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Law of Sines and
Cosines
{
Trigonometry applied to
triangles without right
angles.

You have learned to apply
trigonometry to right angled
triangles.
opp
sinA 
hyp
adj
cosA 
hyp
opp
tanA 
adj
hyp
A
opp
adj
Now we extend our
trigonometry so that we
can deal with triangles
which are not right angled.

First we introduce the following notation.
 We use capital letters for the angles,
and lower case letters for the sides.

B
c
A
b
P
q
r
Q
p
In DABC
 The side opposite angle A
a
is called a.
C  The side opposite angle B
is called b.
In DPQR
 The side opposite angle P
is called p.
R
And so on
There are two new rules.

1. The Law of Sines
C
b
A
.
a
c
B

Find the length of BC.
B
95o
c
a
35o
A
6.2 cm

Substitute A = 35o, B = 95o, b = 6.2:

Multiply by sin35o:
C
Law of Cosines
There are two main ways of
writing the Law of Cosines
one for finding a side,
one for finding an angle.

The Law of Cosines (to find
the length of a side)
B
c
A
a
b
C
The cosine rule for finding
an angle
How do I know whether to use the
sine rule or the cosine rule?
To use the sine rule you need to know an
angle and the side opposite it. You can
use it to find a side (opposite a second
known angle) or an angle (opposite a
second known side).
 To use the cosine rule you need to know
either two sides and the included angle
or all three sides.
