Question

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

Answer 1

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

\[\text { Let }: \]

\[ x = \frac{22}{7} \]

\[x + ∆ x = \frac{22}{14}\]

\[\text { Then,} \]

\[ ∆ x = \frac{- 22}{14}\]

\[\text { For } x = \pi, \]

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

\[\text { Let }: \]

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

\[\text { Now }, y = \sin x\]

\[ \Rightarrow \frac{dy}{dx} = \cos x\]

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

\[ \therefore ∆ y = dy = \frac{dy}{dx}dx = - 1 \times \left( - 1 \right) = 1\]

\[ \Rightarrow ∆ y = 1\]

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