triangle are these triangles?___________________________________________________________________










Student Activity 6

Calculating ratios for similar right angled triangles with angles of 50°Label the 90° and the 50° angles in the following triangles. What is the measure of the third angle? •______________________________________________________________________________________________










Student Activity 7

Calculating ratios for similar right angled triangles with angles of 60°Measure and label the 90° and the 60° angles in the following triangles. What is the measure of the •third angle?__________________________________________________________________________________










Student Activity 8

Calculating ratios for similar right angled triangles with angles of 70°Measure and label the 90° and the 70° angles in the following triangles. What is the measure of the •third angle?___________________________________________________________________________________










Student Activity 9

Master table of class results for ratios of sides in right angled triangles

Angle/° Check Check Check

30°

40°

45°

50°

60°

70°



Student Activity 10

Using the master table of class results answer the following questions

1. What do you notice about sin 30° and cos 60°? __________________________________

_______________________________________________________________________________

2. What do you notice about cos 30° and sin 60°? ___________________________________

_______________________________________________________________________________

3. Can you explain what you have noticed using diagrams?

4. How would you describe angles 30° and 60°? ____________________________________

_______________________________________________________________________________

5. Can you find similar examples in the master table? _______________________________

_______________________________________________________________________________

6. For what angle in a right angled triangle is the opposite side one half of the

hypotenuse? __________________________________________________________________

_______________________________________________________________________________

Draw a diagram to illustrate your answer.

7. For what angle in a right angled triangle are the opposite and adjacent sides equal?

_______________________________________________________________________________

8. Calculate

for each angle A. Compare this to the value of Tan A. What do you

notice? Can you justify the answer? ______________________________________________

_______________________________________________________________________________

_______________________________________________________________________________



Appendix A

Making and using a clinometer to find the height of a tall structure

Materials required: Protractor, sellotape, drinking straw, needle and thread, paper clip.

Work in threes – one holding the clinometer, one

reading the angle of elevation, one recording the

angle of elevation.

Measure the height of the observer from eye to •ground level.Measure the distance from the observer to the •base of the building (under the highest point).Mark the position of the observer on the •ground.

Hold the clinometer so that the string is vertical.•Now tilt the clinometer looking through the •drinking straw so that the highest point on the top of the wall/flagpole/spire is visible.Read the angle of elevation of this highest •point to the nearest degree.Draw a rough sketch of the situation marking •in the distances measured and the angle of elevation.

Finding the height of a wall/spire/ flagpole using a clinometer

Drinking Straw

Thread

A

h1

h2



Appendix A

Rough Sketch

Height from ground to observer’s eye

Distance from observer to the foot of the spire

Angle of elevation to the top of the spire

Back in class:

Calculate the height of a very tall object using a scaled diagram.•

Using graph paper draw a scaled diagram of the above situation. Scale_______________•

Measure the height of the spire from the scaled diagram and using the scale factor convert to its actual •

height.

Height of the spire above the observer’s eye: ____________________•

Height from ground to the observer’s eye: ______________________•

Height of the spire: ___________________________________________•



Appendix A

Calculate the height of a very tall object using trigonometry

Redraw the diagram (doesn’t have to be to scale) marking the right angle, the hypotenuse, the angle of elevation, the side adjacent to the angle of elevation and the side opposite the angle of elevation. Right angled triangle, sides labelled and measurements included

What side do you know the length of? _________________________________________•What side do you require the length of? ________________________________________•What trigonometric ratio in a right-angled triangle uses these 2 sides? ________________•

Using trigonometry, calculate the height of the building.

Angle of elevation of the top of the building

Distance to the base of the building d

Tan A = h1/d h1 Height of the observer h2

Height of the building h1+ h2

QuestionAs you move towards or away from the building while sighting the top of the spire, the angle of elevation of the top of the spire varies. What angle of elevation would allow the height of the building to be measured by using the distance from the observer to the base of the building added to the observer’s height – i.e. no scaled drawing or trigonometry required?



Appendix A

Finding the height of a building with a moat around it

Work in threes – one holding the clinometer, one reading the angles of elevation, one recording the angles of elevation.

Measure the height of the observer from eye to ground level, h• 2, and fill into the table.

h2 ⏐<A⏐ ⏐<B⏐ ⏐d⏐ h1 H = h1+h2

Mark the position of the observer on the ground.•Hold the clinometer so that the string is vertical.•Now tilt the clinometers, looking through the drinking straw so that a point on the top of the wall/•flagpole/spire is visible.Read the angle of elevation, <A, to the nearest degree and fill into the table.•The observer moves closer to the building and again views the top of the building through the •drinking straw on the clinometer.Read the angle of elevation, <B, to the nearest degree and fill into the table.•Measure the distance between the two viewing positions, d, of the observer and fill into the table •below.Draw a rough sketch of the situation marking in the distances measured and the angles of elevations.•

h2

A B

d

h1



Appendix A

Use a scaled diagram to calculate the height of the building with a moat

Using graph paper draw a scaled diagram of the above situation. Scale _____________________

Measure the height of the spire from the scaled diagram and using the scale factor convert to its actual •height.Height of the spire above the observer’s eye: _____________________•Height from ground to the observer’s eye: _______________________•Height of the spire: ______________________________________•



Calculating the height of a very tall object surrounded by a moat using trigonometry

Redraw the diagram (which does not have to be to scale) labelling the sides and angles.Right angled triangle, sides labelled and measurements included

Fill in all the angle and side measurements known for triangles CBD and CAB.•Fill in the other two angle measurements in triangle CAB.•Of the two triangles CBD and CAB, which triangle do you have most information for? ______________•Which side do you require the length of in triangle CBD? _________________________________________•What side is shared by both triangles CAB and CBD? ______________________________________________•What rule can be used to calculate this side? _____________________________________________________•Calculations:•

Label the sides in right angled triangle CBD appropriately as ‘hypotenuse’, ‘adjacent’ and ‘opposite’.•Which side do you know the length of? _________________________________________________________•Which side do you require the length of ? _______________________________________________________•What trigonometric ratio in a right-angled triangle uses these 2 sides? _____________________________•Calculations to find the required length:_________________________________________________________•

____________________________________________________________________________________________________________________________________________________________________

Complete the table below to find the height of the building H•

⏐<A⏐ ⏐<B⏐ ⏐d⏐ h2 h1 H = h1+h2

Appendix A



Appendix B

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Teaching & Learning Plans