<s>[INST] Is there a simple pattern to memorize the sine of $0^\circ$, $15^\circ$, $30^\circ$, $45^\circ$, $60^\circ$, $75^\circ$, $90^\circ$? We know there is a nice pattern to memorize the sine of $0^\circ$, $30^\circ$, $45^\circ$, $60^\circ$,  $90^\circ$ as follows.
\begin{align}
\sin 0^\circ &= \tfrac12\sqrt0\\
\sin 30^\circ &= \tfrac12\sqrt1\\
\sin 45^\circ &= \tfrac12\sqrt2\\
\sin 60^\circ &= \tfrac12\sqrt3\\
\sin 90^\circ &= \tfrac12\sqrt4
\end{align}
We also know that 
\begin{align}
\sin 15^\circ &= \tfrac14(\sqrt6-\sqrt2)\\
\sin 75^\circ &= \tfrac14(\sqrt6+\sqrt2)
\end{align}
Question
If I want to combine these two groups, is there a simple nice pattern available for us to easily rote memorize them?
 [/INST] Hmm, this pattern works:
$$
\sin 0^{\circ} = \frac{1}{2}\sqrt{2-\sqrt{4}}, \\
\sin 15^{\circ} = \frac{1}{2}\sqrt{2-\sqrt{3}}, \\
\color{gray}{\sin 22.5^{\circ} = \frac{1}{2}\sqrt{2-\sqrt{2}},} \\
\sin 30^{\circ} = \frac{1}{2}\sqrt{2-\sqrt{1}}, \\
\sin 45^{\circ} = \frac{1}{2}\sqrt{2\pm\sqrt{0}}, \\
\sin 60^{\circ} = \frac{1}{2}\sqrt{2+\sqrt{1}}, \\
\color{gray}{\sin 67.5^{\circ} = \frac{1}{2}\sqrt{2+\sqrt{2}},} \\
\sin 75^{\circ} = \frac{1}{2}\sqrt{2+\sqrt{3}}, \\
\sin 90^{\circ} = \frac{1}{2}\sqrt{2+\sqrt{4}}.
$$
</s>