The cosine rule

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

triangle notation

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:

cosine ruleUsing 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:

cosine rule 2

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

See also: