Given tan x = 2.
tan(x/8) is the root of f(x), an 8th degree monic polynomial with integer coefficients.
What is the value of f(1)?
It's not entirely clear to me what tan(x/8) means in this problem.
If tan(x) = 2, then geometrically, the corresponding triangle is 1,2, sqrt(5). tan (x/8) is then around 0.139283952438412.
There are some very tedious ways of working back from the root to the polynomial, but there is an implementation RootApproximant in WolframAlpha that does the heavy lifting.
However these figures do not render a monic polynomial in the 8th degree.
On the other hand, using y=x/8 and not tan (x/8) gives the slightly larger RootApproximant[0.1383935897242,8], producing the monic form x^8+15x^7-10x^6-10x^5+4x^4+24x^3-5x^2- 7x + 1, with f(1) worth 13.
Posted by broll
on 2018-11-22 00:16:33