An approximate value of $\cos 40°$ is 

Let $f\left(x\right)=\cos x·40°=45°-5°$

$=\frac{\pi }{4}-\frac{\pi }{180}\times 5=\frac{\pi }{4}-\frac{\pi }{36}$ radians

We use a differential to estimate the change in cos x when  x decreases from

$\pi /4\text{ to }\pi /4-(\pi /36)$

With $x=\pi /4\text{ and }h=-\pi /36 and df$ is given by

$df=-f^{\text{'}}\left(x\right)h=-\left(-\frac{\pi }{36}\right)\sin \left(\frac{\pi }{4}\right)$

$=\frac{\pi }{36}·\frac{\sqrt{2}}{2}=\frac{\pi \sqrt{2}}{72}=0.0617$

$\cos 40\cong \cos 45+0.0617\cong 0.7071+0.0617$

              $=0.7688$