Trigonometry National 4 REL 1.3a REL 1.3b Investigating the Tangent Ratio The Sine Ratio (Finding a length) The Sine Ratio (Finding

ww

w.m

ath

srevis

ion

.com

TrigonomeTrigonometrytry

National 4 REL 1.3a REL 1.3b

Investigating the Tangent Ratio

The Sine Ratio (Finding a length)

The Sine Ratio (Finding an angle)

The Cosine of an Angle

Mixed Problems

The Tangent (Finding lengths)

The Tangent (Finding the Angle)

The Cosine ( Finding a length)

Starter QuestionsStarter Questions

18 Apr 202318 Apr 2023 Created by Mr.Lafferty Maths DeptCreated by Mr.Lafferty Maths Deptww

w.m

ath

srevis

ion

.com

266 % of £ 270

3Q1. Calculate

Q2. Convert 60% to fraction and simplify.

Q3. Convert

Q4. What is the time difference 09:28 and 10:50

45

to a percentage.

Q5. The answer to the question is 90. What is thequestion.


18 Apr 202318 Apr 2023 Created by Mr. Lafferty Maths Dept.Created by Mr. Lafferty Maths Dept.ww

w.m

ath

srevis

ion

.com Learning IntentionLearning Intention Success CriteriaSuccess Criteria

2.2. Work out Tan Ratio.Work out Tan Ratio.

1. To identify the hypotenuse, opposite and adjacent sides in a right angled triangle.

Angles & Angles & Triangles Triangles

1.1. Understand the terms Understand the terms hypotenuse, opposite and hypotenuse, opposite and adjacent in right angled adjacent in right angled triangle.triangle.


ww

w.m

ath

srevis

ion

.com



Trigonometry means “triangle” and “measurement”.

AdjacentO

pp

osit

e

x°x°

hypotenuse

We will be using right-angled triangles.

These triangles have 2 things in common

One is 37o

The other we will investigate

Triangle A B C D

Vertical side 3cm 6cm 4cm

Horizontal side 4cm 8cm 6cmVertical

Horizontal

30.75

4

68

4.56

30.75

4

1.5cm2cm

0.75 0.75

www.mathsrevision.comwww.mathsrevision.com


ww

w.m

ath

srevis

ion

.com For the RAT with angle of 37o

the answer obtained by dividing vertical side by horizontal side was

Right-Angled Triangle

ALWAYS 0.75

The number has a special name

It is called the TANGENT of 37o

tan37o


ww

w.m

ath

srevis

ion

.com


Using a calculator to find tan ratio What Goes In The Box ?

Find the tan ratio to 3 decimal places.

tan 30o tan 0o tan 45o

tan 60o tan 20o tan 70o

= 0.577

= 0.000

= 1.000

= 1.732

= 0.364

= 2.747


ww

w.m

ath

srevis

ion

.com


Adj

x°x°

Tan x° =O

pp

osit

e

Opp

Adjacent


ww

w.m

ath

srevis

ion

.com



60°

12 mAdjacent

Op

posit

e

hypotenuse Copy this!

ww

w.m

ath

srevis

ion

.com



Tan x° =

Opp

Adj

Tan 60° =

Opp12

= Opp

12 x Tan 60°Opp =

12 x Tan 60°= 20.8m (1 d.p.)

Copy this!

ww

w.m

ath

srevis

ion

.com



Tan Ratio Ex1

Ch16 Page 198


www.mathsrevision.comwww.mathsrevision.comww

w.m

ath

srevis

ion

.com 1. True or f alse 30 5 + 6 4 = 48

2. Write in scientifi c notation 0.0456

3. I dentif y the sides of the triangle.

4. A train takes 6 hrs to travel 360 miles apart.

Calculate it's average speed.

xo



w.m

ath

srevis

ion


2.2. Use tan of an angle to Use tan of an angle to solve REAL LIFE problems.solve REAL LIFE problems.

1. To use tan of the angle to solve REAL LIFE problems.


1.1. Write down tan ratio.Write down tan ratio.


ww

w.m

ath

srevis

ion

.com


Adj

x°x°

Tan x° =O

pp

osit

e

Opp

Adjacent


ww

w.m

ath

srevis

ion

.com


Example Example

65°65°

Tan x° =

OppOpp

Adj

Hyp hh

8m8m Tan 65° =

h8

= h

8 x Tan 65°

h =

8 x Tan 65° = 17.2m (1 d.p.)

