S3 3 trigonomet

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TrigonomeTrigonometrytry

S3 Credit

The Tangent Ratio

The Tangent using Angle

The Sine of an Angle

The Sine Ration In Action

The Cosine of an Angle

Mixed Problems

The Tangent Ratio in Action

The Tangent (The Adjacent side)

The Tangent (Finding Angle)

The Sine ( Finding the Hypotenuse)

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Starter QuestionsStarter Questionsw

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1. An F1 car can complete a lap in 2 mins.

A lap is 5 miles in length.

Show that the average speed is 150mph

2. The resistance (R) in copper wire is directly

proportion to its length (L) and inversely to

the square of its radius (r).

Write down an f ormula connecting R, L and r.

S3 Credit

15 Apr 202315 Apr 2023 Created by Mr. Lafferty Maths Dept.Created by Mr. Lafferty Maths Dept.ww

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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.


Let’s Investigate!Let’s Investigate!


S3 Credit

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S3 Credit

Trigonometry means “triangle” and “measurement”.

AdjacentO

pp

osit

e

x°x°

hypotenuse

We will be using right-angled triangles.

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S3 Credit

30°

Adjacent

Op

posit

e

hypotenuse

OppositeAdjacent

= 0.6

Mathemagic!

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S3 Credit

45°

Adjacent

Op

posit

e

hypotenuse

OppositeAdjacent

= 1

Try another!

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S3 Credit

For an angle of 30°, OppositeAdjacent

= 0.6

We write tan 30° = 0.6

OppositeAdjacent

is called the tangent of an angle.

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Tan 25Tan 25°° 0.4660.466

Tan 26Tan 26°° 0.4880.488

Tan 27Tan 27°° 0.5100.510

Tan 28Tan 28°° 0.5320.532

Tan 29Tan 29°° 0.5540.554

Tan 30Tan 30°° 0.5770.577

Tan 31Tan 31°° 0.6010.601

Tan 32Tan 32°° 0.6250.625

Tan 33Tan 33° ° 0.6490.649

Tan 34Tan 34°° 0.6750.675

Tan 30° = 0.577

Accurate to 3 decimal places!

The ancient Greeks discovered this and repeated this for all possible angles.

S3 Credit

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S3 Credit

Now-a-days we can use calculators instead of tables to find the Tan of an angle.

TanOn your calculator press

Notice that your calculator is incredibly accurate!!

Followed by 30, and press

=

Accurate to 9 decimal places!

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S3 Credit

What’s the point of all this???

Don’t worry, you’re about to find out!

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S3 Credit

12 m

How high is the tower?

Opp

60°

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S3 Credit

60°

12 mAdjacent

Op

posit

e

hypotenuse Copy this!

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S3 Credit

Tan x° =

Opp

Adj

Tan 60° =

Opp12

= Opp

12 x Tan 60°Opp =

12 x Tan 60°= 20.8m (1 d.p.)

Copy this!

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S3 Credit

So the tower’s 20.8 m high!

Don’t worry, you’ll be trying plenty of examples!!

20.8m

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Adj

x°x°

Tan x° =O

pp

osit

e

Opp

Adjacent

S3 Credit

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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

S3 Credit

Find the height h

SOH CAH TOA

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S3 Credit

Class GroupIdentifying the

Tan Ratio Ex 3.1 & Ex4.1

MIA Page 203



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22 3 2

1. Find the area and perimeter of the circle.

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

3. Factorise x x

10cm

P

Q

R

6cm

7cm

10cm

S3 Credit


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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

S3 Credit

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1812

ExampleExample

x°x°

Tan x° =

OppOpp

Adj

Hyp

SOH CAH TOA

12m12m Tan x° =

= 1.5 Tan x°

Adj

18m18m

S3 Credit

Calculate the tan xo

ratio

Q

P

R

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S3 Credit

= 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

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S3 Credit

x =

Tan ⁻¹1.5 = 56.3° (1 d.p.)

= 1.5Tan x°

2nd Tan

Tan ⁻¹

Press

Enter =1.5

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S3 Credit

Process

1. Identify Hyp, Opp and Adj

2. Write down ratio Tan xo = Opp Adj

3. Calculate xo 2nd Tan

Tan ⁻¹

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S3 Credit

Now tryExercise 4.2

MIA Page 205



ww

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2. Write in scientifi c notation 0.0456

3. I dentif y the sides of the triangle.

4. The subway train takes 6 mins to travel

between 2 stations 3 miles apar

t.

Show that it's average speed is 30mph.

S3 Credit

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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.



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S3 Credit

SOH CAH TOA

15 Apr 202315 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

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S3 Credit

6o

15 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

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S3 Credit

Now tryExercise 5.1

MIA Page 207



ww

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2. Write in scientifi c notation 32.56

3. I dentif y the sides of the triangle.

4. The train takes 10 mins to travel between 2 stations 6miles apart.

Find the average speed of the train.

S3 Credit

xo


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2.2. Use tan of an angle to Use tan of an angle to solve find adjacent length.solve find adjacent length.

