#### Question

Using differential, find the approximate value of the \[\cos\left( \frac{11\pi}{36} \right)\] ?

#### Solution

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

\[\text { Let }: \]

\[ x = \frac{\pi}{3} \]

\[x + ∆ x = \frac{11\pi}{36}\]

\[\text { Then,} \]

\[ ∆ x = \frac{- \pi}{36} = - 5^\circ\]

\[\text { For } x = \frac{\pi}{3}, \]

\[y = \cos \left( \frac{\pi}{3} \right) = 0 . 5\]

\[\text { Let }: \]

\[ dx = ∆ x = - \sin 5^\circ = - 0 . 08716\]

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

\[ \Rightarrow \frac{dy}{dx} = - \sin x\]

\[ \Rightarrow \left( \frac{dy}{dx} \right)_{x = \frac{\pi}{3}} = - 0 . 86603\]

\[ \therefore ∆ y = dy = \frac{dy}{dx}dx = - 0 . 86603 \times \left( - 0 . 08716 \right) = 0 . 075575\]

\[ \Rightarrow ∆ y = 0 . 075575\]

\[ \therefore \cos\frac{11\pi}{36} = y + ∆ y = 0 . 5 + 0 . 075575 = 0 . 575575\]