SOLUTION 10: Begin with the function
$$f(x)= \sin x$$
and choose
$$x-values: \displaystyle{ \pi \over 6 } \rightarrow \displaystyle{ \pi \over 6 } - 0.09$$
so that
$$\Delta x = (\displaystyle{ \pi \over 6 } - 0.09) - \displaystyle{ \pi \over 6 } = -0.09$$
The derivative of $\ y=f(x) \$ is
$$f'(x)= \cos x$$
The exact change of $y-$values is
$$\Delta y = f(\displaystyle{ \pi \over 6 } - 0.09) - f(\displaystyle{ \pi \over 6 })$$ $$= \sin(\displaystyle{ \pi \over 6 } - 0.09) - \sin(\displaystyle{ \pi \over 6 })$$ $$= \sin(\displaystyle{ \pi \over 6 } - 0.09) - \displaystyle{ 1 \over 2 }$$ $$= \sin(\displaystyle{ \pi \over 6 } - 0.09) - 0.5$$
The Differential is
$$dy = f'(\displaystyle{ \pi \over 6 }) \ \Delta x$$ $$= \cos(\displaystyle{ \pi \over 6 }) \cdot (-0.09)$$ $$= \displaystyle{ \sqrt{3} \over 2 } \cdot (-0.09)$$ $$\approx \displaystyle{ 1.73205 \over 2 } \cdot (-0.09)$$ $$\approx -0.0779$$
We will assume that
$$\Delta y \approx dy \ \ \ \ \longrightarrow$$ $$\sin(\displaystyle{ \pi \over 6 } - 0.09) - 0.5 \approx -0.0779 \ \ \ \ \longrightarrow$$ $$\sin(\displaystyle{ \pi \over 6 } - 0.09) \approx 0.5 - 0.0779 \ \ \ \ \longrightarrow$$ $$\sin(\displaystyle{ \pi \over 6 } - 0.09) \approx 0.4221 \ \ \ \ \longrightarrow$$

NOTE: The number $\displaystyle{ \pi \over 6 }$ was chosen for its proximity to $\displaystyle{ \pi \over 6 }-0.09$ and for it's convenient sine value. Check the accuracy of the final estimate using a CALCULATOR: $\sin(\displaystyle{ \pi \over 6 }-0.09 ) \approx 0.4201$