Question

Using differential, find the approximate value of the $$\sin \left(\frac{22}{14}\right)$$ ?

Answer 1

$$\text{ Consider the function }y=f\left(x\right)=\sin x.$$

$$\text{ Let }:$$

$$x=\frac{22}{7}$$

$$x+∆x=\frac{22}{14}$$

$$\text{ Then,}$$

$$∆x=\frac{-22}{14}$$

$$\text{ For }x=\pi ,$$

$$y=\sin \left(\frac{22}{7}\right)=0$$

$$\text{ Let }:$$

$$dx=∆x=\sin \frac{-22}{14}=-\sin \left(\frac{\pi }{2}\right)=-1$$

$$\text{ Now },y=\sin x$$

$$\Rightarrow \frac{dy}{dx}=\cos x$$

$$\Rightarrow {\left(\frac{dy}{dx}\right)}_{x=\frac{22}{7}}=-1$$

$$\therefore ∆y=dy=\frac{dy}{dx}dx=-1\times (-1)=1$$

$$\Rightarrow ∆y=1$$

$$\therefore \sin \frac{22}{14}=y+∆y=1$$