So here is what I am doing:
I have ($ 215 \times \sin 22^\circ $) which has the correct answer $ \rightarrow 80.5404 \approx 81 $
Now I perform the following procedure:
$ \sin 22^\circ $ is not known, but it lies between $ \sin0^\circ \text{ and } \sin 30^\circ $
i.e $$ \sin0^\circ < \sin22^\circ < \sin30^\circ $$
We know, $ \ \sin0^\circ = 0 \text{ and } \sin30^\circ = \frac1 2$
Taking average of: $\ \sin0^\circ \text{ & } \sin30^\circ $
$$\implies \sin22^\circ \approx \frac{ \sin0^\circ + \sin30^\circ}{2} $$
$$\implies \sin22^\circ \approx \frac{0+ 0.5}{2} = \frac 1 4 $$
$\therefore 215 \times \cfrac 1 4 = 53.75 \text{ ...(i)}$
And $ 215 \times \cfrac 1 2 = 107.5 \text{ ...(ii)} \ \ \ \ $ [ basically, $ \sin30 $]
Taking the average of eq (i) and (ii), i.e
$$ \cfrac{53.75+107.5}{2} = 80.625 \approx 81 $$
Which almost matches with the calculator answer $80.5404$
Can anyone please explain how come my answer was so close, what was I doing and whats happening?
I think the answer is that
$$\sin(22^\circ)\approx0.3746, $$
which you're approximating with the average of the quantities
$$ A=\sin(30^\circ)=\frac{1}{2} $$
and
$$ B = \frac{\sin(0^\circ)+\sin(30^\circ)}{2}=\frac{1}{4}. $$
We have that
$$ \frac{A+B}{2}=\frac{3}{8}=0.375, $$
which is close to $0.3746\approx\sin(22^\circ)$.