basic trig
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SO
HS
OH
SO
HS
OH
SO
H
CAHCAHCAHCAHCAHCAHCAHCAHCAHCAH
TOATOATOATOATOATOATOATOATOATOA
Simple Trigonometry
Moving on from Pythagoras’ Theorem, we can use Trigonometry to find a missing angle of a right
angle triangle, or the length of an unknown side of a right angle triangle, if the angle and the length of
a side are known.
90o Φ
The Formulas
In Simple Trigonometry we use three main formulas and use the acronyms SOH CAH
TOA to remember them.
(SOW – KA – TOE – WA)
SOH
Sine of the Angle = Opposite
Hypotenuse
Also shown as: -
Sin Φ = Opposite
Hypotenuse
SOH
This means the number you get when you divide the length of the opposite side of the triangle, by the length of the hypotenuse side
5
3
3
5= 0.6
This number is the SINE of the angle
CAH
Cosine of the Angle = Adjacent
Hypotenuse
Also shown as: -
Cos Φ = Adjacent
Hypotenuse
CAH
This means the number you get when you divide the length of the adjacent side of the triangle, by the length of the hypotenuse side
5
4
4
5= 0.8
This number is the COSINE of the angle
TOA
Tangent of the Angle = Opposite
Adjacent
Also shown as: -
Tan Φ = Opposite
Adjacent
TOA
This means the number you get when you divide the length of the opposite side of the triangle, by the length of the adjacent side
3
4
3
4= 0.75
This number is the TANGENT of the angle
Try transposing all three formulas
Sin Φ = Opposite
Hypotenuse
Cos Φ = Adjacent
Hypotenuse
Tan Φ = Opposite
Adjacent
The SINE, COSINE and TANGENT of the angle are not a measurement. They are a ratio made up of the lengths of two sides of the triangle.
We can use this number to find the actual angle in degrees. This used to be done using tables, but is now achieved by using a calculator
All three of the SOH CAH TOA formulas are simple divisions, but the answer gives the Sine, Cosine and
Tangent of the angle. To find the angle we need to use the inverse Sine, Cosine and Tangent function to find the actual angle. On a calculator these are shown as: -
Cos-1
Sin-1
Tan-1
These functions are accessed by pressing the SHIFT, or 2nd Function
It is helpful to recognise that the internal angles of triangle will always add up to 180 degrees and in a right angle
triangle one angle will always be 90 degrees. Using these rules we can find any missing angle, or missing side length if there is enough information in the triangle.
90o Φ
Lets try a SIMPLE example
90o Φ
4m
In this example Φ = 30o
Find the length of the Hypotenuse.
Lets try a SIMPLE example
90o Φ
4m
Using the formula Cos 30o = Adjacent
Hypotenuse
We can transpose to find: -
Hypotenuse = 4
Cos 30o
Hypotenuse = 4.62m
There are two ways to find the answer.
Using Pythagoras Theorem
Using Trigonometry Formulas
90o Φ
4m
Now find the length of the opposite side
4.62m
Pythagoras tells us: -
c = √ a2 – b2
90o Φ
4m
Now find the length of the opposite side
4.62m
c = √ (4.622 – 42)
c = 2.31m
We can also say: -
opposite = hypotenuse x Sin 30o
90o Φ
4m
Now find the length of the opposite side
4.62m
opposite = 4.62 x Sin 30o
opposite = 2.31m
Deciding which of the trigonometry formulas to use may seem complicated, but look at the problem and
see what information you have been given. If you have measurements for the opposite and hypotenuse
side, use SOH.
If you have Φ and the adjacent side you can use CAH, or TOA to find the answer.
Try This One
90o Φ
30m
a = ?m
c = ?m
If Φ = 39.64o Find the length of the hypotenuse and opposite sides.
hypotenuse = adjacent
Cos Φ
opposite = hypotenuse x Sin Φ
Try This One
90o 39.64o
30m
39.05m
25m
Based on what you have learned already what is the value of the angle marked θθ
Just One More
90o Φ
125mm
a = ?m
95mm
For this example find the length of the hypotenuse and the angle marked as Φ
a2 = b2 + c2
a2 = 1252 + 952
a = √ 1252 + 952
a = 157mm
Just One More
90o Φ
125mm
157mm
95mm
For this example find the angle marked as Φ
Cos Φ = adjacent
hypotenuse
Cos Φ = = 0.796178 125
157
Cos -1 Φ = 37.23o
Last One I Promise!!!
90o36.87o
125mm
157mm
95mm
θ
What is the value of the angle marked θ?
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