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TrigonomeTrigonometrytry
National 4 REL 1.3a REL 1.3b
Investigating the Tangent Ratio
The Sine Ratio (Finding a length)
The Sine Ratio (Finding an angle)
The Cosine of an Angle
Mixed Problems
The Tangent (Finding lengths)
The Tangent (Finding the Angle)
The Cosine ( Finding a length)
Starter QuestionsStarter Questions
18 Apr 202318 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.
National 4 REL 1.3a REL 1.3b
18 Apr 202318 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.
National 4 REL 1.3a REL 1.3b
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TrigonomeTrigonometrytry
National 4 REL 1.3a REL 1.3b
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
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TrigonomeTrigonometrytry
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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
National 4 REL 1.3a REL 1.3b
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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
National 4 REL 1.3a REL 1.3b
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TrigonomeTrigonometrytry
Adj
x°x°
Tan x° =O
pp
osit
e
Opp
Adjacent
National 4 REL 1.3a REL 1.3b
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TrigonomeTrigonometrytry
National 4 REL 1.3a REL 1.3b
60°
12 mAdjacent
Op
posit
e
hypotenuse Copy this!
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TrigonomeTrigonometrytry
National 4 REL 1.3a REL 1.3b
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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TrigonomeTrigonometrytry
National 4 REL 1.3a REL 1.3b
Tan Ratio Ex1
Ch16 Page 198
Starter QuestionsStarter Questions
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
National 4 REL 1.3a REL 1.3b
18 Apr 202318 Apr 2023 Created by Mr. Lafferty Maths Dept.Created by Mr. Lafferty Maths Dept.ww
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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.
Angles & Angles & Triangles Triangles
1.1. Write down tan ratio.Write down tan ratio.
National 4 REL 1.3a REL 1.3b
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TrigonomeTrigonometrytry
Adj
x°x°
Tan x° =O
pp
osit
e
Opp
Adjacent
National 4 REL 1.3a REL 1.3b
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TrigonomeTrigonometrytry
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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TrigonomeTrigonometrytry
National 4 REL 1.3a REL 1.3b
SOH CAH TOA
18 Apr 202318 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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TrigonomeTrigonometrytry
National 4 REL 1.3a REL 1.3b
6o
18 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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TrigonomeTrigonometrytry
National 4 REL 1.3a REL 1.3b
18 Apr 202318 Apr 2023 Compiled by Mr. Lafferty Maths Dept.Compiled by Mr. Lafferty Maths Dept.
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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TrigonomeTrigonometrytry
National 4 REL 1.3a REL 1.3b
6o
18 Apr 202318 Apr 2023 Compiled by Mr. Lafferty Maths Dept.Compiled by Mr. Lafferty Maths Dept.
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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TrigonomeTrigonometrytry
National 4 REL 1.3a REL 1.3b
Now tryExercise 2
Ch16 (page 199)
Starter QuestionsStarter Questions
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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
National 4 REL 1.3a REL 1.3b
18 Apr 202318 Apr 2023 Created by Mr. Lafferty Maths Dept.Created by Mr. Lafferty Maths Dept.ww
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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.
Angles & Angles & Triangles Triangles
1.1. Write down tan ratio.Write down tan ratio.
National 4 REL 1.3a REL 1.3b
Using Tan to calculate anglesUsing Tan to calculate angles
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National 4 REL 1.3a REL 1.3b
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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
National 4 REL 1.3a REL 1.3b
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TrigonomeTrigonometrytry
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
National 4 REL 1.3a REL 1.3b
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TrigonomeTrigonometrytry
National 4 REL 1.3a REL 1.3b
= 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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TrigonomeTrigonometrytry
National 4 REL 1.3a REL 1.3b
x =
Tan ⁻¹1.5 = 56.3° (1 d.p.)
