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I need to find the value of n but I get an error. I used this:

Solve[
  4*ArcSin[Sqrt[(4 - (n^2))/(3*(n^2))]] - 2*ArcSin[Sqrt[(4 - (n^2))/3]] == 20.2826, 
  n
]

But I get this error:

Solve::inex: Solve was unable to solve the system with inexact coefficients or the system obtained by direct rationalization of inexact numbers present in the system. Since many of the methods used by Solve require exact input, providing Solve with an exact version of the system may help.

Please, help me to get the value of n. I know that the value of n should be about 1 or 2.

Thanks

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  • 1
    $\begingroup$ Btw, Solve::inex is not an error, but a warning about how Solve is solving the equation. $\endgroup$
    – Michael E2
    Commented Apr 3, 2023 at 17:40
  • 1
    $\begingroup$ Perhaps you don't want ArcSin[], which has a restricted range. Alternativie: FindRoot[4*y1 - 2*y2 == 202826/10000 && Sin[y1]^2 == (4 - n^2)/(3 n^2) && Sin[y2]^2 == 1/3 (4 - n^2), {{y1, 2}, {y2, -1}, {n, 1}}, WorkingPrecision -> 24] $\endgroup$
    – Michael E2
    Commented Apr 3, 2023 at 18:06

1 Answer 1

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It seems that there are no solutions. Your left-hand side has the following domain:

FunctionDomain[
 4*ArcSin[Sqrt[(4 - (n^2))/(3*(n^2))]] - 2*ArcSin[Sqrt[(4 - (n^2))/3]],
 n
]

(* Out: -2 <= n <= -1 || 1 <= n <= 2 *)

and plotting over that domain does not indicate that a value of 20 would be attainable anywhere:

Plot[
 4*ArcSin[Sqrt[(4 - (n^2))/(3*(n^2))]] - 
  2*ArcSin[Sqrt[(4 - (n^2))/3]](*-20.2826*),
 {n, -3, 3}, PlotRange -> Full,
]

plot shows value of function between 0 and 3

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  • $\begingroup$ FunctionRange also implies no solutions. So does the replacement _ArcSin :> Interval[{-Pi/2, Pi/2}], which produces an interval much greater than the range yet still not containing a solution. It's possible the OP considers y = arcsin x to be a multivalued function representing any solution to sin y = x. (+1) $\endgroup$
    – Michael E2
    Commented Apr 3, 2023 at 17:51

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