S3 3 trigonometry USES

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TrigonometryS3

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)

www.mathsrevision.com

Starter 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 formula connecting R, L and r.

S3 Credit

7-Sep-14 Created by Mr. Lafferty Maths Dept.

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Learning Intention Success Criteria

2. Work out Tan Ratio.

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

Angles & Triangles

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


Let’s Investigate!

TrigonometryS3

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Trigonometry means “triangle” and “measurement”.

AdjacentOpposite

x°

We will be using right-angled triangles.

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30°

Adjacent

Opposite

OppositeAdjacent

= 0.6

Mathemagic!

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45°

Adjacent

Opp

osite

OppositeAdjacent

= 1

Try another!

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For an angle of 30°, OppositeAdjacent

= 0.6

We write tan 30° = 0.6

Opposite

Adjacentis called the tangent of an angle.

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Trigonometry

Tan 25° 0.466

Tan 26° 0.488

Tan 27° 0.510

Tan 28° 0.532

Tan 29° 0.554

Tan 30° 0.577

Tan 31° 0.601

Tan 32° 0.625

Tan 33° 0.649

Tan 34° 0.675

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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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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What’s the point of all this???

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

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12 m

How high is the tower?

Opp

60°

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60°

12 mAdjacent

Opposite

Copy this!

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TrigonometryS3

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Tan x° =Opp

Adj

Tan 60° =Opp

12

= Opp12 x Tan 60°

Opp =12 x Tan 60° = 20.8m (1 d.p.)

Copy this!

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TrigonometryS3

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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Trigonometry

Adj

x°

Tan x° =

Opposite

Opp

Adjacent

S3 Credit

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TrigonometryExample

65°

Tan x° =

OppOpp

Adjh

8m Tan 65° =h8

= h8 x Tan 65°

h = 8 x Tan 65° = 17.2m (1 d.p.)

S3 Credit

Find the height h

SOH CAH TOA

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TrigonometryS3

Credit

Class GroupIdentifying the

Tan Ratio Ex 3.1 & Ex4.1

MIA Page 203


Starter Questionsw

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22 3 2

1. Find the area and perimeter

of the circle.

2. Is 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. Use tan of an angle to solve problems.

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

Angles & Triangles

1. Write down tan ratio.


Using Tan to calculate angles

S3 Credit

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Trigonometry

1812

Example

x°

Tan x° =

Opp

Opp

Adj

SOH CAH TOA

12m Tan x° =

= 1.5Tan x°

18m

S3 Credit

Calculate the tan xo ratio

Q

P

R

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= 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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TrigonometryS3

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x = Tan ⁻¹1.5 = 56.3° (1 d.p.)

= 1.5Tan x°

2nd Tan

Tan ⁻¹

Press

Enter =1.5

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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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TrigonometryS3

Credit

Now tryExercise 4.2

MIA Page 205


Starter Questionsw

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1. True or false 30 5 + 6 4 = 48

2. Write in scientific notation 0.0456

3. Identify 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

xo


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

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

Angles & Triangles


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TrigonometryS3

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SOH CAH TOA

7-Sep-14 Compiled by Mr. Lafferty Maths Dept.

Use the tan ratio to find the height h of the treeto 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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TrigonometryS3

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6o

7-Sep-14Compiled by Mr. Lafferty Maths

Dept.

Aeroplane

a = 15

c

LennoxtownAirport

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

What is the height of the plane ?

Example 2

o htan 6 =

15

oh = 15× tan 6

h =1.58km

SOH CAH TOA

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TrigonometryS3

Credit

Now tryExercise 5.1

MIA Page 207


Starter Questionsw

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1. Explain why 6 + 9 3 = 9 and not 5

2. Write in scientific notation 32.56

3. Identify 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. Use tan of an angle to solve find adjacent length.

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

Angles & Triangles


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TrigonometryS3

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

45o

12mladder

o opp 12tan 45 = =

adj d

o

12d =

tan 45

d =12m

d m

SOH CAH TOA

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TrigonometryS3

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6o


Aeroplane

a = 1.58 km

LennoxtownAirport

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 2

o 1.58tan 6 =

d

o

1.58d =

tan 6

d =15 km

SOH CAH TOA

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TrigonometryS3

Credit

Now tryExercise 5.2

MIA Page 210


Starter Questionsw

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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. Use tan ratio to find an angle.

