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The Tangent Ratio
The Tangent using Angle
The Tangent Ratio in Action
The Tangent (The Adjacent side)
The Tangent (Finding Angle)
The Sine of an Angle
The Sine Ration In Action
The Sine ( Finding the Hypotenuse)
The Cosine of an Angle
Mixed Problems
Angles & Triangles
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.
Trigonometry means “triangle” and
“measurement”.
We will be using right-angled triangles.
Opposite
x°
Adjacent
Mathemagic!
Opposite
30°
Adjacent
Opposite
= 0.6
Adjacent
Try another!
Opposite
45°
Adjacent
Opposite
= 1
Adjacent
For an angle of 30°,
Opposite
= 0.6
Adjacent
Opposite
is called the tangent of an angle.
Adjacent
We write tan 30° = 0.6
The ancient Greeks
discovered this and
repeated this for
all possible angles.
Tan 25°
0.466
Tan 26°
0.488
Tan 27°
0.510
Tan 28°
0.532
Tan 30° =0.554
0.577
Tan 29°
Tan 30°
0.577
Tan 31°
0.601
Tan 32°
0.625
Tan 33°
0.649
Tan 34°
0.675
Accurate to
3 decimal places!
Now-a-days we can use
calculators instead of tables
to find the Tan of an angle.
On your calculator press
Followed by 30, and press
Tan
=
Notice that your calculator is
incredibly accurate!!
Accurate to 9 decimal places!
What’s the point of all this???
Don’t worry, you’re about to find out!
How high is the tower?
Opp
60°
12 m
Opposite
Copy this!
60°
12 m
Adjacent
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!
So the tower’s
20.8 m high!
20.8m
Don’t worry, you’ll
be trying plenty of
examples!!
Opp
Tan x° =
Adj
Opposite
x°
Adjacent
Example
h
65°
8m
Opp
Find the
height h
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.)
Angles & Triangles
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.
Using Tan to calculate angles
Example
P
SOH CAH TOA
Opp
18m
R
x°
12m
Q
Calculate the
tan xo ratio
Opp
Tan x° =
Adj
Tan x° =
18
12
Tan x° = 1.5
Calculate the
size of
angle xo
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
Tan x° = 1.5
Press
2nd
Enter 1.5
Tan ⁻¹
Tan
=
x = Tan ⁻¹1.5 = 56.3° (1 d.p.)
Process
1.
Identify Hyp, Opp and Adj
2. Write down ratio Tan xo = Opp
Adj
3.
Calculate xo
2nd
Tan ⁻¹
Tan
Angles & Triangles
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.
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
47o
8m
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.
What is the height of the plane ?
tan 6o =
h
15
h = 15 × tan 6o
h = 1.58km
24-May-17
Aeroplane
c
6o
Airport
a = 15
Lennoxtown
Angles & Triangles
Learning Intention
1. To use tan of the angle to
find adjacent length.
Success Criteria
1. Write down tan ratio.
2. Use tan of an angle to solve
find adjacent length.
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
d = 12m
SOH CAH TOA
ladder
45o
dm
12m
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
Aeroplane
a = 1.58 km
6o
Airport
Lennoxtown
Angles & Triangles
Learning Intention
1. To show how to find an
angle using tan ratio.
Success Criteria
1. Write down tan ratio.
2. Use tan ratio to find an angle.
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
x o = 70o
11m
xo
4m
Use the tan ratio to find the angle of take-off.
SOH CAH TOA
opp
88
tan x =
=
adj 500
o
tan x o = 0.176
o
-1
o
x = tan (0.176) = 10
88m
xo
500 m
Angles & Triangles
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.
The Sine Ratio
Sin x° =
Opposite
x°
Opp
Hyp
Example
Find the
height h
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.)
Using Sin to calculate angles
Example
6m
Opp
Find the xo
9m
Opp
Sin x° =
Hyp
6
Sin x° =
9
x°
SOH CAH TOA
Sin x° = 0.667 (3 d.p.)
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
Sin x° = 0.667 (3 d.p.)
Press
2nd
Enter 0.667
Sin ⁻¹
Sin
=
x = Sin ⁻¹0.667 = 41.8° (1 d.p.)
Angles & Triangles
Learning Intention
1. To show how to use the
sine ratio to solve
Success Criteria
1. Write down sine ratio.
REAL-LIFE problems.
2. Use sine ratio to solve
REAL-LIFE problems.
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
h = 11.7  sin70o
SOH CAH TOA
11.7m
h = 11m
70o
h
Use the sine ratio to find the angle of the ramp.
opp 10
sin x =
=
hyp 20
o
SOH CAH TOA
10
sin x =
20
o
 10 
o
x = sin 
=
30

 20 
o
-1
20 m
xo
10m
Angles & Triangles
Learning Intention
1. To show how to calculate
the hypotenuse using the
sine ratio.
Success Criteria
1. Write down sine ratio.
2. Use sine ratio to find the
hypotenuse.
Example
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
Angles & Triangles
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.
The Cosine Ratio
Cos x° =
x°
Adjacent
Adj
Hyp
Find the
adjacent
length b
Example
b
b
Cos 40° =
35
40°
Opp
Adj
Cos x° =
Hyp
35mm
SOH CAH TOA
35 x Cos 40° = b
b = 35 x Cos 40°= 26.8mm (1 d.p.)
Using Cos to calculate angles
Example
Find the
angle xo
34cm
Cos x° =
34
45
Opp
Adj
Cos x° =
Hyp
x°
45cm
SOH CAH TOA
Cos x° = 0.756 (3 d.p.)
x = Cos ⁻¹0.756 =41°
The Three Ratios
adjacent
opposite
Sine
Tangent
Cosine
hypotenuse
adjacent
Sine
adjacent
Cosine
opposite
Cosine
Tangent
Sine
hypotenuse
opposite
Sine
hypotenuse
Sin x° =
Opp
Hyp
Cos x° =
Adj
Hyp
O
A
S HC H
Tan x° =
O
T A
Opp
Adj
Process
1.
Write down
SOH CAH TOA
2.
3.
Identify what you want to find
what you know
Copy this!
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
SOH CAH TOA
4 marks
Past Paper Type Questions
SOH CAH TOA
Past Paper Type Questions
SOH CAH TOA
(4marks)
Past Paper Type Questions
SOH CAH TOA
Past Paper Type Questions
SOH CAH TOA
(4marks)