I've noticed some relationships with cosine and square root.

real-analysis calculus trigonometry algebra-precalculus square-numbers

Your list is sometimes given in trig books as a mnemonic device. Looking a little deeper the pattern that the angles in a regular $n$-gon give algebraic values for the $\cos$ function is showing itself, though the pattern of simple surds does not continue, e.g. $\cos (\frac{2\pi}{5})=\frac{1}{4}(\sqrt{5}-1)$


This rules we used in high school with fingers do apply for both sine and cosine see the pictures

enter image description here

See also the following

enter image description here

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