Cosine Rule - SS2 Mathematics Lesson Note
In any triangleĀ \(ABC\)Ā with conventional sidesĀ \(a,b,c\)Ā and corresponding anglesĀ \(A,B,C\), 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:Ā Given aĀ \(\mathrm{\Delta}ABC\)Ā as shown above, whereĀ \(b = 7.23cm,\ \angle A = 42{^\circ}\)Ā andĀ \(c = 5.46cm\). Find the length of sideĀ \(a\)
Solution
\[a^{2} = b^{2} + c^{2} - 2bc\cos A\]
\[a^{2} = {(7.23)}^{2} + {(5.46)}^{2} - 2(7.23)(5.46)\cos 42\]
\[a^{2} = 52.2729 + 29.8116 - 78.9516(0.7431)\]
\[a^{2} = 52.2729 + 29.8116 - 58.6689\]
\[a^{2} = 23.4156\]
\[a = \sqrt{23.4156} = \ 4.8cm\]