Finding the exact value of $\sin1^\circ$

The exact values of sine, such as $\sin 30^\circ = 1/2$ and $\sin 45^\circ = 1/\sqrt{2}$ are well known, but the exact value for sine for other angles such as $\sin 1^\circ$ and $\sin 7^\circ$ are not widely known. In this note, we find the exact values for $\sin 15^\circ$, $\sin 18^\circ$, $\sin 3^\circ$ and $\sin 1^\circ$.