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Trigonometry
Trigonometry is used to find the length of sides and the size of 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.
Non - right angled triangles
Solve by using the Sine Rule or the Cosine Rule
In these rules the small case letters (a, b, c) represent the sides and the large case letters (A, B, C) represent
the angle opposite the side with the same letter
A
c
B
b
a
The Sine Rule is:
C
a
b
c


sinA
sinB
sinC
In words this means that in a triangle the relationship between a side and the sine of
the angle opposite it is the same for each of the three side and angles
The Cosine Rule is:
or
a
b 2  c 2  2bccosA
b2  c2  a2
cosA 
2bc
for finding a side
for finding an angle
Think about what this means in relation to a triangle
To find a side or angle in a non-right angled triangle
1. Decide whether you are using the Sine or Cosine Rule
The Sine Rule is used when you are dealing with two angles
For example:
3.4 m
a
35 
4.2 m
23 
6.3 m

35 
(Given 2 angles and finding a side)
or
(Given 1 angle and finding an angle)
The Cosine Rule is used when you are dealing with one angle
For example:
6.3 m
30 
3.2 m
x
2.9 m

5.6 m
(Finding one angle, given none)
5.2 m
or
(Given one angle and finding a side)
2. Label the triangle
The triangle must be labelled A, B, C at the angles and a, b, c at the sides with the
small case letter opposite the angle labelled with the same upper case letter
For the Sine Rule you can label any angle A, B or C
For the Cosine Rule you must label the angle you are dealing with A. It does not matter
what you label the other two.
3.
Write down the rule in the form you are going to use it.
Sine Rule
a)
Only two parts of the Sine Rule are used.
sin C
sin A
a
b

or

sin A sinB
c
a
For example:
b)
The Sine Rule may be used with either the sides or the angles on the top. (as
above)
c)
It is best to put what you are trying to find on the top left hand corner.
Note: If you need information that you are not given remember the angles of a
triangle sum to 180
Cosine Rule
a 2  b 2  c 2  2bccosA
a)
cosA 
b)
4.
b2  c2  a2
2bc
- in this form used to find a side
- in this form used to find an angle
Substitute into the formula and solve
Note: If the angle is given in degrees your calculator must be set to degrees
Summary
1.
Check that it is a non-right angled triangle
2.
Label the sides A, B, C at angles and a, b, c at sides
3.
Choose the Sine Rule or the Cosine Rule and write in the form that best suits
4.
Substitute in the values you are given
5.
Solve the equation
6.
Give correct units of measurement, check your answer is sensible and think
about rounding.
Examples:
A
3.4 m
B
A
c
b
35 
4.2 m
23 
a
C
B
c
b 6.3 m
35 

a
C
Find the length of AC (b)
Find the size of angle BCA ()
b
c

sinB sin C
sin C sin B

c
b
b
3.4

sin 35 sin 23
sin C sin 35

4.2
6.3
3.4
x sin 35
sin 23
b 
sin C 
b  4.99 m
sin 35
x 4.2
6.3
sin C  0.3823842
C = sin1 0.3823842
C = 22.5
C
C
b
3.2 m
a
2.9 m
6.3 m
b
a
A

c
B
A
30
c
5.6 m
B
5.2 m
Find the size of angle BAC ()
Find the length of BC
b2  c 2  a2
2bc
2
3.2  5.6 2  2.9 2
cos A =
2 x 3.2 x 5.6
a 2  b 2  c 2  2bccosA
cos A =
a 2  6.3 2  5.2 2  2 x 6.3 x 5.2 x cos30
a 2  9.9880155
cos A = 0.9260602
a  9.9880155
A = cos
1
0.9260602
a = 3.16 m
A = 22.2
Exercises
1.
Find the length of the side marked x
80
2.7 m
Ans: 4.64m
35
xm
2.
Find the size of the angle marked 
4.7m
3m
Ans: 33.7

5.4m
3.
Find the length of the side marked x
3.7m
x
22
Ans: 1.79m
2.3m
4.
Find the size of the angle marked 
4.5m
30
2.3m

Ans 78.0\