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National 4
REL 1.3a
REL 1.3b
Trigonometry
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Investigating the Tangent Ratio
The Tangent (Finding lengths)
The Tangent (Finding the Angle)
The Sine Ratio (Finding a length)
The Sine Ratio (Finding an angle)
The Cosine ( Finding a length)
The Cosine of an Angle
Mixed Problems
Starter Questions
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National 4
REL 1.3a
REL 1.3b
2
66 % of £270
3
Q1.
Calculate
Q2.
Convert 60% to fraction and simplify.
Q3.
Convert
Q4.
What is the time difference 09:28 and 10:50
Q5.
The answer to the question is 90. What is the
question.
5-May-17
4
5
to a percentage.
Created by Mr.Lafferty Maths Dept
Angles & Triangles
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National 4
REL 1.3a
REL 1.3b
Learning Intention
1. To identify the
hypotenuse, opposite and
adjacent sides in a right
angled triangle.
Success Criteria
1. Understand the terms
hypotenuse, opposite and
adjacent in right angled
triangle.
2. Work out Tan Ratio.
5-May-17
Created by Mr. Lafferty Maths Dept.
Trigonometry
Trigonometry means “triangle” and
“measurement”.
We will be using right-angled triangles.
Opposite
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National 4
REL 1.3a
REL 1.3b
x°
Adjacent
These
One triangles
is 37o
have
The 2other
things
wein
will common
investigate
Triangle
Vertical side
A
3cm
B
6cm
Horizontal side
4cm
8cm
Vertical
Horizontal
3
6
 0.75
 0.75
4
8
C
4cm
D
1.5cm
2cm
6cm
4.5
3
 0.75
 0.75
6
4
Right-Angled Triangle
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National 4
REL 1.3a
REL 1.3b
Trigonometry
For the RAT with angle of 37o
the answer obtained by dividing vertical side
by horizontal side was
ALWAYS 0.75
tan37o
The number has a special name
It is called the TANGENT of 37o
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National 4
REL 1.3a
REL 1.3b
Using a calculator to find tan ratio
What Goes In The Box ?
Find the tan ratio to 3 decimal places.
tan 0o
tan 45o
= 0.577
= 0.000
= 1.000
tan 60o
tan 20o
tan 70o
= 1.732
= 0.364
= 2.747
tan
30o
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Trigonometry
Opp
Tan x° =
Adj
Opposite
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National 4
REL 1.3a
REL 1.3b
x°
Adjacent
Trigonometry
Opposite
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National 4
REL 1.3a
REL 1.3b
Copy this!
60°
12 m
Adjacent
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National 4
REL 1.3a
REL 1.3b
Trigonometry
Opp
Tan x° =
Adj
Opp
Tan 60° =
12
12 x Tan 60° = Opp
Opp =12 x Tan 60° = 20.8m (1 d.p.)
Copy this!
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National 4
REL 1.3a
REL 1.3b
Trigonometry
Tan Ratio Ex1
Ch16 Page 198
Starter Questions
National 4
REL 1.3a
REL 1.3b
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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. A train takes 6 hrs to travel
360 miles apart.
Calculate it's average speed.
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xo
Angles & Triangles
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National 4
REL 1.3a
REL 1.3b
Learning Intention
1. To use tan of the angle to
solve REAL LIFE problems.
Success Criteria
1. Write down tan ratio.
2. Use tan of an angle to solve
REAL LIFE problems.
5-May-17
Created by Mr. Lafferty Maths Dept.
Trigonometry
Opp
Tan x° =
Adj
Opposite
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National 4
REL 1.3a
REL 1.3b
x°
Adjacent
Example
Find the
height h
Trigonometry
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National 4 REL 1.3 a + 1.3b
h
65°
8m
Opp
SOH CAH TOA
Opp
Tan x° =
Adj
Tan 65° =
h
8
8 x Tan 65° = h
h = 8 x Tan 65° = 17.2m (1 d.p.)
Trigonometry
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National 4
REL 1.3a
REL 1.3b
Use the tan ratio to find the height h of the tree
to 2 decimal places.
tan 47o =
opp h
=
adj 8
tan 47o =
h
8
SOH CAH TOA
rod
h = 8 × tan 47o
h = 8.58m
5-May-17
47o
Compiled by Mr. Lafferty Maths Dept.
8m
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National 4
REL 1.3a
REL 1.3b
Trigonometry
SOH CAH TOA
Example 2
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.
