Is this number meaningful?

I was looking at numbers of the form

x^n =cos x

and noticed that the smallest value of n for which x is real is about 0.1714, with x=~0.487.

I know that x can only be found numerically, but is there a way to derive n, or find it some way besides trial and error?

Is there anything special about this number besides the fact that it does the thing I just mentioned?

Extrapolating to hyperbolic land, the minimal value of n for real x in the equation

x^n =cosh x

is just under 2 (about 1.93).