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

Angles & Triangles


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TrigonometryS3

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

xo

11m

o opp 11tan x = =

adj 4

11

4

o -1x = tan

o ox = 70

4 m

SOH CAH TOA

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TrigonometryS3

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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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TrigonometryS3

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Now tryExercise 6.1

MIA Page 211


Starter Questionsw

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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. Use sine ratio to find an angle.

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

Angles & Triangles

1. Write down sine ratio.

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TrigonometryThe Sine Ratio

x°

Sin x° =

Opposite

OppHyp

S3 Credit

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TrigonometryExample

34°Sin x° =OppOpp

Hyp

h 11cm

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 angles

S3 Credit

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TrigonometryExample

x°

Sin x° =

Opp

Opp

Hyp

6m 9m

Sin x° =69

= 0.667 (3 d.p.)Sin x°

S3 Credit

Find the xo

SOH CAH TOA

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TrigonometryS3

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=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 SinSin ⁻¹

To get this press SinFollowed by

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TrigonometryS3

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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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TrigonometryS3

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Now tryExercise 7.1

MIA Page 212


Starter Questionsw

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1 2 31. Explain why we can simply pick out Q

the 5 figure summary for the data

19, 15, 11, 22, 9, 12, 11

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

If it cos

, Q and Q

then find

ts £8100 after a discount of 10%

3. A lorry is travelling at 40mph.

It has travelled 60 miles.

How long has it taken to travel 60 miles.

S3 Credit


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2. Use sine ratio to solve

REAL-LIFE problems.

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

REAL-LIFE problems.

Angles & Triangles


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TrigonometryS3

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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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TrigonometryS3

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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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TrigonometryS3

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Now tryExercise 7.2

MIA Page 214


Starter Questionsw

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2

1. Fill in the ? marks.

2. Calculate A when w= (-10)

A = w

4

x x x x

S3 Credit


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2. Use sine ratio to find the hypotenuse.

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

Angles & Triangles


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Trigonometry

SOH CAH TOA

Example

72°

Sin x° =Opp

Hyp

Sin 72° =5

r

r =

r = 5.3 km

5km

S3 Credit

AB

C

r5

sin72o

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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TrigonometryS3

Credit

Now tryExercise 8.1

MIA Page 215


Starter Questionsw

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1 31. Explain why we can simply pick out Q

then find the 5 figure summary for the data

9, 5, 11, 2, 9, 2

2. Find the original price of a football

If it costs £20 after a discoun

and Q

t of 80%

3. A lorry is travelling at 50mph.

It has travelled 75 miles.

Show that the time taken is 1hr 30 mins.

S3 Credit


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

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

Angles & Triangles

1. Write down cosine ratio.

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TrigonometryThe Cosine Ratio

Cos x° =

Adjacent

Adj

x°

Hyp

S3 Credit

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Trigonometry

SOH CAH TOA

Example

40°

Cos x° = Opp

Adj

Hyp

b

35mm

Cos 40° =b

35

= b35 x Cos 40°

b = 35 x Cos 40°= 26.8mm (1 d.p.)

S3 Credit

Find the adjacent length b


Using Cos to calculate angles

S3 Credit

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Trigonometry

SOH CAH TOA

Example

x°Cos x° = O

pp

Adj

Hyp45cm

Cos x° =3445

= 0.756 (3 d.p.)Cos x°

x = Cos ⁻¹0.756 =41°

34cm

S3 Credit

Find the angle xo

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TrigonometryS3

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Now tryExercise 9.1

MIA Page 216


Starter Questionsw

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o

1. Calculate 104 x 100

putting your answer in standard form.

2. Is this triangle right angled ?

If yes, find the size of angle x .

If no find the area of the triangle.

S3 Credit

xo

6

8

10


The 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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Trigonometry

Sin x° =Opp

HypCos x° =

Adj

HypTan x° =

Opp

Adj

CAH TOASOH

S3 Credit

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Trigonometry

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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TrigonometryPast 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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Trigonometry

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Trigonometry

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TrigonometryS3

Credit

Now try Exercise 10.1 & 10.2

MIA Page 218

Education

S3 3 trigonometry USES