Cosine Rule - SS2 Mathematics Lesson Note

In any triangle \(ABC\) with conventional sides \(a,b,c\) and corresponding angles \(A,B,C\), the cosine rule states that:

\[a^{2} = b^{2} + c^{2} - 2bc\cos A\]

\[b^{2} = a^{2} + c^{2} - 2ac\cos B\]

\[c^{2} = a^{2} + b^{2} - 2ab\cos C\]

Example: Given a \(\mathrm{\Delta}ABC\) as shown above, where \(b = 7.23cm,\ \angle A = 42{^\circ}\) and \(c = 5.46cm\). Find the length of side \(a\)

Solution

\[a^{2} = b^{2} + c^{2} - 2bc\cos A\]

\[a^{2} = {(7.23)}^{2} + {(5.46)}^{2} - 2(7.23)(5.46)\cos 42\]

\[a^{2} = 52.2729 + 29.8116 - 78.9516(0.7431)\]

\[a^{2} = 52.2729 + 29.8116 - 58.6689\]

\[a^{2} = 23.4156\]

\[a = \sqrt{23.4156} = \ 4.8cm\]