= 1.5Tan x°
2nd Tan
Tan ⁻¹
Press
Enter =1.5
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TrigonomeTrigonometrytry
National 4 REL 1.3a REL 1.3b
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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TrigonomeTrigonometrytry
National 4 REL 1.3a REL 1.3b
18 Apr 202318 Apr 2023 Compiled by Mr. Lafferty Maths Dept.Compiled by Mr. Lafferty Maths Dept.
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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TrigonomeTrigonometrytry
National 4 REL 1.3a REL 1.3b
Now tryExercise 3
Ch16 Page 202
Starter QuestionsStarter Questions
18 Apr 202318 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
National 4 REL 1.3a REL 1.3b
18 Apr 202318 Apr 2023 Created by Mr. Lafferty Maths Dept.Created by Mr. Lafferty Maths Dept.ww
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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.
Angles & Angles & Triangles Triangles
1.1. Write down sine ratio.Write down sine ratio.
National 4 REL 1.3a REL 1.3b
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TrigonomeTrigonometrytryThe Sine RatioThe Sine Ratio
x°x°
Sin x° =O
pp
osit
e
OppHyp
hypotenuse
National 4 REL 1.3a REL 1.3b
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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
National 4 REL 1.3a REL 1.3b
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TrigonomeTrigonometrytry
National 4 REL 1.3a REL 1.3b
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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TrigonomeTrigonometrytry
National 4 REL 1.3a REL 1.3b
SOH CAH TOA
18 Apr 202318 Apr 2023 Compiled by Mr. Lafferty Maths Dept.Compiled by Mr. Lafferty Maths Dept.
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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TrigonomeTrigonometrytry
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.
National 4 REL 1.3a REL 1.3b
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TrigonomeTrigonometrytry
National 4 REL 1.3a REL 1.3b
Now tryExtension Booklet
Ex4E (page 83)
Starter QuestionsStarter Questions
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1. Calculate the mean median and mode f or the data below 19, 15, 11, 22, 9, 12, 11
2. The price of a car is £ 9000
I t is reduced by 20% in a sale.
What is the price of the car now?
3. A lorry is travelling at 40mph.
I t has travelled 60 miles. How long has it taken to travel 60 miles.
National 4 REL 1.3a REL 1.3b
18 Apr 202318 Apr 2023 Created by Mr. Lafferty Maths Dept.Created by Mr. Lafferty Maths Dept.ww
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2.2. Use sine ratio to find angle Use sine ratio to find angle REAL-LIFE problems.REAL-LIFE problems.
1. To show how to use the sine ratio to find angle in REAL-LIFE problems.
Angles & Angles & Triangles Triangles
1.1. Write down sine ratio.Write down sine ratio.
National 4 REL 1.3a REL 1.3b
Using Sin to calculate anglesUsing Sin to calculate angles
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National 4 REL 1.3a REL 1.3b
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Using a calculator to find anglegiven sin ratio
What Goes In The Box ?
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
National 4 REL 1.3a REL 1.3b
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TrigonomeTrigonometrytry
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
National 4 REL 1.3a REL 1.3b
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TrigonomeTrigonometrytry
National 4 REL 1.3a REL 1.3b
=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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TrigonomeTrigonometrytry
National 4 REL 1.3a REL 1.3b
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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TrigonomeTrigonometrytry
National 4 REL 1.3a REL 1.3b
SOH CAH TOA
18 Apr 202318 Apr 2023 Compiled by Mr. Lafferty Maths Dept.Compiled by Mr. Lafferty Maths Dept.
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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TrigonomeTrigonometrytry
National 4 REL 1.3a REL 1.3b
Now tryExtension Booklet
Ex5E (page 85)
Starter QuestionsStarter Questions
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1. Calculate the mean median and mode f or the data below 9, 5, 11, 2, 9, 2
2. The price of a f ootball is £ 20.
I t is reduced in a sale by 80%.
What is the sale price.
3. A lorry is travelling at 50mph.
I t has travelled 75 miles.
National 4 REL 1.3a REL 1.3b
18 Apr 202318 Apr 2023 Created by Mr. Lafferty Maths Dept.Created by Mr. Lafferty Maths Dept.ww
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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.
