A recent question had a comment which intrigued me, so I went to Wolfram Alpha and put in $\cos(36°)$ and it was half the golden ratio? Why is $\cos(36^\circ) = \frac{\phi}{2}$? Is there a nice geometric proof? I have never heard this before and it's quite fascinating. Is it a coincidence?
Is there something special about the angle $\frac{\pi}{5}$?
Because it's $\frac{\sin 72^\circ}{2\sin 36^\circ}=\frac{\sin 108^\circ}{2\sin 36^\circ}$, which by the sine rule, applied to a side-side-diagonal isosceles triangle of a rectangular pentagon, is $\frac{\varphi}{2}$.