Adj

National 4 REL 1.3 a + 1.3b

Find the height h

SOH CAH TOA

ww

w.m

ath

srevis

ion

.com



SOH CAH TOA

18 Apr 202318 Apr 2023 Compiled by Mr. Lafferty Maths Dept.Compiled by Mr. Lafferty Maths Dept.

Use the tan ratio to find the height h of the tree

to 2 decimal places.

47o

8m

rod

o opp htan 47 = =

adj 8

o htan 47 =

8

oh = 8 × tan 47

h = 8.58m

ww

w.m

ath

srevis

ion

.com



6o

18 Apr 2023Compiled by Mr. Lafferty Maths

Dept.

Aeroplane

a = 15

c

Lennoxtown

Airport

Q1.Q1. An aeroplane is preparing to land at Glasgow An aeroplane is preparing to land at Glasgow Airport. Airport. It is over Lennoxtown at present which is It is over Lennoxtown at present which is 15km from 15km from the airport. The angle of descent is 6the airport. The angle of descent is 6oo. .

What is the height of the plane ?What is the height of the plane ?

Example 2Example 2

o htan 6 =

15

oh = 15 × tan 6

h = 1.58km

SOH CAH TOA

ww

w.m

ath

srevis

ion

.com




Use the tan ratio to calculate how far the ladder is away from the building.

45o

12m

ladder

o opp 12tan 45 = =

adj d

o

12d =

tan 45

d = 12m

d m

SOH CAH TOA

ww

w.m

ath

srevis

ion

.com



6o


Aeroplane

a = 1.58 km

Lennoxtown

Airport

Q1. An aeroplane is preparing to land at Glasgow Airport. It is over Lennoxtown at present. It is at a height of 1.58 km above the ground. It ‘s angle of descent is 6o.

How far is it from the airport to Lennoxtown?

Example 2Example 2

o 1.58tan 6 =

d

o

1.58d =

tan 6

d = 15 km

SOH CAH TOA

ww

w.m

ath

srevis

ion

.com



Now tryExercise 2

Ch16 (page 199)



w.m

ath

srevis

ion

.com 1. Find the area and perimeter

of the circle.

2. I s the triangle right -angled at P. Explain your answer.

10cm

P

Q

R

6cm

7cm

10cm



w.m

ath

srevis

ion


2.2. Use tan of an angle to Use tan of an angle to solve problems.solve problems.

1. To use tan of the angle to solve problems.


1.1. Write down tan ratio.Write down tan ratio.


Using Tan to calculate anglesUsing Tan to calculate angles


w.m

ath

srevis

ion

.com


ww

w.m

ath

srevis

ion

.com


Using a calculator to find anglegiven tan ratio

What Goes In The Box ?

Find the angle given the tan ratio to 1 decimal places.

tan-1 1 tan-1 0.5 tan-1 0.8

tan-1 0.6 tan-1 2 tan-1 3

= 45.0o = 26.6o

= 38.7o

= 31.0o = 64.4o = 71.6o


ww

w.m

ath

srevis

ion

.com


1812

ExampleExample

x°Tan x° =

OppOpp

Adj

Hyp

SOH CAH TOA

12m Tan x° =

= 1.5 Tan x°

Adj

18m

Calculate the tan xo

ratio

Q

P

R


ww

w.m

ath

srevis

ion

.com



= 1.5Tan x°How do we find x°?

We need to use Tan ⁻¹on the calculator.

2nd

Tan ⁻¹is written above Tan

Tan ⁻¹

To get this press TanFollowed by

Calculate the size of angle xo

ww

w.m

ath

srevis

ion

.com



x =

Tan ⁻¹1.5 = 56.3° (1 d.p.)

= 1.5Tan x°

2nd Tan

Tan ⁻¹

Press

Enter =1.5

ww

w.m

ath

srevis

ion

.com



Process

1. Identify Hyp, Opp and Adj

2. Write down ratio Tan xo = Opp Adj

3. Calculate xo 2nd Tan

Tan ⁻¹

ww

w.m

ath

srevis

ion

.com




Use the tan ratio to calculate the angle that the support wire makes with the ground.

xo

11m

o opp 11tan x = =

adj 4

114

o -1x = tan

o ox = 70

4 m

SOH CAH TOA

ww

w.m

ath

srevis

ion

.com



Now tryExercise 3

Ch16 Page 202


18 Apr 202318 Apr 2023 Created by Mr.Lafferty Maths DeptCreated by Mr.Lafferty Maths Deptww

w.m

ath

srevis

ion

.com 4 13 11 3

2. The ratio of apples to pears is 2:5.

I f there are 28 pieces of f ruit in total.

