Chapter 4 - Applications of the Derivative - 4.1 Linear Approximation and Applications - Exercises - Page 172: 13

$\Delta y\approx-0.007$

For small $\Delta x$, $\Delta y\approx dy$. That is, $\Delta y\approx f'(a) dx$. $f'(a)=-\sin a=-\sin \frac{\pi}{6}= -\frac{1}{2}$ $dx=0.014$ $\implies \Delta y\approx -\frac{1}{2}\times0.014=-0.007$