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Trigonometry
Trigonometry is a method of finding out an unknown angle or side in a right angled triangle
Both the triangles below are similar because: The angles are the same but the
sides are different
Trigonometry is a method of finding out an unknown angle or side in a right angled triangle
Both the triangles below are similar because: The angles are the same but the
sides are different
Trigonometry is a method of finding out an unknown angle or side in a right angled triangle
Both the triangles below are similar because: The angles are the same but the
sides are different
If we measure the height and the base:
Height Base
Small triangle
Large triangle
height
base5
0.6258
100.625
16
5 cm
10 cm
8 cm
16 cm
For both trianglesheight
0.625base
This angle is in fact 320
So as long as the value of
then this angle will always be 320
height0.625
base
This is the idea behind trigonometry
If we know 2 sides then we can find the angles in the triangle
How do we know the angle is 320 ?
We can use our calculator which has been programmed to work out the angle.
We don`t have to know the height and the base it can be any 2 sides
Depending on which 2 sides are known then we use a different button on the calculator
Names are given to the 3 sides which all refer to the angle we are trying to find
The names are:
Opposite, Adjacent and Hypotenuse
Opposite means on the other side from the angle we need.
Adjacent means next to the angle we need.
Hypotenuse means the side opposite the right angle
XOpposite
Adjacent
Hypotenuse
• Identify the names of the sides of these right angled-triangles given angle k
hypotenuse
opposite
adjacent
b
b
hypotenuse
hypotenusehypotenuse
a
a
a
b
b
opposite
oppositeopposite
c
c
c
c
adjacent
adjacent
adjacenta
k
k
k
k
Opposite, Adjacent and HypotenuseIn each case label all the sides of the triangles as Opposite (O), Adjacent (A) and Hypotenuse (H) with relation to the angle marked as “X”.
X
X
X
X
X
X
X
X
X
X
x
Using the Opposite (O), Adjacent (A) and Hypotenuse (H) to work out the missing angle
The calculator has 3 buttons which are used to find the missing angle:
Sin – short for Sine
Cos – short for Cosine
Tan – short for Tangent
Deciding which button to use depends on which sides are given
• SOH CAH TOA
• Memory Aid
• Some Old Horses Sin Opposite Hypotenuse• Can Always Hear Cos Adjacent Hypotenuse• Their Owners Approaching Tan Opposite Adjacent
• Or invent one of your own
SOH CAH TOA Divide it up into three groups Place each group of three in a triangle
starting in the bottom left of each triangle
S
O
H C
A
H T
O
A
Trigonometric Ratios
TOA
CAH
SOHoppositex
hypotenusesin
oppositex
adjacenttan
adjacentx
hypotenusecos
S
O
H
C
A
H
T
O
A
What have we got and need to find?We need an angle – x.We have the Hypotenuse and Adjacent side.
Looking at the phrase, we can use C A H
10 cm
25 cm
x
Cos (x) =
Adjacent
Hypotenuse
Example 1
Hypotenuse
Adjacent
SOH CAH TOA
10 cm
25cm
x
Cos (x) =
Adjacent
Hypotenuse
Replace A and H by 10 and 25
Cos (x) = 25
10= 0.4
We now need to convert this to an angle in degrees using the Cos-1 button!!!
x = Cos –1(0.4) = 66.42oWe always find the angle using either the Cos–
1, Sin–1 or Tan–1 buttons.
Hypotenuse
Adjacent
What have we got and need to find?We need an angle – x.We have the Opposite and Adjacent side.
Looking at the phrase, we can use TOA
20 cm
15 cm
x
Tan (x) =
Opposite
Adjacent
Example 2
Opposite
Adjacent
SOH CAH TOA
Replace O and A by 15 and 20
Tan (x) =
15
20= 0.75
We now need to convert this to an angle in degrees using the Tan-1 button!!!
x = Tan –1(0.75) = 36.67o
Tan (x) =
Opposite
Adjacent
20 cm
15 cm
x
Opposite
Adjacent
We always find the angle using either the Cos–1, Sin–1 or Tan–1 buttons.
What have we got and need to find?We need an angle – x.We have the Hypotenuse and Opposite side.Looking at the phrase, we can use S O H 8 cm
12 cm
x
Sin (x) = Opposite
Hypotenuse
Example 3
Hypotenuse
Opposite
SOH CAH TOA
Sin (x) = Opposite
Hypotenuse
Replace O and H by 8 and 12
Sin (x) =
8
12= 0.666
We now need to convert this to an angle in degrees using the Sin-1 button!!!
x = Sin –1(0.666) = 41.81o
8 cm12 cm
x
Hypotenuse
Opposite
We always find the angle using either the Cos–1, Sin–1 or Tan–1 buttons.
Using Trigonometry to Find a Missing Side
Trigonometric Ratios
TOA
CAH
SOHoppositex
hypotenusesin
oppositetanx=
adjacent
adjacentx
hypotenusecos
S
O
H
C
A
H
T
O
A
Trigonometric Ratios
SOH
oppositesinx=
hypotenuse S
O
H
The triangle can also be used to find either the opposite side or the hypotenuse
opposite x hypotenusesin
oppositehypotenuse =
sinx
CAH
adjacentx
hypotenusecos C
A
H
Trigonometric Ratios
The triangle can also be used to find either the adjacent side or the hypotenuse
adjacent x hypotenusecos
adjacenthypotenuse =
cosx
TOA
oppositex
adjacenttan
T
O
A
Trigonometric Ratios
The triangle can also be used to find either the adjacent side or the opposite
opposite x adjacenttan
oppositeadjacent =
tanx
Example 1 SOH CAH TOA
60o
3 m
H
What have we got and need to find?
We need the Hypotenuse H
We have an angle and the Opposite O
Looking at the phrase we can use S O H
Hypotenuse = Opposite
Sin (angle)
Opposite
Hypotenuse
S
O
H
Hypotenuse = Opposite
Sin (angle)
60o
3 m
H
Replace O by 3 and (angle) by 60o
H =3
Sin (60o)Use the Sin button on your calculator to find this value
H = 8660.03
H = 3.46410…..
H = 3.46 m to 2 d.p.Opposite
Hypotenuse
Example 2 SOH CAH TOA
40o
3 m
A
What have we got and need to find?
We need the Adjacent A
We have an angle and the Opposite O
Looking at the phrase we can use T O A
Opposite
Adjacent
T
O
Aopposite
adjacent = tanx
Replace O by 3 and (angle) by 40o
H =3
Tan (40o)Use the Tan button on your calculator to find this value
H = 3
0.8390
H = 3.575…..
H = 3.58 m to 2 d.p.
oppositeadjacent =
tanx
3 m
A
Opposite
Adjacent
40o
Example 3 SOH CAH TOA
70oA
8
What have we got and need to find?
We need the Adjacent A
We have an angle and the Hypotenuse H
Looking at the phrase we can use C A H
Adjacent
Hypotenuse
C
A
H adjacent x hypotenusecos
70o
A
8
Replace H by 8 and (angle) by 70o
A = cos70 x 8 Use the Cos button on your calculator to find this value
H = 0.342 x 8
H = 2.736…..
H = 2.74 m to 2 d.p.
Hypotenuse
adjacent x hypotenusecos
Adjacent