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

Posted 7 months 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