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

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a

\frac{S}{2 a} Δ A

b

\frac{2 S}{a} Δ A

c

b c \sin A Δ A

d

none of these

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detailed solution

Correct option is B

In $∆ A B C$ , we have

$\begin{array}{l} 2 b c \cos A = b^{2} + c^{2} - a^{2} \\ \Rightarrow d (2 b c \cos A) = d (b^{2} + c^{2} - a^{2}) \\ \Rightarrow - 2 b c \sin A d A = - 2 a d a \\ \begin{array}{l} \Rightarrow b c \sin A d A = a d a \\ \Rightarrow \frac{2}{a} (\frac{1}{2} b c \sin A) d A = d a \\ \Rightarrow d a = \frac{2 S}{a} d A \\ \Rightarrow Δ a = \frac{2 S}{a} Δ A [∵ d a ≅ Δ a and d A ≅ Δ A] \end{array} \end{array}$

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