Download S3 Trigonometry

KS4 Mathematics
S3 Trigonometry
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Contents
S3 Trigonometry
S3.1 Right-angled triangles
S3.2 The three trigonometric ratios
S3.3 Finding side lengths
S3.4 Finding angles
S3.5 Angles of elevation and depression
S3.6 Trigonometry in 3-D
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Right-angled triangles
A right-angled triangle contains a right angle.
The longest side opposite
the right angle is called the
hypotenuse.
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The opposite and adjacent sides
The two shorter sides of a right-angled triangle are named
with respect to one of the acute angles.
The side opposite the
marked angle is called
the opposite side.
The side between the
marked angle and the
right angle is called
the adjacent side.
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x
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Label the sides
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Similar right-angled triangles
If two right-angled triangles have an acute angle of the same
size they must be similar.
For example, two triangles with an acute angle of 37° are
similar.
5 cm
3 cm
6 cm
37°
10 cm
37°
4 cm
8 cm
The ratio of the side lengths in each triangle is the same.
opp
3
6
=
=
adj
4
8
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opp
3
6
=
=
hyp
5
10
adj
4
8
=
=
hyp
5
10
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Similar right-angled triangles
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Contents
S3 Trigonometry
S3.1 Right-angled triangles
S3.2 The three trigonometric ratios
S3.3 Finding side lengths
S3.4 Finding angles
S3.5 Angles of elevation and depression
S3.6 Trigonometry in 3-D
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Trigonometry
The word trigonometry comes from the Greek meaning
‘triangle measurement’.
Trigonometry uses the fact that the side lengths of similar
triangles are always in the same ratio to find unknown sides
and angles.
For example, when one of the angles in a right-angled triangle
is 30° the side opposite this angle is always half the length of
the hypotenuse.
12 cm
8 cm
6 cm
?
4 cm
30°
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?
30°
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The sine ratio
the length of the opposite side
is the sine ratio.
the length of the hypotenuse
The ratio of
The value of the sine ratio depends on the size of the angles
in the triangle.
O
P
P
O
S
I
T
E
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H
Y
P
We say:
O
T
E
N
U
S
E
opposite
sin θ =
hypotenuse
θ
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The sine ratio
What is the value of sin 65°?
This is the same as asking:
In a right-angled triangle with
an angle of 65°, what is the
ratio of the opposite side to
the hypotenuse?
To work this out we can accurately draw a right-angled
triangle with a 65° angle and measure the lengths of the
opposite side and the hypotenuse.
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The sine ratio
What is the value of sin 65°?
It doesn’t matter how big the triangle is because all rightangled triangles with an angle of 65° are similar.
The length of the opposite side divided by the length of the
hypotenuse will always be the same value as long as the
angle is the same.
In this triangle,
opposite
sin 65° =
11 cm
65°
hypotenuse
10 cm
= 10
11
= 0.91 (to 2 d.p.)
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The sine ratio using a table
What is the value of sin 65°?
It is not practical to draw a diagram each time.
Before the widespread use of scientific calculators, people
would use a table of values to work this out.
Here is an extract from a table of sines:
Angle in
degrees
63
.0
.1
.2
.3
.4
.5
0.891
0.892
0.893
0.893
0.894
0.895
64
0.899
0.900
0.900
0.901
0.902
0.903
65
0.906
0.907
0.908
0.909
0.909
0.910
66
0.914
0.914
0.915
0.916
0.916
0.917
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The sine ratio using a calculator
What is the value of sin 65°?
To find the value of sin 65° using a scientific calculator, start
by making sure that your calculator is set to work in degrees.
Key in:
sin
6
5
=
Your calculator should display 0.906307787
This is 0.906 to 3 significant figures.
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The cosine ratio
the length of the adjacent side
The ratio of
is the cosine ratio.
the length of the hypotenuse
The value of the cosine ratio depends on the size of the
angles in the triangle.