How many apples are there.

a a d a a d 1. Simplify

3. Find HCF of 8 and 18



w.m

ath

srevis

ion


2.2. Use sine ratio to find an Use sine ratio to find an angle.angle.

1. Definite the sine ratio and show how to find an angle using this ratio.


1.1. Write down sine ratio.Write down sine ratio.


ww

w.m

ath

srevis

ion

.com

TrigonomeTrigonometrytryThe Sine RatioThe Sine Ratio

x°x°

Sin x° =O

pp

osit

e

OppHyp

hypotenuse


ww

w.m

ath

srevis

ion

.com


Using a calculator to find sin ratio What Goes In The Box ?

Find the sin ratio to 3 decimal places.

sin 30o sin 0o sin 45o

sin 60o sin 20o sin 90o

= 0.500

= 0.000

= 0.707

= 0.866

= 0.342

= 1.000


ww

w.m

ath

srevis

ion

.com



ExampleExample

34°34°Sin x° =

OppOpp

Hyp

Hyphh

11c11cmm

Sin 34° =

h11= h11 x Sin

34°h = 11 x Sin

34°= 6.2cm (1 d.p.)

Find the height h

SOH CAH TOA

ww

w.m

ath

srevis

ion

.com



SOH CAH TOA


The support rope is 11.7m long. The angle between the rope and ground is 70o. Use the sine

ratio to calculate the height of the flag pole.

70o

h

o opp hsin 70 = =

hyp 11.7

h o= 11.7 sin70

h = 11 m

11.7m

ww

w.m

ath

srevis

ion

.com


SOH CAH TOA

ExampleExample

72°72°

Sin x° =Opp

Hyp

Sin 72° =

5r

r =

r =

5.3 km

5km5km

AB

C

r5sin72o

A road AB is right angled at B. The road BC is 5 km.

Calculate the length of the new road AC.


ww

w.m

ath

srevis

ion

.com



Now tryExtension Booklet

Ex4E (page 83)



w.m

ath

srevis

ion

.com

1. Calculate the mean median and mode f or the data below 19, 15, 11, 22, 9, 12, 11

2. The price of a car is £ 9000

I t is reduced by 20% in a sale.

What is the price of the car now?

3. A lorry is travelling at 40mph.

I t has travelled 60 miles. How long has it taken to travel 60 miles.



w.m

ath

srevis

ion


2.2. Use sine ratio to find angle Use sine ratio to find angle REAL-LIFE problems.REAL-LIFE problems.

1. To show how to use the sine ratio to find angle in REAL-LIFE problems.


1.1. Write down sine ratio.Write down sine ratio.


Using Sin to calculate anglesUsing Sin to calculate angles


w.m

ath

srevis

ion

.com


ww

w.m

ath

srevis

ion

.com


Using a calculator to find anglegiven sin ratio


Find the angle given the sin ratio to 1 decimal places.

sin-1 1 sin-1 0.5 sin-1 0.8

sin-1 0.6 sin-1 0.1 sin-1 0.9

= 90.0o = 30.0o

= 53.1o

= 36.9o = 5.7o = 64.2o


ww

w.m

ath

srevis

ion

.com


ExampleExample

x°x°

Sin x° =

Opp

Opp

Hyp

Hyp6m6m 9m9m

Sin x° =

69

= 0.667 (3 d.p.)

Sin x°

Find the xo

SOH CAH TOA


ww

w.m

ath

srevis

ion

.com



=0.667 (3 d.p.)Sin x°How do we find x°?

We need to use Sin ⁻¹on the calculator.

2nd

Sin ⁻¹is written above Sin

Sin ⁻¹

To get this press SinFollowed by

ww

w.m

ath

srevis

ion

.com



x =

Sin ⁻¹0.667 = 41.8° (1 d.p.)

= 0.667 (3 d.p.)

Sin x°

2nd Sin

Sin ⁻¹

Press

Enter =0.667

ww

w.m

ath

srevis

ion

.com



SOH CAH TOA


Use the sine ratio to find the angle of the ramp.

xo10m

o opp 10sin x = =

hyp 20

o 10sin x =

20

o -1 o10x = sin = 30

20

20 m

ww

w.m

ath

srevis

ion

.com




Ex5E (page 85)



w.m

ath

srevis

ion

.com

1. Calculate the mean median and mode f or the data below 9, 5, 11, 2, 9, 2

2. The price of a f ootball is £ 20.

I t is reduced in a sale by 80%.

What is the sale price.

3. A lorry is travelling at 50mph.

I t has travelled 75 miles.



w.m

ath

srevis

ion


2.2. Use cosine ratio to find a Use cosine ratio to find a length or angle.length or angle.

1. Definite the cosine ratio and show how to find an length or angle using this ratio.


1.1. Write down cosine ratio.Write down cosine ratio.


ww

w.m

ath

srevis

ion

.com


Using a calculator to find cos ratio What Goes In The Box ?

