Trigonometry
Trigonometry is used to find unknown sides and angles in triangles.
The first question you need to ask is:
Is the triangle a right angled or non-right angled triangle?
Different methods are used for each.
Right angled triangles
Solve by using the sine, cosine and tangent ratios.
The triangle must be labelled H, O and A.
H is the hypotenuse – the side opposite the right angle and the longest side.
O is the side opposite the angle you are dealing with.
A is the side adjacent to the angle you are dealing with.
The sine ratio is the ratio of the side opposite the angle () to the hypotenuse:
sin Θ 
O
H
The cosine ratio is the ratio of the side adjacent to the angle () to the hypotenuse:
cos Θ 
A
H
The tangent ratio is the ratio of the side opposite the angle () to the side adjacent to
the angle ():
O
tan Θ 
A
SOH CAH TOA helps us remember this.
The second question you need to ask is:
What two sides am I dealing with?
The answer to this question will tell you which ratio to use.
(Note: If you are given two sides and finding a third use Pythagoras’ Theorem)
Write down the ratio in its complete form and substitute the values you are given into it.
There are three possible equations to solve:
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Example 1: The unknown is on the top
sin 
a cm
O

H
10 cm
a
10
sin 30 
30º
sin 30 x 10  a
a  5 cm
Example 2: The unknown is on the bottom
5m
cos Θ 
42
A
H
am
cos 42 =
5
a
cos 42 x a = 5
a =
5
cos42
a = 6.7 m
Example 3: The unknown is an angle

tan  =
3.5
4.2
4.2 mm
3.5 mm
 = tan 1 (
3.5
)
4.2
 = 39.8
Summary
1.
Check that it is a right angled triangle
2.
Label the sides H, O, A
3.
Choose sin, cos or tan and write it down in full
4.
Substitute in the values you are given
5.
Solve for what you are trying to find – notice the position of the unknown –
this will tell you which way to solve the equation
6.
Give correct units of measurement, check your answer is sensible and
think about rounding.
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Exercise
1.
Find the angle 

10 m
Ans: 23.6 
4m
2.
Find the length of the side marked x
12.4 mm
x
37 
3.
Ans: 9.90 mm
Find the length of the side marked x
23
x
2.7 m
Ans: 6.36 m
Notes:
1.
If the angle is given in degrees your calculator must be set to degrees
2.
Angles of a triangle add up to 180
Unitec D:\687288935.doc