S3 3 trigonometry USES

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

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.

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

Credit

TrigonometryS3

Credit

Trigonometry means “triangle” and “measurement”.

AdjacentOpposite

We will be using right-angled triangles.

TrigonometryS3

Credit

Adjacent

Opposite

OppositeAdjacent

Mathemagic!

TrigonometryS3

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Adjacent

OppositeAdjacent

Try another!

TrigonometryS3

Credit

For an angle of 30°, OppositeAdjacent

We write tan 30° = 0.6

Opposite

Adjacentis called the tangent of an angle.

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

TrigonometryS3

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!

TrigonometryS3

Credit

What’s the point of all this???

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

TrigonometryS3

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How high is the tower?

TrigonometryS3

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12 mAdjacent

Opposite

Copy this!

TrigonometryS3

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Tan x° =Opp

Tan 60° =Opp

= Opp12 x Tan 60°

Opp =12 x Tan 60° = 20.8m (1 d.p.)

Copy this!

TrigonometryS3

Credit

So the tower’s 20.8 m high!

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

Trigonometry

Tan x° =

Opposite

Adjacent

S3 Credit

TrigonometryExample

Tan x° =

OppOpp

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

TrigonometryS3

Credit

Class GroupIdentifying the

Tan Ratio Ex 3.1 & Ex4.1

MIA Page 203

Starter Questionsw

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

S3 Credit

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

Trigonometry

Example

Tan x° =

SOH CAH TOA

12m Tan x° =

= 1.5Tan x°

S3 Credit

Calculate the tan xo ratio

TrigonometryS3

Credit

= 1.5Tan x°

How do we find x°?

We need to use Tan ⁻¹on the calculator.

Tan ⁻¹is written above Tan

Tan ⁻¹

To get this press TanFollowed by

Calculate the size of angle xo

TrigonometryS3

Credit

x = Tan ⁻¹1.5 = 56.3° (1 d.p.)

= 1.5Tan x°

2nd Tan

Tan ⁻¹

Enter =1.5

TrigonometryS3

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Process

1. Identify Hyp, Opp and Adj

2. Write down ratio Tan xo = Opp

3. Calculate xo 2nd Tan

Tan ⁻¹

TrigonometryS3

Credit

Now tryExercise 4.2

MIA Page 205

Starter Questionsw

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

Show that it's average speed is 30mph.

S3 Credit

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

TrigonometryS3

Credit

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.

o opp htan 47 = =

o htan 47 =

oh = 8× tan 47

h = 8.58m

TrigonometryS3

Credit

7-Sep-14Compiled by Mr. Lafferty Maths

Aeroplane

a = 15

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 =

oh = 15× tan 6

h =1.58km

SOH CAH TOA

TrigonometryS3

Credit

Now tryExercise 5.1

MIA Page 207

Starter Questionsw

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

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

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

Angles & Triangles

TrigonometryS3

Credit

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

12mladder

o opp 12tan 45 = =

tan 45

d =12m

SOH CAH TOA

TrigonometryS3

Credit

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 =

1.58d =

d =15 km

SOH CAH TOA

TrigonometryS3

Credit

Now tryExercise 5.2

MIA Page 210

Starter Questionsw

3 3 51. +

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

2. Use tan ratio to find an angle.

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

Angles & Triangles

TrigonometryS3

Credit

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

o opp 11tan x = =

o -1x = tan

o ox = 70

SOH CAH TOA

TrigonometryS3

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

SOH CAH TOA

TrigonometryS3

Credit

Now tryExercise 6.1

MIA Page 211

Starter Questionsw

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

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.

TrigonometryThe Sine Ratio

Sin x° =

Opposite

OppHyp

S3 Credit

TrigonometryExample

34°Sin x° =OppOpp

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

TrigonometryExample

Sin x° =

Sin x° =69

= 0.667 (3 d.p.)Sin x°

S3 Credit

Find the xo

SOH CAH TOA

TrigonometryS3

Credit

=0.667 (3 d.p.)Sin x°

How do we find x°?

We need to use Sin ⁻¹on the calculator.

Sin ⁻¹is written above SinSin ⁻¹

To get this press SinFollowed by

TrigonometryS3

Credit

x = Sin ⁻¹0.667 = 41.8° (1 d.p.)

= 0.667 (3 d.p.)Sin x°

2nd Sin

Sin ⁻¹

Enter =0.667

TrigonometryS3

Credit

Now tryExercise 7.1

MIA Page 212

Starter Questionsw

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

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

TrigonometryS3

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.

o opp hsin 70 = =

hyp 11.7

h o= 11.7 sin70

h =11 m

TrigonometryS3

Credit

SOH CAH TOA

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

o opp 10sin x = =

hyp 20

o 10sin x =

o -1 o10x = sin = 30

TrigonometryS3

Credit

Now tryExercise 7.2

MIA Page 214

Starter Questionsw

.com 22 9 7 = (2 ?)( ?)

1. Fill in the ? marks.

2. Calculate A when w= (-10)

x x x x

S3 Credit

2. Use sine ratio to find the hypotenuse.

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

Angles & Triangles

Trigonometry

SOH CAH TOA

Example

Sin x° =Opp

Sin 72° =5

r = 5.3 km

S3 Credit

sin72o

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

Calculate the length of the new road AC.

TrigonometryS3

Credit

Now tryExercise 8.1

MIA Page 215

Starter Questionsw

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

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

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.

TrigonometryThe Cosine Ratio

Cos x° =

Adjacent

S3 Credit

Trigonometry

SOH CAH TOA

Example

Cos x° = Opp

Cos 40° =b

= 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

Trigonometry

SOH CAH TOA

Example

x°Cos x° = O

Hyp45cm

Cos x° =3445

= 0.756 (3 d.p.)Cos x°

x = Cos ⁻¹0.756 =41°

S3 Credit

Find the angle xo

TrigonometryS3

Credit

Now tryExercise 9.1

MIA Page 216

Starter Questionsw

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

The Three Ratios

Cosine

Tangent

Cosine

S3 Credit

opposite

opposite opposite

adjacent

hypotenuse

Trigonometry

Sin x° =Opp

HypCos x° =

HypTan x° =

CAH TOASOH

S3 Credit

Trigonometry

SOH CAH TOA

Copy this!

S3 Credit

1. Write down

Process

Identify what you want to find

what you know3.

TrigonometryPast Paper Type Questions

S3 Credit

SOH CAH TOA

S3 Credit

(4 marks)

SOH CAH TOA

S3 Credit

SOH CAH TOA

S3 Credit

SOH CAH TOA

4 marks

S3 Credit

SOH CAH TOA

S3 Credit

(4marks)

SOH CAH TOA

S3 Credit

SOH CAH TOA

S3 Credit

(4marks)

SOH CAH TOA

Trigonometry

TrigonometryS3

Credit

Now try Exercise 10.1 & 10.2

MIA Page 218

S3 3 trigonometry USES

oppadjh8m tan

h88 x tan

opp1212 x tan

x xwww

tangent ratiothe tangent

actionthe tangent

anglethe tangent ratio

adjacent sidethe tangent

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