Angles & Angles & Triangles Triangles
1.1. Write down cosine ratio.Write down cosine ratio.
National 4 REL 1.3a REL 1.3b
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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
National 4 REL 1.3a REL 1.3b
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TrigonomeTrigonometrytry
The Cosine The Cosine RatioRatio
Cos x° =
Adjacent
Adj
x°x°
Hyp
hypotenuse
National 4 REL 1.3a REL 1.3b
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TrigonomeTrigonometrytry
National 4 REL 1.3a REL 1.3b
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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TrigonomeTrigonometrytry
National 4 REL 1.3a REL 1.3b
Now tryExtension BookletEx6E (page 88 Q1 to
Q5)
18 Apr 202318 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
2. The ratio of apples to pears is 4:5.
I f there are 72 pieces of f ruit in total.
How many apples are there.
y x x x xy 1. Simplify
3. Find HCF of 15 and 21
National 4 REL 1.3a REL 1.3b
18 Apr 202318 Apr 2023 Created by Mr. Lafferty Maths Dept.Created by Mr. Lafferty Maths Dept.ww
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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.
Angles & Angles & Triangles Triangles
1.1. Write down cosine ratio.Write down cosine ratio.
National 4 REL 1.3a REL 1.3b
Using Cos to calculate anglesUsing Cos to calculate angles
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National 4 REL 1.3a REL 1.3b
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Using a calculator to find anglegiven cos ratio
What Goes In The Box ?
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
National 4 REL 1.3a REL 1.3b
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TrigonomeTrigonometrytry
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
National 4 REL 1.3a REL 1.3b
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TrigonomeTrigonometrytry
National 4 REL 1.3a REL 1.3b
Now tryExtension BookletEx6E (page 89 Q6 to
Q8)
Starter QuestionsStarter Questions
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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
National 4 REL 1.3a REL 1.3b
The Three RatiosThe Three Ratios
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Cosine
Sine
Tangent
Sine
Sine
Tangent
Cosine
Cosine
Sine
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opposite
opposite opposite
adjacent
adjacent
adjacent
hypotenuse
hypotenuse
hypotenuse
National 4 REL 1.3a REL 1.3b
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TrigonomeTrigonometrytry
Sin x° =Opp
HypCos x° =
Adj
HypTan x° =
Opp
Adj
CAH TOASOH
National 4 REL 1.3a REL 1.3b
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TrigonomeTrigonometrytry
SOH CAH TOA
Copy this!
1. Write down
Process
Identify what you want to find
what you know3.
2.
National 4 REL 1.3a REL 1.3b
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TrigonomeTrigonometrytryPast Paper Type Questions
SOH CAH TOA
National 4 REL 1.3a REL 1.3b
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TrigonomeTrigonometrytryPast Paper Type Questions
(4 marks)
SOH CAH TOA
National 4 REL 1.3a REL 1.3b
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TrigonomeTrigonometrytryPast Paper Type Questions
SOH CAH TOA
National 4 REL 1.3a REL 1.3b
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TrigonomeTrigonometrytryPast Paper Type Questions
SOH CAH TOA
4 marks
National 4 REL 1.3a REL 1.3b
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TrigonomeTrigonometrytryPast Paper Type Questions
SOH CAH TOA
National 4 REL 1.3a REL 1.3b
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TrigonomeTrigonometrytryPast Paper Type Questions
(4marks)
SOH CAH TOA
National 4 REL 1.3a REL 1.3b
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TrigonomeTrigonometrytryPast Paper Type Questions
SOH CAH TOA
National 4 REL 1.3a REL 1.3b
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TrigonomeTrigonometrytryPast Paper Type Questions
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SOH CAH TOA
National 4 REL 1.3a REL 1.3b
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TrigonomeTrigonometrytry
National 4 REL 1.3a REL 1.3b
Now tryExtension Booklet
Ex7E (page 91)