Answer
$f(x)=\sin x$ and $a=\frac{\pi}{6}$
Work Step by Step
The definition of derivative at point $a$ is
$f'(a)=\lim\limits_{h \to 0}\frac{f(a+h)-f(a)}{h}$
But given that
$f'(a)=\lim\limits_{h \to 0}\frac{\sin(\frac{\pi}{6}+h)-0.5}{h}$
$\implies f(a+h)=\sin(\frac{\pi}{6}+h)$ and $f(a)=0.5=\sin \frac{\pi}{6}$
which gives $f(x)=\sin x$ and $a=\frac{\pi}{6}$.
