MATHEMATICS

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TrigonometryTrigonometryS3
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)

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Starter QuestionsStarter Questions
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) andinversely to
the square ofits radius (r).
Write down an formula connecting R, L and r.
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20 Aug 201720 Aug 2017 Created by Mr. Lafferty Maths Dept.Created by Mr. Lafferty Maths Dept.
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 & TrianglesAngles & Triangles
1.1. Understand the termsUnderstand the terms
hypotenuse, opposite andhypotenuse, opposite and
adjacent in right angledadjacent in right angled
triangle.triangle.

Let’s Investigate!Let’s Investigate!
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Trigonometry means “triangle” and
“measurement”.
Adjacent
Opposite
x°x°
hypotenuse
We will be using right-angled triangles.

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30°
Adjacent
Opposite
hypotenuse
Opposite
Adjacent
= 0.6
Mathemagic!

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45°
Adjacent
Opposite
hypotenuse
Opposite
Adjacent
= 1
Try another!

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For an angle of 30°,
Opposite
Adjacent
= 0.6
We write tan 30° = 0.6
Opposite
Adjacent
is called the tangent of an angle.

TrigonometryTrigonometry
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.
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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!

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What’s the point of all this???
Don’t worry, you’re about to find out!

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12 m
How high is the tower?
Opp
60°

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60°
12 m
Adjacent
Opposite
hypotenuse Copy this!

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Tan x° =
Opp
Adj
Tan 60° =
Opp
12
= Opp12 x Tan 60°
Opp =12 x Tan 60° = 20.8m (1 d.p.)
Copy this!

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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° =
Opposite
Opp
Adjacent
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ExampleExample
65°65°
Tan x° =
Opp
Opp
Adj
Hyp hh
8m8m Tan 65° =
h
8
= h8 x Tan 65°
h = 8 x Tan 65° = 17.2m (1 d.p.)
Adj
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Find the
height h
SO
H CA
H TO
A

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Class Group
Identifying the
Tan Ratio Ex 3.1 & Ex4.1
MIA Page 203

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
10cm
P
Q
R
6cm
7cm
10cm
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2.2. Use tan of an angle to solveUse tan of an angle to solve
problems.problems.
1. To use tan of the angle to
solve problems.
1.1. Write down tan ratio.Write down tan ratio.

Using Tan to calculate anglesUsing Tan to calculate angles
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18
12
ExampleExample
x°x°
Tan x° =
Opp
Opp
Adj
Hyp
SO
H CA
H TO
A
12m12m Tan x° =
= 1.5Tan x°
Adj
18m18m
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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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Now try
Exercise 4.2
MIA Page 205

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
÷ ×
t.
Show that it's average speed is 30mph.
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xo

REAL LIFE problems.REAL LIFE problems.
solve REAL LIFE problems.

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SO
H CA
H TO
A
20 Aug 201720 Aug 2017 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 h
tan 47 = =
adj 8
o h
tan 47 =
8
o
h = 8× tan 47
h = 8.58m

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6o
20 Aug 2017
Compiled by Mr. Lafferty Maths
Dept.
Aeroplane
a = 15
c
LennoxtownAirport
Q1.Q1. An aeroplane is preparing to land at Glasgow Airport.An aeroplane is preparing to land at Glasgow Airport.
It is over Lennoxtown at present which is 15km fromIt is over Lennoxtown at present which is 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 h
tan 6 =
15
o
h = 15× tan 6
h = 1.58km
SO
H CA
H TO
A

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Now try
Exercise 5.1
MIA Page 207

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.
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xo

find adjacent length.find adjacent length.
find adjacent length.

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Use the tan ratio to calculate how far the ladder
is away from the building.
45o
12m
ladder
o opp 12
tan 45 = =
adj d
o
12
d =
tan 45
d = 12m
d m
SO
H CA
H TO
A

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6o
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 2Example 2
o 1.58
tan 6 =
d
o
1.58
d =
tan 6
d = 15 km
SO
H CA
H TO
A

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Now try
Exercise 5.2
MIA Page 210

3 3 5
1. +
4 5 7
2. Explain why y = 6 when a = (-1) b = 2
y = (a -b)(b - a)(2a -b)
p
Q3. Given T =k .
v
Find k when T = 6 P = 18 and v = 9.
×
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2.2. Use tan ratio to find an angle.Use tan ratio to find an angle.
1. To show how to find an
angle using tan ratio.

