An Opening Note : First of all, I want to make this very clear that by the phrase "without using trigonometry tables", I mean without using them to find $\sin$ values of the "non-standard angles" (For example $73.5^\circ$).
Now, its obviously easy to find a somewhat "broader" range for the answer. Taking the example of $73.5^\circ$, its obvious that $\sin 73.5^\circ$ will lie between $\sin 60^\circ$ and $\sin 90^\circ$. But, how can this range be narrowed ?
One answer I think would be between $sin 60^\circ$ and $sin(60^\circ+18^\circ)$.
What to do next ?
Is there a way to find an even better approximation without using "much" calculations ?