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Interpret the results obtained by Solve on the following trigonometric eq?

Posted 12 days ago
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Hello, could you help with the interpretation of the results of solving this equation.

Solve[(Cos[f]Cos[b]+Sin[f]Cos[v])Cos[d]Cos[x]+Cos[d]Sin[x]Sin[b]Sin[v]+Sin[d](Sin[f]Cos[b]-Cos[f]Sin[f]Cos[v])==0,x]
{{x -> ConditionalExpression[
    ArcTan[(-Cos[b]^2 Cos[d] Cos[f] Sin[d] Sin[f] + 
         Cos[b] Cos[d] Cos[f]^2 Cos[v] Sin[d] Sin[f] - 
         Cos[b] Cos[d] Cos[v] Sin[d] Sin[f]^2 + 
         Cos[d] Cos[f] Cos[v]^2 Sin[d] Sin[
           f]^2 - \[Sqrt](Cos[b]^2 Cos[d]^4 Cos[f]^2 Sin[b]^2 Sin[
              v]^2 + 2 Cos[b] Cos[d]^4 Cos[f] Cos[v] Sin[b]^2 Sin[
              f] Sin[v]^2 + 
            Cos[d]^4 Cos[v]^2 Sin[b]^2 Sin[f]^2 Sin[v]^2 - 
            Cos[b]^2 Cos[d]^2 Sin[b]^2 Sin[d]^2 Sin[f]^2 Sin[v]^2 + 
            2 Cos[b] Cos[d]^2 Cos[f] Cos[v] Sin[b]^2 Sin[d]^2 Sin[
              f]^2 Sin[v]^2 - 
            Cos[d]^2 Cos[f]^2 Cos[v]^2 Sin[b]^2 Sin[d]^2 Sin[f]^2 Sin[
              v]^2 + Cos[d]^4 Sin[b]^4 Sin[v]^4))/(Cos[b]^2 Cos[
           d]^2 Cos[f]^2 + 2 Cos[b] Cos[d]^2 Cos[f] Cos[v] Sin[f] + 
         Cos[d]^2 Cos[v]^2 Sin[f]^2 + Cos[d]^2 Sin[b]^2 Sin[v]^2), 
      Csc[b] Csc[v] Sec[
        d] (-Cos[b] Sin[d] Sin[f] + Cos[f] Cos[v] Sin[d] Sin[f] + (
         Cos[b]^3 Cos[d]^2 Cos[f]^2 Sin[d] Sin[f])/(
         Cos[b]^2 Cos[d]^2 Cos[f]^2 + 
          2 Cos[b] Cos[d]^2 Cos[f] Cos[v] Sin[f] + 
          Cos[d]^2 Cos[v]^2 Sin[f]^2 + Cos[d]^2 Sin[b]^2 Sin[v]^2) - (
         Cos[b]^2 Cos[d]^2 Cos[f]^3 Cos[v] Sin[d] Sin[f])/(
         Cos[b]^2 Cos[d]^2 Cos[f]^2 + 
          2 Cos[b] Cos[d]^2 Cos[f] Cos[v] Sin[f] + 
          Cos[d]^2 Cos[v]^2 Sin[f]^2 + Cos[d]^2 Sin[b]^2 Sin[v]^2) + (
         2 Cos[b]^2 Cos[d]^2 Cos[f] Cos[v] Sin[d] Sin[f]^2)/(
         Cos[b]^2 Cos[d]^2 Cos[f]^2 + 
          2 Cos[b] Cos[d]^2 Cos[f] Cos[v] Sin[f] + 
          Cos[d]^2 Cos[v]^2 Sin[f]^2 + Cos[d]^2 Sin[b]^2 Sin[v]^2) - (
         2 Cos[b] Cos[d]^2 Cos[f]^2 Cos[v]^2 Sin[d] Sin[f]^2)/(
         Cos[b]^2 Cos[d]^2 Cos[f]^2 + 
          2 Cos[b] Cos[d]^2 Cos[f] Cos[v] Sin[f] + 
          Cos[d]^2 Cos[v]^2 Sin[f]^2 + Cos[d]^2 Sin[b]^2 Sin[v]^2) + (
         Cos[b] Cos[d]^2 Cos[v]^2 Sin[d] Sin[f]^3)/(
         Cos[b]^2 Cos[d]^2 Cos[f]^2 + 
          2 Cos[b] Cos[d]^2 Cos[f] Cos[v] Sin[f] + 
          Cos[d]^2 Cos[v]^2 Sin[f]^2 + Cos[d]^2 Sin[b]^2 Sin[v]^2) - (
         Cos[d]^2 Cos[f] Cos[v]^3 Sin[d] Sin[f]^3)/(
         Cos[b]^2 Cos[d]^2 Cos[f]^2 + 
          2 Cos[b] Cos[d]^2 Cos[f] Cos[v] Sin[f] + 
          Cos[d]^2 Cos[v]^2 Sin[f]^2 + Cos[d]^2 Sin[b]^2 Sin[v]^2) +
          (Cos[b] Cos[d] Cos[
             f] \[Sqrt](-Cos[d]^2 Sin[b]^2 Sin[
                v]^2 (-Cos[b]^2 Cos[d]^2 Cos[f]^2 - 
                 2 Cos[b] Cos[d]^2 Cos[f] Cos[v] Sin[f] - 
                 Cos[d]^2 Cos[v]^2 Sin[f]^2 + 
                 Cos[b]^2 Sin[d]^2 Sin[f]^2 - 
                 2 Cos[b] Cos[f] Cos[v] Sin[d]^2 Sin[f]^2 + 
                 Cos[f]^2 Cos[v]^2 Sin[d]^2 Sin[f]^2 - 
                 Cos[d]^2 Sin[b]^2 Sin[v]^2)))/(Cos[b]^2 Cos[d]^2 Cos[
