Understanding Engineering Mathematics

wheres=(a+b+c)/2, the semi-perimeter of the triangle. This result is very useful in
surveying, for example.
Note that the sine rule can only be used if we know at least one side and the angle
opposite to that side. Failing such information, we may be able to use thecosine rule,
which states that

a^2 =b^2 +c^2 − 2 bccosA

The proof for this is instructive and illustrates yet again the power of Pythagoras – we
only consider the case of acute angles, but the rule holds for all angles.
Consider the triangle shown in Figure 6.7, with altitudeh(154

➤

).

C B

A

b c

h

a

x

Figure 6.7Proof of cosine rule.

We have, by Pythagoras’ theorem (154

➤

):

h^2 =b^2 −x^2

=c^2 −(a−x)^2

which simplifies to (42

➤

)

c^2 =a^2 +b^2 − 2 ax

Butx=bcosC,so

c^2 =a^2 +b^2 − 2 abcosC

The result is clearly ‘symmetric’ under the rotation of labels of the sides and corresponding
angles, so we get also

a^2 =b^2 +c^2 − 2 bccosAandb^2 =a^2 +c^2 − 2 accosB

It may help to think of it is as

(side)^2 = sum of squares of opposite sides

− 2 ×product of opposite sides×cos of their included angle.