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
Answer & explanation
Correct answer: option 2
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 $d A \cong \Delta A]$