Find a root for this expression:

[MaxwellPhill]

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)

 

[Zurtex]

I don't think there is an anaylitical solution and I am not sure that there even is a root.

x = 0 is closer than your x = 3.764350600, but that's only taken it down to 4 rather than 8.0003108.

 

[MaxwellPhill]

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) ?

 

[Zurtex]

I did some 3D Plots in mathematica to get a rough idea where a root might be and then used the FindRoot function and got that 1.4405686011239758 - 3.477840254362339i is approximatly a root.

I don't think there is any simplification like you want.