              f]^2 + 2 Cos[b] Cos[d]^2 Cos[f] Cos[v] Sin[f] + 
            Cos[d]^2 Cos[v]^2 Sin[f]^2 + 
            Cos[d]^2 Sin[b]^2 Sin[v]^2) + (Cos[d] Cos[v] Sin[
             f] \[Sqrt](-Cos[d]^2 Sin[b]^2 Sin[
                v]^2 (-Cos[b]^2 Cos[d]^2 Cos[f]^2 - 
                 2 Cos[b] Cos[d]^2 Cos[f] Cos[v] Sin[f] - 
                 Cos[d]^2 Cos[v]^2 Sin[f]^2 + 
                 Cos[b]^2 Sin[d]^2 Sin[f]^2 - 
                 2 Cos[b] Cos[f] Cos[v] Sin[d]^2 Sin[f]^2 + 
                 Cos[f]^2 Cos[v]^2 Sin[d]^2 Sin[f]^2 - 
                 Cos[d]^2 Sin[b]^2 Sin[v]^2)))/(Cos[b]^2 Cos[d]^2 Cos[
              f]^2 + 2 Cos[b] Cos[d]^2 Cos[f] Cos[v] Sin[f] + 
            Cos[d]^2 Cos[v]^2 Sin[f]^2 + 
            Cos[d]^2 Sin[b]^2 Sin[v]^2))] + 2 \[Pi] C[1], 
    C[1] \[Element] Integers]}, {x -> 
   ConditionalExpression[
    ArcTan[(-Cos[b]^2 Cos[d] Cos[f] Sin[d] Sin[f] + 
         Cos[b] Cos[d] Cos[f]^2 Cos[v] Sin[d] Sin[f] - 
         Cos[b] Cos[d] Cos[v] Sin[d] Sin[f]^2 + 
         Cos[d] Cos[f] Cos[v]^2 Sin[d] Sin[
           f]^2 + \[Sqrt](Cos[b]^2 Cos[d]^4 Cos[f]^2 Sin[b]^2 Sin[
              v]^2 + 2 Cos[b] Cos[d]^4 Cos[f] Cos[v] Sin[b]^2 Sin[
              f] Sin[v]^2 + 
            Cos[d]^4 Cos[v]^2 Sin[b]^2 Sin[f]^2 Sin[v]^2 - 
            Cos[b]^2 Cos[d]^2 Sin[b]^2 Sin[d]^2 Sin[f]^2 Sin[v]^2 + 
            2 Cos[b] Cos[d]^2 Cos[f] Cos[v] Sin[b]^2 Sin[d]^2 Sin[
              f]^2 Sin[v]^2 - 
            Cos[d]^2 Cos[f]^2 Cos[v]^2 Sin[b]^2 Sin[d]^2 Sin[f]^2 Sin[
              v]^2 + Cos[d]^4 Sin[b]^4 Sin[v]^4))/(Cos[b]^2 Cos[
           d]^2 Cos[f]^2 + 2 Cos[b] Cos[d]^2 Cos[f] Cos[v] Sin[f] + 
         Cos[d]^2 Cos[v]^2 Sin[f]^2 + Cos[d]^2 Sin[b]^2 Sin[v]^2), 
      Csc[b] Csc[v] Sec[
        d] (-Cos[b] Sin[d] Sin[f] + Cos[f] Cos[v] Sin[d] Sin[f] + (
         Cos[b]^3 Cos[d]^2 Cos[f]^2 Sin[d] Sin[f])/(
         Cos[b]^2 Cos[d]^2 Cos[f]^2 + 
          2 Cos[b] Cos[d]^2 Cos[f] Cos[v] Sin[f] + 
          Cos[d]^2 Cos[v]^2 Sin[f]^2 + Cos[d]^2 Sin[b]^2 Sin[v]^2) - (
         Cos[b]^2 Cos[d]^2 Cos[f]^3 Cos[v] Sin[d] Sin[f])/(
         Cos[b]^2 Cos[d]^2 Cos[f]^2 + 
          2 Cos[b] Cos[d]^2 Cos[f] Cos[v] Sin[f] + 
          Cos[d]^2 Cos[v]^2 Sin[f]^2 + Cos[d]^2 Sin[b]^2 Sin[v]^2) + (
         2 Cos[b]^2 Cos[d]^2 Cos[f] Cos[v] Sin[d] Sin[f]^2)/(
         Cos[b]^2 Cos[d]^2 Cos[f]^2 + 
          2 Cos[b] Cos[d]^2 Cos[f] Cos[v] Sin[f] + 
          Cos[d]^2 Cos[v]^2 Sin[f]^2 + Cos[d]^2 Sin[b]^2 Sin[v]^2) - (
         2 Cos[b] Cos[d]^2 Cos[f]^2 Cos[v]^2 Sin[d] Sin[f]^2)/(
         Cos[b]^2 Cos[d]^2 Cos[f]^2 + 
          2 Cos[b] Cos[d]^2 Cos[f] Cos[v] Sin[f] + 
          Cos[d]^2 Cos[v]^2 Sin[f]^2 + Cos[d]^2 Sin[b]^2 Sin[v]^2) + (
         Cos[b] Cos[d]^2 Cos[v]^2 Sin[d] Sin[f]^3)/(
         Cos[b]^2 Cos[d]^2 Cos[f]^2 + 
          2 Cos[b] Cos[d]^2 Cos[f] Cos[v] Sin[f] + 
          Cos[d]^2 Cos[v]^2 Sin[f]^2 + Cos[d]^2 Sin[b]^2 Sin[v]^2) - (
         Cos[d]^2 Cos[f] Cos[v]^3 Sin[d] Sin[f]^3)/(
         Cos[b]^2 Cos[d]^2 Cos[f]^2 + 
          2 Cos[b] Cos[d]^2 Cos[f] Cos[v] Sin[f] + 
          Cos[d]^2 Cos[v]^2 Sin[f]^2 + 
          Cos[d]^2 Sin[b]^2 Sin[
            v]^2) - (Cos[b] Cos[d] Cos[