H
Y
P
O
We say,
T
E
N
U
S
E
adjacent
cos θ =
hypotenuse
θ
ADJACENT
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The cosine ratio
What is the value of cos 53°?
This is the same as asking:
In a right-angled triangle with
an angle of 53°, what is the
ratio of the adjacent side to
the hypotenuse?
To work this out we can accurately draw a right-angled
triangle with a 53° angle and measure the lengths of the
adjacent side and the hypotenuse.
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The cosine ratio
What is the value of cos 53°?
It doesn’t matter how big the triangle is because all rightangled triangles with an angle of 53° are similar.
The length of the opposite side divided by the length of the
hypotenuse will always be the same value as long as the
angle is the same.
In this triangle,
adjacent
cos 53° =
hypotenuse
10 cm
53°
6 cm
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= 6
10
= 0.6
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The cosine ratio using a table
What is the value of cos 53°?
Here is an extract from a table of cosines:
Angle in
degrees
50
.0
.1
.2
.3
.4
.5
0.643
0.641
0.640
0.639
0.637
0.636
51
0.629
0.628
0.627
0.625
0.624
0.623
52
0.616
0.614
0.613
0.612
0.610
0.609
53
0.602
0.600
0.599
0.598
0.596
0.595
54
0.588
0.586
0.585
0.584
0.582
0.581
55
0.574
0.572
0.571
0.569
0.568
0.566
56
0.559
0.558
0.556
0.555
0.553
0.552
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The cosine ratio using a calculator
What is the value of cos 25°?
To find the value of cos 25° using a scientific calculator, start
by making sure that your calculator is set to work in degrees.
Key in:
cos
2
5
=
Your calculator should display 0.906307787
This is 0.906 to 3 significant figures.
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The tangent ratio
the length of the opposite side
The ratio of
is the tangent ratio.
the length of the adjacent side
The value of the tangent ratio depends on the size of the
angles in the triangle.
O
P
P
O
S
I
T
E
We say,
tan θ =
opposite
adjacent
θ
ADJACENT
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The tangent ratio
What is the value of tan 71°?
This is the same as asking:
In a right-angled triangle with
an angle of 71°, what is the
ratio of the opposite side to
the adjacent side?
To work this out we can accurately draw a right-angled
triangle with a 71° angle and measure the lengths of the
opposite side and the adjacent side.
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The tangent ratio
What is the value of tan 71°?
It doesn’t matter how big the triangle is because all rightangled triangles with an angle of 71° are similar.
The length of the opposite side divided by the length of the
adjacent side will always be the same value as long as the
angle is the same.
In this triangle,
opposite
tan 71° =
71°
adjacent
4 cm
11.6 cm
= 11.6
4
= 2.9
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The tangent ratio using a table
What is the value of tan 71°?
Here is an extract from a table of tangents:
Angle in
degrees
70
.0
.1
.2
.3
.4
.5
2.75
2.76
2.78
2.79
2.81
2.82
71
2.90
2.92
2.94
2.95
2.97
2.99
72
3.08
3.10
3.11
3.13
3.15
3.17
73
3.27
3.29
3.31
3.33
3.35
3.38
74
3.49
3.51
3.53
3.56
3.58
3.61
75
3.73
3.76
3.78
3.81
3.84
3.87
76
4.01
4.04
4.07
4.10
4.13
4.17
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The tangent ratio using a calculator
What is the value of tan 71°?
To find the value of tan 71° using a scientific calculator, start
by making sure that your calculator is set to work in degrees.
Key in:
tan
7
1
=
Your calculator should display 2.904210878
This is 2.90 to 3 significant figures.
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Calculate the following ratios
Use your calculator to find the following to 3 significant figures.
