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Differentials, Errors and Approximations — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsDifferentials, Errors and Approximations4 Marks
Question
Using differentials, find the approximate values of the following:
$\sin\Big(\frac{22}{14}\Big)$
✓

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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