Why is $\cos(36°) = \frac{\phi}{2}$ where $\phi$ is the golden ratio?

Question

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 $\frac{\pi}{5}$?

https://math.stackexchange.com/questions/827540/proving-trigonometric-equation-cos36-circ-cos72-circ-1-2 — lab bhattacharjee, Mar 29 '19 at 17:37
From https://en.wikipedia.org/wiki/Golden_ratio#Alternative_forms : "These correspond to the fact that the length of the diagonal of a regular pentagon is φ times the length of its side" — amsmath, Mar 29 '19 at 17:38
@labbhattacharjee I don't think that the question you cite is a good duplicate target. First off, it is itself marked as a duplicate of another question. Second, while computing $\cos(36^\circ)$ is a step in the argument that $\cos(36^\circ) - \cos(72^\circ) = 1/2$, that is, I think, not quite what this question is asking about. — Xander Henderson, Mar 29 '19 at 17:46
Possible duplicate of Exact value for $\cos 36°$ (I think that this is a better dupe target.) — Xander Henderson, Mar 29 '19 at 17:47
@Xander, I have not proposed it to be duplicate, it just shows how to calculate $\cos36^\circ$ — lab bhattacharjee, Mar 29 '19 at 18:31

score 0 · Answer 1 · answered Mar 29 '19 at 17:39

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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 $\frac{\varphi}{2}$.

answered Mar 29 '19 at 17:39

J.G.

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Why is $\cos(36°) = \frac{\phi}{2}$ where $\phi$ is the golden ratio?

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