5-May-17
What is the height of the plane ?
tan 6o =
h
15
c
h = 15 × tan 6o
h = 1.58km
Aeroplane
6o
Airport
Compiled by Mr. Lafferty Maths
Dept.
a = 15
Lennoxtown
Trigonometry
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National 4
REL 1.3a
REL 1.3b
Use the tan ratio to calculate how far the ladder
is away from the building.
opp 12
tan 45 =
=
adj
d
o
12
d=
tan 45o
SOH CAH TOA
ladder
45o
d = 12m
5-May-17
Compiled by Mr. Lafferty Maths Dept.
dm
12m
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National 4
REL 1.3a
REL 1.3b
Trigonometry
Example 2
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?
tan 6o =
1.58
d
SOH CAH TOA
1.58
d=
tan 6o
d = 15 km
5-May-17
Aeroplane
a = 1.58 km
6o
Airport
Compiled by Mr. Lafferty Maths Dept.
Lennoxtown
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National 4
REL 1.3a
REL 1.3b
Trigonometry
Now try
Exercise 2
Ch16 (page 199)
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National 4
REL 1.3a
REL 1.3b
Starter Questions
1. Find the area and perimeter
10cm
of the circle.
2. Is the triangle right-angled at P.
Q
Explain your answer.
6cm
P
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10cm
7cm
R
Angles & Triangles
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National 4
REL 1.3a
REL 1.3b
Learning Intention
1. To use tan of the angle to
solve problems.
Success Criteria
1. Write down tan ratio.
2. Use tan of an angle to solve
problems.
5-May-17
Created by Mr. Lafferty Maths Dept.
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National 4
REL 1.3a
REL 1.3b
Using Tan to calculate angles
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National 4
REL 1.3a
REL 1.3b
Using a calculator to find angle
given tan ratio
What Goes In The Box ?
Find the angle given the tan ratio to 1 decimal
places.
tan-1 0.5
tan-1 0.8
= 45.0o
= 26.6o
= 38.7o
tan-1 0.6
tan-1 2
tan-1 3
= 31.0o
= 64.4o
= 71.6o
tan-1
1
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Example
Trigonometry
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National 4
REL 1.3a
REL 1.3b
P
SOH CAH TOA
Opp
18m
R
Calculate the
tan xo ratio
x°
12m
Q
Opp
Tan x° =
Adj
Tan x° =
18
12
Tan x° = 1.5
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National 4
REL 1.3a
REL 1.3b
Calculate the
size of
angle xo
Trigonometry
Tan x° = 1.5
How do we find x°?
We need to use Tan ⁻¹on the
calculator.
Tan ⁻¹is written above
To get this press
2nd
Tan ⁻¹
Tan
Followed by
Tan
National 4
REL 1.3a
REL 1.3b
Trigonometry
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Tan x° = 1.5
Press
2nd
Enter 1.5
Tan ⁻¹
Tan
=
x = Tan ⁻¹1.5 = 56.3° (1 d.p.)
Trigonometry
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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 ⁻¹
Tan
Trigonometry
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National 4
REL 1.3a
REL 1.3b
Use the tan ratio to calculate the angle that the
support wire makes with the ground.
opp 11
tan x =
=
adj
4
o
SOH CAH TOA
 11 
x = tan  
4
o
-1
11m
x o = 70o
5-May-17
Compiled by Mr. Lafferty Maths Dept.
xo
4m
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National 4
REL 1.3a
REL 1.3b
Trigonometry
Now try
Exercise 3
Ch16 Page 202
Starter Questions
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National 4
REL 1.3a
REL 1.3b
1. Simplify a  4a  13d  11a  a  3d
2. The ratio of apples to pears is 2 : 5.
If there are 28 pieces of fruit in total.
How many apples are there.
3.
5-May-17
Find HCF of 8 and 18
Created by Mr.Lafferty Maths Dept
Angles & Triangles
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National 4
REL 1.3a
REL 1.3b
Learning Intention
1. Definite the sine ratio and
show how to find an angle
using this ratio.
Success Criteria
1. Write down sine ratio.
2. Use sine ratio to find an
angle.
5-May-17
Created by Mr. Lafferty Maths Dept.
Trigonometry
The Sine Ratio
Sin x° =
Opposite
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National 4
REL 1.3a
REL 1.3b
x°
Opp
Hyp
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National 4
REL 1.3a
REL 1.3b
Using a calculator to find sin ratio
What Goes In The Box ?
