While exploring a relatively simple sin equation, depending on how I process I get different forms of the root. Numerically, they appear to be the same, but I am having difficult figuring out how they could be, and would appreciate some guidance.
The more standard root is
21*arccos(RootOf(448*_Z^7+192*_Z^6-784*_Z^5-288*_Z^4+392*_Z^3+108*_Z^2-49*_Z-6, index = 2))/Pi
The less standard approach is
-(21*I)*ln(RootOf(7*_Z^14+6*_Z^13+6*_Z+7, index = 2))/Pi
Both of the RootOf appear to be irreducible, and it is not clear to me how you could transform the more complicated degree 7 polynomial into the simpler degree 14 polynomial while still retaining exactly the same roots?
If the equivalence holds up then it would be much easier for me to generalize the second form than the first.
I was, by the way, looking at minimizing sin(2/7*Pi*x) _+ sin(1/3*Pi*x) over its first complete cycle, as part of working up to a general rule for minimizing sum of sin of different amplitude and periods. diff(), then standard form is solve(), and less standard form is solve() of convert/exp() of the diff()