In a $\triangle A B C$ the sides b and c are given. If there is an error $\triangle A$ in measuring angle A, then the error $\triangle a$ in side a is given by

$\frac{2 S}{a} \Delta A$

In $\triangle A B C$, we have

$2 b c \cos A=b^2+c^2-a^2$

$\Rightarrow d(2 b c \cos A)=d\left(b^2+c^2-a^2\right)$

$\Rightarrow -2 b c \sin A~ d A=-2 a ~d a$

$\Rightarrow b c \sin A~ d A=a ~d a$

$\Rightarrow \frac{2}{a}\left(\frac{1}{2} b c \sin A\right) d A=d a$

$\Rightarrow d a=\frac{2 S}{a} d A$

$\Rightarrow \Delta a=\frac{2 S}{a} \Delta A$           $[∵ d a \cong \Delta a$ and $d A \cong \Delta A]$