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?
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}}
n == 93.6922 you can get with NSolve[{(Cos[\[Pi]/n])^(2 n)==0.90,n>1},n,Reals]. It looks like there are none for n<0
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Commented
Jul 30, 2020 at 8:43
n->0
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Commented
Jul 30, 2020 at 9:03
0 < n < 0.1 you get even more solutions. There's probably an infinite number of them.
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Commented
Jul 30, 2020 at 14:38
LogLinearPlot[Cos[\[Pi]/x]^(2 x), {x, 0, 10}] shows that there is an infinite number of solutions, which shouldn't be surprising, as Cos[π/x] has an essential singularity.
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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 ...
