Is this (1.11716..) a known/named constant?

While looking at the Wikipedia entry for the Supergolden ratio aka Narayana's cow constant (cf. https://en.wikipedia.org/wiki/Supergolden_ratio), I noticed the following construction involving that constant and a 120-degree ($$2\pi/3$$ radian) triangle on that page:

I then began to wonder about similar constructions with other fractional angles: $$2\pi/2$$, $$2\pi/4$$, ...

I found that an angle of $$2\pi/2$$ gives a degenerate triangle based on the golden ratio:

$$2\pi/4$$ gives a right triangle based on the square-root of the golden ratio:

And $$2\pi/6$$ gives an equilateral triangle based on unity:

Nothing too surprising or new in the results so far, but an angle of $$2\pi/5$$ (72-degrees) produced the following:

So, the positive solution to $$-2 - x + \sqrt 5 x - 2 x^2 + 2 x^4=0$$, approximate value $$1.11716..$$, minimal polynomial $$1 + x + x^2 + x^3 - x^4 - x^5 - 2 x^6 + x^8$$

Is this a known/named constant?

• It is not a constant from this list, but this is only a small list... Commented Jan 14, 2019 at 20:58
• The Inverse Symbolic Calculator doesn't have a match for the longer decimal expansion, $1.11715586807111\ldots$.
– Blue
Commented Jan 14, 2019 at 21:05
• I was getting around to this question myself. Pi/4 is related to the square root of $1 - x^2 - 2 x^3 + x^4=0$. I can also relate Pi/3 to the square root of the root for $1 - x - x^2 - 2 x^3 + x^4=0$. Commented Jan 14, 2019 at 22:06
• The polynomial is known. lmfdb.org/NumberField/8.4.42903125.1 Commented Jan 14, 2019 at 22:24
• The polynomials involved for Pi/7 and Pi/9 are not known at lmfdb.org. Commented Jan 14, 2019 at 23:11