Area of a triangle is given by $A = \dfrac{1}{2}ab \sin C$ where the symbols have their usual meanings. If $C = \dfrac{\pi}{3}$ and there is an error in measuring C by $x \%$, calculate the approximate percentage error in the area if a and b are constant.
Given: $C=\dfrac{\pi}{3}$; $\dfrac{\Delta C}{C} = x\% $ $\implies$ $\Delta C = \dfrac{x}{100} \times C = \dfrac{\pi x}{300}$
Area $A = \dfrac{1}{2}ab\sin C = \dfrac{1}{2}ab \sin \left(\dfrac{\pi}{3}\right) = \dfrac{\sqrt{3}}{4}ab$
$\dfrac{dA}{dC} = \dfrac{1}{2}ab \cos C$
$\begin{aligned}
\text{Now, } \Delta A & = \left(\dfrac{dA}{dC}\right)\times \Delta C \\
& = \dfrac{1}{2}ab \cos C \times \dfrac{\pi x}{300} \\
& = \dfrac{1}{2}ab \times \cos \left(\dfrac{\pi}{3}\right) \times \dfrac{\pi x}{300} \\
& = \dfrac{1}{4} ab \times \dfrac{\pi x}{300}
\end{aligned}$
$\begin{aligned}
\therefore \% \; \text{Error in area} & = \dfrac{\Delta A}{A} \times 100 \; \% \\
& = \dfrac{\dfrac{1}{4}ab \times \dfrac{\pi x}{300}}{\dfrac{\sqrt{3}}{4}ab} \times 100 \; \% \\
& = \dfrac{\pi x}{3\sqrt{3}} \; \%
\end{aligned}$