Find the sin ratio to 3 decimal places.
sin 0o
sin 45o
= 0.500
= 0.000
= 0.707
sin 60o
sin 20o
sin 90o
= 0.866
= 0.342
= 1.000
sin
30o
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Find the
National 4 height h
Example
Trigonometry
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REL 1.3a
REL 1.3b
h
Opp
Opp
Sin x° =
Hyp
Sin 34° =
11cm
34°
h
11
SOH CAH TOA
11 x Sin 34° = h
h = 11 x Sin 34° = 6.2cm (1 d.p.)
Trigonometry
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National 4
REL 1.3a
REL 1.3b
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.
sin 70o =
opp
h
=
hyp 11.7
SOH CAH TOA
h = 11.7  sin70o
11.7m
h = 11m
5-May-17
Compiled by Mr. Lafferty Maths Dept.
70o
h
Example
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National 4
REL 1.3a
REL 1.3b
Trigonometry
SOH CAH TOA
Opp
Sin x° =
Hyp
Sin 72° =
r=
5
r
5
sin 72o
r = 5.3 km
A road AB is right angled at B.
The road BC is 5 km.
Calculate the length of
the new road AC.
B
5km
C
72°
r
A
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National 4
REL 1.3a
REL 1.3b
Trigonometry
Now try
Extension Booklet
Ex4E (page 83)
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National 4
REL 1.3a
REL 1.3b
Starter Questions
1. Calculate the mean median and mode
for the data below
19, 15, 11, 22, 9, 12, 11
2. The price of a car is £9000
It is reduced by 20% in a sale.
What is the price of the car now?
3. A lorry is travelling at 40mph.
It has travelled 60 miles.
How long has it taken to travel 60 miles.
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Angles & Triangles
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National 4
REL 1.3a
REL 1.3b
Learning Intention
1. To show how to use the
sine ratio to find angle in
REAL-LIFE problems.
Success Criteria
1. Write down sine ratio.
2. Use sine ratio to find angle
REAL-LIFE problems.
5-May-17
Created by Mr. Lafferty Maths Dept.
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National 4
REL 1.3a
REL 1.3b
Using Sin to calculate angles
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National 4
REL 1.3a
REL 1.3b
Using a calculator to find angle
given sin ratio
What Goes In The Box ?
Find the angle given the sin ratio to 1 decimal places.
sin-1 0.5
sin-1 0.8
= 90.0o
= 30.0o
= 53.1o
sin-1 0.6
sin-1 0.1
sin-1 0.9
= 36.9o
= 5.7o
= 64.2o
sin-1
1
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Example
Trigonometry
National 4
REL 1.3a
REL 1.3b
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Find the xo
6m
Opp
9m
Opp
Sin x° =
Hyp
6
Sin x° =
9
x°
SOH CAH TOA
Sin x° = 0.667 (3 d.p.)
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National 4
REL 1.3a
REL 1.3b
Trigonometry
Sin x° =0.667
(3 d.p.)
How do we find x°?
We need to use Sin ⁻¹on the
calculator.
Sin ⁻¹is written above
To get this press
2nd
Sin ⁻¹
Sin
Followed by
Sin
Trigonometry
National 4
REL 1.3a
REL 1.3b
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Sin x° = 0.667 (3 d.p.)
Press
2nd
Enter 0.667
Sin ⁻¹
Sin
=
x = Sin ⁻¹0.667 = 41.8° (1 d.p.)
Trigonometry
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National 4
REL 1.3a
REL 1.3b
Use the sine ratio to find the angle of the ramp.
SOH CAH TOA
opp 10
sin x =
=
hyp 20
o
10
sin x =
20
o
 10 
o
x = sin 
=
30

 20 
o
5-May-17
-1
20 m
xo
Compiled by Mr. Lafferty Maths Dept.
10m
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National 4
REL 1.3a
REL 1.3b
Trigonometry
Now try
Extension Booklet
Ex5E (page 85)
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National 4
REL 1.3a
REL 1.3b
Starter Questions
1. Calculate the mean median and mode
for the data below
9, 5, 11, 2, 9, 2
2. The price of a football is £20.
It is reduced in a sale by 80%.
What is the sale price.
3. A lorry is travelling at 50mph.
It has travelled 75 miles.
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Angles & Triangles
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National 4
REL 1.3a
REL 1.3b
Learning Intention
1. Definite the cosine ratio
and show how to find an
length or angle using this
ratio.
Success Criteria
1. Write down cosine ratio.
2. Use cosine ratio to find a
length or angle.
5-May-17
Created by Mr. Lafferty Maths Dept.
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National 4
REL 1.3a
REL 1.3b
Using a calculator to find cos ratio
What Goes In The Box ?
