The approximate value of $\cos {31}^{\circ }$ is

 

Known value near to $\cos {31}^{\circ }$ is $\cos {30}^{\circ }$ 

$=\cos \frac{\pi }{6}=\frac{\sqrt{3}}{2}$

Let    $y=\cos x{y}^{\mathrm{\prime }}\left(x\right)=-\sin x$

$\cos 31=\cos \left(\frac{\pi }{6}+\frac{\pi }{180}\right)$ . So putting $x=\frac{\pi }{6}\text{ and }\mathrm{\Delta }x=\frac{\pi }{180}$

we have $y^{\mathrm{\prime }}\left(\frac{\pi }{6}\right)=-\sin \frac{\pi }{6}=-\frac{1}{2}$

$\begin{array}{l}\cos (x+\mathrm{\Delta }x)\simeq \cos x+f^{\mathrm{\prime }}\left(x\right)\mathrm{\Delta }x\text{. Putting }x=\frac{\pi }{6}\text{ and }\mathrm{\Delta }x=\frac{\pi }{180},\\ \text{ we have }\end{array}$

$\begin{array}{l}\cos 31\simeq \frac{\sqrt{3}}{2}-\frac{1}{2}\frac{\pi }{180}=\frac{1.732}{2}-\frac{1}{2}\times \frac{3.14}{180}\\ =0.851\end{array}$