             f] \[Sqrt](-Cos[d]^2 Sin[b]^2 Sin[
                v]^2 (-Cos[b]^2 Cos[d]^2 Cos[f]^2 - 
                 2 Cos[b] Cos[d]^2 Cos[f] Cos[v] Sin[f] - 
                 Cos[d]^2 Cos[v]^2 Sin[f]^2 + 
                 Cos[b]^2 Sin[d]^2 Sin[f]^2 - 
                 2 Cos[b] Cos[f] Cos[v] Sin[d]^2 Sin[f]^2 + 
                 Cos[f]^2 Cos[v]^2 Sin[d]^2 Sin[f]^2 - 
                 Cos[d]^2 Sin[b]^2 Sin[v]^2)))/(Cos[b]^2 Cos[d]^2 Cos[
              f]^2 + 2 Cos[b] Cos[d]^2 Cos[f] Cos[v] Sin[f] + 
            Cos[d]^2 Cos[v]^2 Sin[f]^2 + 
            Cos[d]^2 Sin[b]^2 Sin[v]^2) - (Cos[d] Cos[v] Sin[
             f] \[Sqrt](-Cos[d]^2 Sin[b]^2 Sin[
                v]^2 (-Cos[b]^2 Cos[d]^2 Cos[f]^2 - 
                 2 Cos[b] Cos[d]^2 Cos[f] Cos[v] Sin[f] - 
                 Cos[d]^2 Cos[v]^2 Sin[f]^2 + 
                 Cos[b]^2 Sin[d]^2 Sin[f]^2 - 
                 2 Cos[b] Cos[f] Cos[v] Sin[d]^2 Sin[f]^2 + 
                 Cos[f]^2 Cos[v]^2 Sin[d]^2 Sin[f]^2 - 
                 Cos[d]^2 Sin[b]^2 Sin[v]^2)))/(Cos[b]^2 Cos[d]^2 Cos[
              f]^2 + 2 Cos[b] Cos[d]^2 Cos[f] Cos[v] Sin[f] + 
            Cos[d]^2 Cos[v]^2 Sin[f]^2 + 
            Cos[d]^2 Sin[b]^2 Sin[v]^2))] + 2 \[Pi] C[1], 
    C[1] \[Element] Integers]}}

The solution has the form

ConditionalExpression[something +2Pi*C[1],
 Element[C[1], Integers]]

which means that the expression something +2Pi*C[1] is a solution provided that C[1] is an integer. Try replacing C[1] with numbers:

sol = Solve[(Cos[f] Cos[b] + Sin[f] Cos[v]) Cos[d] Cos[x] + 
     Cos[d] Sin[x] Sin[b] Sin[v] + 
     Sin[d] (Sin[f] Cos[b] - Cos[f] Sin[f] Cos[v]) == 0, x];
sol /. C[1] -> 1
sol /. C[1] -> 1/2
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