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Use the tan ratio to calculate the angle that the
support wire makes with the ground.
xo
11m
o opp 11
tan x = =
adj 4
11
4
 
 ÷
 
o -1
x = tan
o o
x = 70
4 m
SO
H CA
H TO
A

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Use the tan ratio to find the angle of take-off.
xo
88m
o opp 88
tan x = =
adj 500
o
tan x = 0.176
o -1 o
x = tan (0.176) = 10
500 m
SO
H CA
H TO
A

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Now try
Exercise 6.1
MIA Page 211

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)
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2.2. Use sine ratio to find anUse sine ratio to find an
angle.angle.
1. Definite the sine ratio and
show how to find an angle
using this ratio.
1.1. Write down sine ratio.Write down sine ratio.

The Sine RatioThe Sine Ratio
x°x°
Sin x° =
Opposite
Opp
Hyp
hypotenuse
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ExampleExample
34°34°Sin x° =
Opp
Opp
Hyp
Hyp
hh 11cm11cm
Sin 34° =
h
11
= h11 x Sin 34°
h = 11 x Sin 34° = 6.2cm (1 d.p.)
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Find the
height h
SO
H CA
H TO
A

Using Sin to calculate anglesUsing Sin to calculate angles
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ExampleExample
x°x°
Sin x° =
Opp
Opp
Hyp
Hyp
6m6m 9m9m
Sin x° =
6
9
= 0.667 (3 d.p.)Sin x°
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Find the xo
SO
H CA
H TO
A

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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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Now try
Exercise 7.1
MIA Page 212

1 2 3
1. 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. Alorry is travelling at 40mph.
It has travelled 60 miles.
How long has it taken to travel 60 miles.
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2.2. Use sine ratio to solveUse sine ratio to solve
REAL-LIFE problems.REAL-LIFE problems.
1. To show how to use the
sine ratio to solve
REAL-LIFE problems.

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SO
H CA
H TO
A
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 h
sin 70 = =
hyp 11.7
h × o
= 11.7 sin70
h = 11m
11.7m

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SO
H CA
H TO
A
Use the sine ratio to find the angle of the ramp.
xo
10m
o opp 10
sin x = =
hyp 20
o 10
sin x =
20
 
 ÷
 
o -1 o10
x = sin = 30
20
20 m

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Now try
Exercise 7.2
MIA Page 214

22 9 7 = (2 ?)( ?)
2
1. Fill in the ? marks.
2. Calculate A when w = (-10)
A =
w
4
+ + + +x x x x
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2.2. Use sine ratio to find theUse sine ratio to find the
hypotenuse.hypotenuse.
1. To show how to calculate
the hypotenuse using the
sine ratio.

SO
H CA
H TO
A
ExampleExample
72°72°
Sin x° =
Opp
Hyp
Sin 72° =
5
r
r =
r = 5.3 km
5km5km
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AB
C
r5
sin72o
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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Now try
Exercise 8.1
MIA Page 215

1 3
1. 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
and Q
t of 80%
3. Alorry is travelling at 50mph.
It has travelled 75 miles.
Show that the time taken is 1hr 30 mins.
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2.2. Use cosine ratio to find aUse 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.
1.1. Write down cosine ratio.Write down cosine ratio.

The Cosine RatioThe Cosine Ratio
Cos x° =
Adjacent
Adj
x°x°
Hyp
hypotenuse
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SO
H CA
H TO
A
ExampleExample
40°40°
Cos x° =
Opp
Adj
Hyp Hyp
b
35mm
Cos 40° =
b
35
= b35 x Cos 40°
b = 35 x Cos 40°= 26.8mm (1 d.p.)
Adj
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Find the
adjacent
length b

Using Cos to calculate anglesUsing Cos to calculate angles
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SO
H CA
H TO
A
ExampleExample
x°x°
Cos x° =
Opp
Adj
Hyp Hyp
45cm
Cos x° =
34
45
= 0.756 (3 d.p.)Cos x°
x = Cos ⁻¹0.756 =41°
Adj34cm34cm
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Find the
angle xo

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Now try
Exercise 9.1
MIA Page 216

o
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.
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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

Sin x° =
Opp
Hyp
Cos x° =
Adj
Hyp
Tan x° =
Opp
Adj
CA
H TO
ASO
H
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SO
H CA
H TO
A
Copy this!
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1. Write down
Process
Identify what you want to find
what you know3.
2.

Past Paper Type Questions
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SO
H CA
H TO
A

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(4 marks)
SO
H CA
H TO
A

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SO
H CA
H TO
A
4 marks

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(4marks)
SO
H CA
H TO
A

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Now try
Exercise 10.1 & 10.2
MIA Page 218

MATHEMATICS

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