Investigating the Tangent Ratio

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TrigonomeTrigonometrytry

National 4 REL 1.3a REL 1.3b

Investigating the Tangent Ratio

The Sine Ratio

Trickier Questions

Mixed Problems / Exam Type Questions

The Tangent (Finding lengths)

The Tangent (Finding the Angle)

The Cosine

Starter QuestionsStarter Questions

21 Apr 202321 Apr 2023 Created by Mr.Lafferty Maths DeptCreated by Mr.Lafferty Maths Deptww

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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.


21 Apr 202321 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. We are learning how 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.


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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


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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


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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


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Adj

x°x°

Tan x° =O

pp

osit

e

Opp

Adjacent


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60°

12 mAdjacent

Op

posit

e

hypotenuse Copy this!

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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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Now Try TJ 4+

Ex19.1

Ch19 (Page 151)


www.mathsrevision.comwww.mathsrevision.comww

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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



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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. We are learning how to find a length using the tan ratio.


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


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Adj

x°x°

Tan x° =O

pp

osit

e

Opp

Adjacent


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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

National 4 REL 1.3 a + 1.3b

Find the height h

SOH CAH TOA

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SOH CAH TOA

21 Apr 202321 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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6o

21 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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Now Try TJ 4+

Ex19.2

Ch19 (Page 151)



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of the circle.

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

10cm

P

Q

R

6cm

7cm

10cm



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

1. We are learning how to find an angle using the tan ratio.


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


Using Tan to calculate anglesUsing Tan to calculate angles


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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


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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


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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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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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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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Now Try TJ 4+

Ex19.3

Ch19 (Page 153)


21 Apr 202321 Apr 2023 Created by Mr.Lafferty Maths DeptCreated by Mr.Lafferty Maths Deptww

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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



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

1. We are learning the sine ratio and how to find a length or 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


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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


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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


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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.)

Find the height h

SOH CAH TOA

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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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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.


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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


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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 Sin

Sin ⁻¹

To get this press SinFollowed by

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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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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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Now Try TJ 4+

Ex19.4

Ch19 (Page 155)

21 Apr 202321 Apr 2023 Created by Mr.Lafferty Maths DeptCreated by Mr.Lafferty Maths Dept

Starter QuestionsStarter Questionsw

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5 6 7 3 3




y x x x xy 1. Simplify




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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. We are learning the cosine ratio and how to find a length or angle using this ratio.


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


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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


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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


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The Cosine The Cosine RatioRatio

Cos x° =

Adjacent

Adj

x°x°

Hyp

hypotenuse


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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.)

Adj

Find the adjacent length b

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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

Find the angle xo


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Now Try TJ 4+

Ex19.5

Ch19 (Page 158)

21 Apr 202321 Apr 2023 Created by Mr.Lafferty Maths DeptCreated by Mr.Lafferty Maths Dept

Starter QuestionsStarter Questionsw

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5 6 7 3 3




y x x x xy 1. Simplify




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2.2. Use balancing method to Use balancing method to solve.solve.

1. We are learning to solve trickier problems using trig ratios.


1.1. Identify correct ratio.Identify correct ratio.


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SOH CAH TOA

ExampleExample

30°30°Tan x° =

12cm

Opp

Adj

b

Tan 30° =

12b

b =

20. 8 cm ( to 1 decimal place)

Find the length b

b = 12Tan 30o

Balancing

Method

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10


SOH CAH TOA

ExampleExample

20°20°Sin x° = 10m

Opp

Hypy

Sin 20° =y

y =

29.2 m ( to 1 decimal place)

Find the length y

y = 10Sin 20o

Balancing

Method

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14


SOH CAH TOA

ExampleExample

25°25°Cos x° =

14m

Adj

Hypk

Cos 25° =k

k =

15.4 m ( to 1 decimal place)

Find the length k

k = 14Cos 25o

Balancing

Method

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Now Try TJ 4+

Ex19.6

Ch19 (Page 160)



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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.

xo

6

8

10


The Three RatiosThe Three Ratios


Cosine

Sine

Tangent

Sine

Sine

Tangent

Cosine

Cosine

Sine

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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


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SOH CAH TOA

Copy this!

1. Write down

Process

Identify what you want to find

what you know3.

2.


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TrigonomeTrigonometrytryPast Paper Type Questions

SOH CAH TOA


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(4 marks)

SOH CAH TOA


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SOH CAH TOA


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SOH CAH TOA

4 marks


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SOH CAH TOA


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SOH CAH TOA


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(4marks)

SOH CAH TOA


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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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Now Try TJ 4+

Ex19.7

Ch19 (Page 162)

Documents

Investigating the Tangent Ratio