1) sin 79° = 0.982
2) cos 28° = 0.883
3) tan 65° =
4) cos 11° = 0.982
2.14
5) sin 34° = 0.559
6) tan 84° =
7) tan 49° =
8) sin 62° = 0.883
1.15
9) tan 6° = 0.105
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9.51
10) cos 56° = 0.559
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The relationship between sine and cosine
The sine of a given angle is equal to the
cosine of the complement of that angle.
We can write this as,
sin θ = cos (90 – θ)
We can show this as follows,
a
sin θ =
b
a
b
a
90 – θ
b
cos (90 – θ) =
a
b
θ
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The three trigonometric ratios
O
P
P
O
S
I
T
E
H
Y
P
O
T
E
N
U
S
E
θ
ADJACENT
Opposite
Sin θ =
Hypotenuse
SOH
Adjacent
Cos θ =
Hypotenuse
CAH
Opposite
Tan θ =
Adjacent
TOA
Remember: S O H C A H
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TOA
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Contents
S3 Trigonometry
S3.1 Right-angled triangles
S3.2 The three trigonometric ratios
S3.3 Finding side lengths
S3.4 Finding angles
S3.5 Angles of elevation and depression
S3.6 Trigonometry in 3-D
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Finding side lengths
If we are given one side and one acute angle in a right-angled
triangle we can use one of the three trigonometric ratios to find
the lengths of other sides. For example,
Find x to 2 decimal places.
12 cm
56°
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x
We are given the hypotenuse and we want
to find the length of the side opposite the
angle, so we use:
opposite
sin θ =
hypotenuse
x
sin 56° =
12
x = 12 × sin 56°
= 9.95 cm
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Finding side lengths
A 5 m long ladder is resting against a wall. It makes an
angle of 70° with the ground.
What is the distance between the
base of the ladder and the wall?
5m
70°
x
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We are given the hypotenuse and we want
to find the length of the side adjacent to
the angle, so we use:
adjacent
cos θ =
hypotenuse
x
cos 70° =
5
x = 5 × cos 70°
= 1.71 m (to 2d.p.)
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Finding side lengths
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Contents
S3 Trigonometry
S3.1 Right-angled triangles
S3.2 The three trigonometric ratios
S3.3 Finding side lengths
S3.4 Finding angles
S3.5 Angles of elevation and depression
S3.6 Trigonometry in 3-D
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The inverse of sin
sin θ = 0.5, what is the value of θ?
To work this out use the sin–1 key on the calculator.
sin–1 0.5 =
30°
sin–1 is the inverse of sin. It is sometimes called arcsin.
sin
30°
0.5
sin–1
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The inverse of cos
Cos θ = 0.5, what is the value of θ?
To work this out use the cos–1 key on the calculator.
cos–1 0.5 =
60°
Cos–1 is the inverse of cos. It is sometimes called arccos.
cos
60°
0.5
cos–1
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The inverse of tan
tan θ = 1, what is the value of θ?
To work this out use the tan–1 key on the calculator.
tan–1 1 =
45°
tan–1 is the inverse of tan. It is sometimes called arctan.
tan
45°
1
tan–1
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Finding angles
8 cm
Find θ to 2 decimal places.
5 cm
θ
We are given the lengths of the sides opposite and the adjacent
to the angle, so we use:
opposite
tan θ =
adjacent
8
tan θ =
5
θ = tan–1 (8 ÷ 5)
= 57.99° (to 2 d.p.)
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Finding angles
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Contents
S3 Trigonometry
S3.1 Right-angled triangles
S3.2 The three trigonometric ratios
S3.3 Finding side lengths
S3.4 Finding angles
S3.5 Angles of elevation and depression
S3.6 Trigonometry in 3-D
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Angles of elevation
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Angles of depression
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Contents
S3 Trigonometry
S3.1 Right-angled triangles
S3.2 The three trigonometric ratios
S3.3 Finding side lengths
S3.4 Finding angles
S3.5 Angles of elevation and depression
S3.6 Trigonometry in 3-D
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Angles in a cuboid
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Lengths in a square-based pyramid
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