Answer
$$ 0.023,\ \ 0.02299797 ,\ \ 0.00000203$$
Work Step by Step
Given $$\sin (0.023)$$
Consider $f(x)=\sin x $, $a= 0$, $\Delta x=0.023$, since
\begin{align*}
f'(x) &= \cos x \\
f'(0)&=1
\end{align*}
Then the linear approximation is given by
\begin{align*}
\Delta &f \approx f^{\prime}(a) \Delta x\\
&= (1)(0.023)\\
&= 0.023
\end{align*}
and the actual change is given by
\begin{align*}
\Delta f&=f(a+\Delta x)-f(a)\\
&=f(0.023)-f(0)\\
& = 0.02299797
\end{align*}
Hence the error is
$$ |0.02299797- 0.023| \approx 0.00000203$$
