Why is Mathematica taking so long to solve this equation?

Question

I was trying to solve a polarization problem where I needed to solve this transcendental equation.
NSolve[(Cos[[Pi]/n])^(2 n) == 0.90, n]

I don't know why but Mathematica is taking a very long time trying to solve it while Desmos gives the answer instantly. What should I do to speed it up?

Ulrich Neumann · Accepted Answer

If you give a range for n NSolve evaluates all solutions immediately: NSolve[{(Cos[[Pi]/n])^(2 n) == 0.90, 0 < n < 1}, n, Reals] {{n -> 0.000444741}, {n -> 0.000460511}, {n -> 0.00134138}, {n -> 0.00169924}, {n -> 0.00223964}, {n -> 0.00294551}, {n -> 0.00330579}, {n -> 0.00391389}, {n -> 0.004158}, {n -> 0.00468384}, {n -> 0.00519481}, {n -> 0.0056338}, {n -> 0.00573066}, {n -> 0.00677965}, {n -> 0.00754719}, {n -> 0.0079681}, {n -> 0.0081633}, {n -> 0.00947856}, {n -> 0.00995041}, {n -> 0.0102562}, {n -> 0.0112998}, {n -> 0.0114282}, {n -> 0.0130729}, {n -> 0.013244}, {n -> 0.0146004}, {n -> 0.0152648}, {n -> 0.016003}, {n -> 0.0162569}, {n -> 0.0165325}, {n -> 0.0168028}, {n -> 0.0173867}, {n -> 0.0177042}, {n -> 0.0180125}, {n -> 0.0183547}, {n -> 0.018685}, {n -> 0.0190549}, {n -> 0.0194095}, {n -> 0.0198107}, {n -> 0.0201925}, {n -> 0.0206291}, {n -> 0.0210411}, {n -> 0.0215181}, {n -> 0.021964}, {n -> 0.0224874}, {n -> 0.0229715}, {n -> 0.0235483}, {n -> 0.0240756}, {n -> 0.0247145}, {n -> 0.025291}, {n -> 0.0260025}, {n -> 0.0266353}, {n -> 0.0274324}, {n -> 0.0281301}, {n -> 0.0290293}, {n -> 0.0298022}, {n -> 0.0308242}, {n -> 0.0316849}, {n -> 0.0328564}, {n -> 0.0338209}, {n -> 0.0351763}, {n -> 0.0362645}, {n -> 0.0378501}, {n -> 0.0390874}, {n -> 0.0409654}, {n -> 0.0423851}, {n -> 0.0446417}, {n -> 0.0462881}, {n -> 0.049046}, {n -> 0.0509793}, {n -> 0.0544189}, {n -> 0.0567238}, {n -> 0.0611202}, {n -> 0.0639201}, {n -> 0.0697138}, {n -> 0.0731964}, {n -> 0.0811357}, {n -> 0.0856037}, {n -> 0.0970629}, {n -> 0.103043}, {n -> 0.120829}, {n -> 0.129337}, {n -> 0.160146}, {n -> 0.173482}, {n -> 0.237862}, {n -> 0.262804}, {n -> 0.465417}, {n -> 0.53725}}

Bob Hanlon · Answer

Re Ulrich's comment that there are probably more solutions as n approaches zero.

Length[sol = Join[
   NSolve[{(Cos[π/n])^(2 n) == 9/10, 0 <= n <= 1}, n, Reals, 
    WorkingPrecision -> 40],
   NSolve[{(Cos[π/n])^(2 n) == 9/10, n > 1}, n, Reals, 
    WorkingPrecision -> 40]]]

(* 86 *)

({min, max} = MinMax[sol[[All, 1, -1]]]) // N

(* {0.00134138, 93.6922} *)

Looking in the region {0, min}

Length[sol2 = 
  NSolve[{(Cos[π/n])^(2 n) == 9/10, 0 <= n <= min}, n, Reals, 
   WorkingPrecision -> 40]]

(* 967 *)

({min2, max2} = MinMax[sol2[[All, 1, -1]]]) // N

(* {2.40645*10^-6, 0.00134138} *)

Zooming in again,

Length[sol3 = 
  NSolve[{(Cos[π/n])^(2 n) == 9/10, 0 <= n <= min2}, n, Reals, 
   WorkingPrecision -> 40]]

(* 831 *)

and so on ...

Why is Mathematica taking so long to solve this equation?

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