The area of a triangle is computed using the formula $S=\frac{1}{2} b c \sin A$. If the relative errors made in measuring $b, c$ and calculating $S$ are respectively $0.02,0.01$ and 0.13 the approximate error in $A$ when $A=\pi / 6$, is

0.05 radians

We have,

$S =\frac{1}{2} b c \sin A$

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

$\Rightarrow d S =\frac{1}{2}\{c \sin A ~d b+b \sin A ~b c+b c \cos A ~d A\}$

$\Rightarrow \frac{d S}{S} =\frac{d b}{b}+\frac{d c}{c}+\cot A d A$

$\Rightarrow 0.13=0.02+0.01+\cot \frac{\pi}{6} d A $          $\left[\begin{array}{r}∵ \frac{d S}{S}=0.13, \frac{d b}{b}=0.02 \\ \text { and } \frac{d c}{c}=0.01\end{array}\right]$

$\Rightarrow 0.10=\sqrt{3} d A$

$\Rightarrow d A=\frac{0.10}{\sqrt{3}}$ = 0.05 radians