Transcript S3 Trigonometry

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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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 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 2 d.p.)
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Finding side lengths
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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 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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