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)

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.

Trigonometry means “triangle” and “measurement”.

AdjacentO

pp

osit

e

x°x°

hypotenuse

We will be using right-angled triangles.

30°

Adjacent

Op

posit

e

hypotenuse

OppositeAdjacent

= 0.6

Mathemagic!

45°

Adjacent

Op

posit

e

hypotenuse

OppositeAdjacent

= 1

Try another!

For an angle of 30°, OppositeAdjacent

= 0.6

We write tan 30° = 0.6

OppositeAdjacent

is called the tangent of an angle.

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.

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!

What’s the point of all this???

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

12 m

How high is the tower?

Opp

60°

60°

12 mAdjacent

Op

posit

e

hypotenuse Copy this!

Tan x° =

Opp

Adj

Tan 60° =

Opp12

= Opp

12 x Tan 60°Opp =

12 x Tan 60°= 20.8m (1 d.p.)

Copy this!

So the tower’s 20.8 m high!

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

20.8m

Adj

x°x°

Tan x° =O

pp

osit

e

Opp

Adjacent

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

Find the height h

SOH CAH TOA

Learning IntentionLearning Intention Success CriteriaSuccess Criteria

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.

Angles & Angles & Triangles Triangles

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

Using Tan to calculate anglesUsing Tan to calculate angles

1812

ExampleExample

x°x°

Tan x° =

OppOpp

Adj

Hyp

SOH CAH TOA

12m12m Tan x° =

= 1.5 Tan x°

Adj

18m18m

Calculate the tan xo ratio

Q

P

R

= 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

x =

Tan ⁻¹1.5 = 56.3° (1 d.p.)

= 1.5Tan x°

2nd Tan

Tan ⁻¹

Press

Enter =1.5

Process

1. Identify Hyp, Opp and Adj

2. Write down ratio Tan xo = Opp Adj

3. Calculate xo 2nd Tan

Tan ⁻¹

Learning IntentionLearning Intention Success CriteriaSuccess Criteria

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.

Angles & Angles & Triangles Triangles

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

SOH CAH TOA

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

6o

20 Apr 2023

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

Learning IntentionLearning Intention Success CriteriaSuccess Criteria

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.

Angles & Angles & Triangles Triangles

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

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

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

Learning IntentionLearning Intention Success CriteriaSuccess Criteria

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.

Angles & Angles & Triangles Triangles

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

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

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

Learning IntentionLearning Intention Success CriteriaSuccess Criteria

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.

Angles & Angles & Triangles Triangles

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

The Sine RatioThe Sine Ratio

x°x°

Sin x° =O

pp

osit

e

OppHyp

hypotenuse

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

Using Sin to calculate anglesUsing Sin to calculate angles

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

=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

x =

Sin ⁻¹0.667 = 41.8° (1 d.p.)

= 0.667 (3 d.p.)

Sin x°

2nd Sin

Sin ⁻¹

Press

Enter =0.667

Learning IntentionLearning Intention Success CriteriaSuccess Criteria

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.

Angles & Angles & Triangles Triangles

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

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

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

Learning IntentionLearning Intention Success CriteriaSuccess Criteria

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.

Angles & Angles & Triangles Triangles

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

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.

Learning IntentionLearning Intention Success CriteriaSuccess Criteria

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.

Angles & Angles & Triangles Triangles

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

The Cosine The Cosine RatioRatio

Cos x° =

Adjacent

Adj

x°x°

Hyp

hypotenuse

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

Using Cos to calculate anglesUsing Cos to calculate angles

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

The Three RatiosThe Three Ratios

Cosine

Sine

Tangent

Sine

Sine

Tangent

Cosine

Cosine

Sine

opposite

opposite opposite

adjacent

adjacent

adjacent

hypotenuse

hypotenuse

hypotenuse

Sin x° =Opp

HypCos x° =

Adj

HypTan x° =

Opp

Adj

CAH TOASOH

SOH CAH TOA

Copy this!

1. Write down

Process

Identify what you want to find

what you know3.

2.

Past Paper Type Questions

SOH CAH TOA

Past Paper Type Questions

(4 marks)

SOH CAH TOA

Past Paper Type Questions

SOH CAH TOA

Past Paper Type Questions

SOH CAH TOA

4 marks

Past Paper Type Questions

SOH CAH TOA

Past Paper Type Questions

(4marks)

SOH CAH TOA

Past Paper Type Questions

SOH CAH TOA

Past Paper Type Questions

(4marks)

SOH CAH TOA