Find the cos ratio to 3 decimal places.
cos 0o
cos 45o
= 0.866
= 1.000
= 0.707
cos 60o
cos 20o
cos 90o
= 0.500
= 0.940
= 0.000
cos
30o
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The Cosine Ratio
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National 4
REL 1.3a
REL 1.3b
Trigonometry
Cos x° =
x°
Adjacent
Adj
Hyp
Find the
adjacent
National 4 length b
REL 1.3a
Example
Trigonometry
b
Adj
Cos x° =
Hyp
b
Cos 40° =
35
40°
Opp
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REL 1.3b
35mm
SOH CAH TOA
35 x Cos 40° = b
b = 35 x Cos 40°= 26.8mm (1 d.p.)
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National 4
REL 1.3a
REL 1.3b
Trigonometry
Now try
Extension Booklet
Ex6E (page 88 Q1 to Q5)
Starter Questions
National 4
REL 1.3a
REL 1.3b
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1. Simplify 5 y  6 x  7 x  3 x  3 xy
2. The ratio of apples to pears is 4 : 5.
If there are 72 pieces of fruit in total.
How many apples are there.
3.
5-May-17
Find HCF of 15 and 21
Created by Mr.Lafferty Maths Dept
Angles & Triangles
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National 4
REL 1.3a
REL 1.3b
Learning Intention
1. Definite the cosine ratio
and show how to find an
length or angle using this
ratio.
Success Criteria
1. Write down cosine ratio.
2. Use cosine ratio to find a
length or angle.
5-May-17
Created by Mr. Lafferty Maths Dept.
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National 4
REL 1.3a
REL 1.3b
Using Cos to calculate angles
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National 4
REL 1.3a
REL 1.3b
Using a calculator to find angle
given cos ratio
What Goes In The Box ?
Find the angle given the cos ratio to 1 decimal places.
cos-1 0
cos-1 0.8
= 0.0o
= 90.0o
= 36.9o
cos-1 0.6
cos-1 0.1
cos-1 0.9
= 53.1o
= 84.3o
= 25.8o
cos-1
1
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Example
Trigonometry
34cm
Adj
Cos x° =
Hyp
Cos x° =
34
45
Opp
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National 4
REL 1.3a
REL 1.3b
Find the
angle xo
x°
45cm
SOH CAH TOA
Cos x° = 0.756 (3 d.p.)
x = Cos ⁻¹0.756 =41°
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National 4
REL 1.3a
REL 1.3b
Trigonometry
Now try
Extension Booklet
Ex6E (page 89 Q6 to Q8)
National 4
REL 1.3a
REL 1.3b
Starter Questions
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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 .
o
If no find the area of the triangle.
6
xo
10
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8
The Three Ratios
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National 4
REL 1.3a
REL 1.3b
adjacent
opposite
Sine
Tangent
Cosine
hypotenuse
adjacent
Sine
adjacent
Cosine
opposite
Cosine
Tangent
Sine
hypotenuse
opposite
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Sine
hypotenuse
Trigonometry
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National 4
REL 1.3a
REL 1.3b
Sin x° =
Opp
Hyp
Cos x° =
Adj
Hyp
O
A
S HC H
Tan x° =
O
T A
Opp
Adj
Trigonometry
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National 4
REL 1.3a
REL 1.3b
Process
1.
Write down
SOH CAH TOA
2.
3.
Identify what you want to find
what you know
Copy this!
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National 4
REL 1.3a
REL 1.3b
Trigonometry
Past Paper Type Questions
SOH CAH TOA
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National 4
REL 1.3a
REL 1.3b
Trigonometry
Past Paper Type Questions
SOH CAH TOA
(4 marks)
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National 4
REL 1.3a
REL 1.3b
Trigonometry
Past Paper Type Questions
SOH CAH TOA
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National 4
REL 1.3a
REL 1.3b
Trigonometry
Past Paper Type Questions
SOH CAH TOA
4 marks
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National 4
REL 1.3a
REL 1.3b
Trigonometry
Past Paper Type Questions
SOH CAH TOA
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National 4
REL 1.3a
REL 1.3b
Trigonometry
Past Paper Type Questions
SOH CAH TOA
(4marks)
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National 4
REL 1.3a
REL 1.3b
Trigonometry
Past Paper Type Questions
SOH CAH TOA
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National 4
REL 1.3a
REL 1.3b
Trigonometry
Past Paper Type Questions
SOH CAH TOA
(4marks)
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National 4
REL 1.3a
REL 1.3b
Trigonometry
Now try
Extension Booklet
Ex7E (page 91)