The cosine rule is used to find unknown lengths or angles in triangles. It is particularly useful in non right-angled triangles, and when two sides and a given angle are given, or three sides with the objective of finding an angle.

Labelling convention

The capital letters denote the interior angle at that point. In other words, B is the angle ABC, and A is the angle BAC. The lower-case letters are the lengths of the side opposite to the corresponding angle. That means that a, in this case is opposite to A and represents the length BC.

The cosine rule

The cosine rule states that for a triangle ABC,

$a^2 = b^2 + c^2 - 2bc cos(A)$
or, if transformed into the convenience of finding an angle,
$cos(A) = \dfrac{b^2+c^2 - a^2}{2bc}$

Application

Find the value of x in the following triangle:

Using the cosine rule,
$a^2 = b^2 + c^2 - 2bc cos(A)$
$x^2 = 9.1^2 + 9.2^2 - 2*9.1*9.2*cos(110)$
$x = \sqrt{9.1^2 + 9.2^2 - 2*9.1*9.2*cos(110)}$
$x = 15$

Find the value of angle A in the following triangle:

Using the cosine rule,
$cos(A) = \dfrac{b^2+c^2 - a^2}{2bc}$
$cos (A) = \dfrac{14^2 + 10^2 - 7^2}{2*14*10}$
$A = 28.10^o$