Answer
If $A$, $B$ and $C$ are the measures of the angle of a triangle, and $a$, $b$ and $c$ are the lengths of the sides opposite to these angles, then the Law of Cosines states that ${{a}^{2}}={{b}^{2}}+{{c}^{2}}-2bc\cos A$.
Work Step by Step
Consider the triangles given below and use the basic trigonometric ratios to obtain:
From the figure, we get
$a=c\cos B+b\cos C$
Similarly we get
$\begin{align}
& b=c\cos A+a\cos C \\
& c=a\cos B+b\cos A \\
\end{align}$
Therefore,
$\begin{align}
& {{a}^{2}}=ac\cos B+ab\operatorname{cosC} \\
& {{b}^{2}}=bc\cos A+ab\cos C \\
& {{c}^{2}}=ac\cos B+bc\cos A \\
\end{align}$
So,
$\begin{align}
& {{b}^{2}}+{{c}^{2}}-{{a}^{2}}=ab\cos C+bc\cos A+bc\cos A+ac\cos B-ac\cos B-ab\cos C \\
& =2bc\cos A
\end{align}$
Hence,
${{a}^{2}}={{b}^{2}}+{{c}^{2}}-2bc\cos A$.
