Using differentials, find the approximate values of the following…

Question

Using differentials, find the approximate values of the following:
$\sin\Big(\frac{22}{14}\Big)$

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Answer

Consider the function $\text{y}=\text{f} (\text{x})=\sin\text{x}^\circ$ Let:
$\text{x}=\frac{22}{7}$
$\text{x}+\triangle \text{x}=\frac{22}{14}$ Then,
$\triangle\text{x}= \frac{-22}{14}$For $\text{x}=\pi$
$\text{y}=\sin\Big (\frac{22}{7}\Big)=0$ Let:
$\text{dx}=\triangle \text{x}=\sin\frac{-22}{14}=-\sin\Big (\frac{\pi}{2}\Big)=-1$ Now, $\text{y}=\sin\text {x}$
$\Rightarrow\frac {\text{dy}}{\text{dx}}=\cos\text{x}$ $\Rightarrow\Big(\frac {\text{dy}}{\text{dx}}\Big)_{\text{x}= \frac{22}{7}}=-1$ $\therefore\triangle \text{y}=\text{dy}=\frac{\text{dy}} {\text{dx}}\text{dx}=-1\times(-1)=1$ $\Rightarrow\triangle \text{y} =1$ $\therefore\sin\frac {22}{14}=\text{y}+\triangle\text{y}=1$

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