Hmm..I ust realized for the expression above the root must be complex of the form i^1/4.
Trig functions of the form cosx+cosy =0 can be solved because of the product formula:
cos(x)+cox(y)=2*(cos((x+y)/2))*(cos((x-y)/2)))
but is there such a formula for cos(x)+cos(y)+cos(z) ?
cos(x)+cos(ix)+cos(x*i^3/2)+cos(x*i^1/2)=0 for x
I have spent a lot of time finding an analytic root to this equation without success. An analytic root may not exist. I don't know. It is roughly equal to (8facorial)^(1/8)