Find the cos ratio to 3 decimal places.

cos 30o cos 0o cos 45o

cos 60o cos 20o cos 90o

= 0.866

= 1.000

= 0.707

= 0.500

= 0.940

= 0.000


ww

w.m

ath

srevis

ion

.com


The Cosine The Cosine RatioRatio

Cos x° =

Adjacent

Adj

x°x°

Hyp

hypotenuse


ww

w.m

ath

srevis

ion

.com



SOH CAH TOA

ExampleExample

40°40°Cos x° =

Opp

Adj

Hyp Hyp

b

35mm

Cos 40° =

b35

= b

35 x Cos 40°

b =

35 x Cos 40°

= 26.8mm (1 d.p.)

Adj

Find the adjacent length b

ww

w.m

ath

srevis

ion

.com



Now tryExtension BookletEx6E (page 88 Q1 to

Q5)

18 Apr 202318 Apr 2023 Created by Mr.Lafferty Maths DeptCreated by Mr.Lafferty Maths Dept

Starter QuestionsStarter Questionsw

ww

.math

srevis

ion

.com

5 6 7 3 3

2. The ratio of apples to pears is 4:5.

I f there are 72 pieces of f ruit in total.

How many apples are there.

y x x x xy 1. Simplify

3. Find HCF of 15 and 21



w.m

ath

srevis

ion


2.2. Use cosine ratio to find a Use cosine ratio to find a length or angle.length or angle.

1. Definite the cosine ratio and show how to find an length or angle using this ratio.


1.1. Write down cosine ratio.Write down cosine ratio.


Using Cos to calculate anglesUsing Cos to calculate angles


w.m

ath

srevis

ion

.com


ww

w.m

ath

srevis

ion

.com


Using a calculator to find anglegiven cos ratio


Find the angle given the cos ratio to 1 decimal places.

cos-1 1 cos-1 0 cos-1 0.8

cos-1 0.6 cos-1 0.1 cos-1 0.9

= 0.0o = 90.0o

= 36.9o

= 53.1o = 84.3o = 25.8o


ww

w.m

ath

srevis

ion

.com


SOH CAH TOA

ExampleExample

x°x°Cos x° =

Opp

Adj

Hyp Hyp45cm

Cos x° = 3445= 0.756 (3 d.p.)Cos

x°x =

Cos ⁻¹0.756 =41°

Adj34cm34cm

Find the angle xo


ww

w.m

ath

srevis

ion

.com



Now tryExtension BookletEx6E (page 89 Q6 to

Q8)



w.m

ath

srevis

ion

.com

o

1. Calculate 104 x 100

putting your answer in standard f orm.

2. I s this triangle right angled ?

I f yes, fi nd the size of angle x .

I f no fi nd the area of the triangle.

xo

6

8

10


The Three RatiosThe Three Ratios


Cosine

Sine

Tangent

Sine

Sine

Tangent

Cosine

Cosine

Sine

ww

w.m

ath

srevis

ion

.com

opposite

opposite opposite

adjacent

adjacent

adjacent

hypotenuse

hypotenuse

hypotenuse


ww

w.m

ath

srevis

ion

.com


Sin x° =Opp

HypCos x° =

Adj

HypTan x° =

Opp

Adj

CAH TOASOH


ww

w.m

ath

srevis

ion

.com


SOH CAH TOA

Copy this!

1. Write down

Process

Identify what you want to find

what you know3.

2.


ww

w.m

ath

srevis

ion

.com

TrigonomeTrigonometrytryPast Paper Type Questions

SOH CAH TOA


ww

w.m

ath

srevis

ion

.com


(4 marks)

SOH CAH TOA


ww

w.m

ath

srevis

ion

.com


SOH CAH TOA


ww

w.m

ath

srevis

ion

.com


SOH CAH TOA

4 marks


ww

w.m

ath

srevis

ion

.com


SOH CAH TOA


ww

w.m

ath

srevis

ion

.com


(4marks)

SOH CAH TOA


ww

w.m

ath

srevis

ion

.com


SOH CAH TOA


ww

w.m

ath

srevis

ion

.com


(4marks)

SOH CAH TOA


ww

w.m

ath

srevis

ion

.com




Ex7E (page 91)

Documents

Trigonometry National 4 REL 1.3a REL 1.3b Investigating the Tangent Ratio The Sine Ratio (Finding a length) The Sine Ratio (Finding