1. To use tan of the angle to find adjacent length.



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S3 Credit


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

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S3 Credit

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

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S3 Credit

Now tryExercise 5.2

MIA Page 210



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3 3 51. +

4 5 7

2. Explain why y=6 when a = (-1) b = 2

y = (a- b)(b- a)(2a- b)

pQ3. Given T =k .

v Find k when T =6 P = 18 and v =9.

S3 Credit


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

1. To show how to find an angle using tan ratio.



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S3 Credit


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

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S3 Credit


Use the tan ratio to find the angle of take-off.

xo 88m

o opp 88tan x = =

adj 500

otan x = 0.176

o -1 ox = tan (0.176) = 10

500 m

SOH CAH TOA

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S3 Credit

Now tryExercise 6.1

MIA Page 211



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1. A train takes 12 minutes to travel between 2 stations. Show that the average speed is 60km/ hr

if the stations are 6miles apart.

2. Calculate A when w= (-3) y = 4

A = (w - y) + (y - w)

S3 Credit


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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.

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TrigonomeTrigonometrytryThe Sine RatioThe Sine Ratio

x°x°

Sin x° =O

pp

osit

e

OppHyp

hypotenuse

S3 Credit

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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.)

S3 Credit

Find the height h

SOH CAH TOA


Using Sin to calculate anglesUsing Sin to calculate angles

S3 Credit

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ExampleExample

x°x°

Sin x° =

Opp

Opp

Hyp

Hyp6m6m 9m9m

Sin x° =

69

= 0.667 (3 d.p.)

Sin x°

S3 Credit

Find the xo

SOH CAH TOA

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S3 Credit

=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

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S3 Credit

x =

Sin ⁻¹0.667 = 41.8° (1 d.p.)

= 0.667 (3 d.p.)

Sin x°

2nd Sin

Sin ⁻¹

Press

Enter =0.667

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S3 Credit

Now tryExercise 7.1

MIA Page 212



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1. Explain why we can simply pick out Q

the 5 fi gure summary f or the data 19, 15, 11, 22, 9, 12, 11

2. Show that the original price of a car is £ 9000

I f it cos

, Q and Q

then fi nd

ts £ 8100 af ter a discount of 10%

3. A lorry is travelling at 40mph.

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

S3 Credit


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2.2. Use sine ratio to solve Use sine ratio to solve

REAL-LIFE problems.REAL-LIFE problems.

1. To show how to use the sine ratio to solve

REAL-LIFE problems.



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S3 Credit

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

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S3 Credit

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

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S3 Credit

Now tryExercise 7.2

MIA Page 214



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22 9 7 = (2 ?)( ?)

2

1. Fill in the ? marks.

2. Calculate A when w= (-10)

A = w4

x x x x

S3 Credit


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2.2. Use sine ratio to find the Use sine ratio to find the hypotenuse.hypotenuse.

1. To show how to calculate the hypotenuse using the sine ratio.



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SOH CAH TOA

ExampleExample

72°72°

Sin x° =Opp

Hyp

Sin 72° =

5r

r =

r =

5.3 km

5km5km

S3 Credit

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.

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S3 Credit

Now tryExercise 8.1

MIA Page 215



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1. Explain why we can simply pick out Q

then fi nd the 5 fi gure summary f or the data 9, 5, 11, 2, 9, 2

2. Find the original price of a f ootball

I f it costs £ 20 af ter a discoun

and Q

t of 80%

3. A lorry is travelling at 50mph.

I t has travelled 75 miles. Show that the time taken is 1hr 30 mins.

S3 Credit


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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.

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The Cosine The Cosine RatioRatio

Cos x° =

Adjacent

Adj

x°x°

Hyp

hypotenuse

S3 Credit

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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.)

AdjS3

Credit

Find the adjacent length b


Using Cos to calculate anglesUsing Cos to calculate angles

S3 Credit

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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

S3 Credit

Find the angle xo

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S3 Credit

Now tryExercise 9.1

MIA Page 216



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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.

S3 Credit

xo

6

8

10


The Three RatiosThe Three Ratios

Cosine

Sine

Tangent

Sine

Sine

Tangent

Cosine

Cosine

Sine

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S3 Credit

opposite

opposite opposite

adjacent

adjacent

adjacent

hypotenuse

hypotenuse

hypotenuse

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Sin x° =Opp

HypCos x° =

Adj

HypTan x° =

Opp

Adj

CAH TOASOH

S3 Credit

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SOH CAH TOA

Copy this!

S3 Credit

1. Write down

Process

Identify what you want to find

what you know3.

2.

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TrigonomeTrigonometrytryPast Paper Type Questions

S3 Credit

SOH CAH TOA

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S3 Credit

(4 marks)

SOH CAH TOA

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S3 Credit

SOH CAH TOA

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S3 Credit

SOH CAH TOA

4 marks

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S3 Credit

SOH CAH TOA

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S3 Credit

(4marks)

SOH CAH TOA

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S3 Credit

SOH CAH TOA

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S3 Credit

(4marks)

SOH CAH TOA

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S3 Credit

Now try Exercise 10.1 & 10.2

MIA Page 218

Self Improvement

S3 3 trigonomet