# Does anyone here know how can we extract Lambda from the following equation

I am working on an estimation problem where I have the following

Code is

A = 1 + Pi*Csc[(1 + Pi)*\[Lambda]]^2*
(Pi + Sin[2*(1 + Pi)*\[Lambda]])


Can anyone help me how can we extract $$\lambda$$ from this equation in Mathematica?

• Try : Solve or Reduce ? Feb 1 '21 at 8:18
• @MariuszIwaniuk i want lemda on one side and rest on other side so that I can calculate lemda which is my requirement Feb 1 '21 at 8:25
• Execute this in Mathematica:Solve[A == 1 + Pi*Csc[(1 + Pi)*\[Lambda]]^2*(Pi + Sin[2*(1 + Pi)*\[Lambda]]), \[Lambda]] then you have the answer? Feb 1 '21 at 8:28
• You might get a simpler result if there are known constraints on A and/or lambda Feb 1 '21 at 12:56
• Look at the plot of lambda as a function of A, i.e., ParametricPlot[{1 + Pi*Csc[(1 + Pi)*\[Lambda]]^2*(Pi + Sin[2*(1 + Pi)*\[Lambda]]), \[Lambda]}, {\[Lambda], 0, 2 Pi}, WorkingPrecision -> 15, AspectRatio -> 1, Frame -> True, FrameLabel -> {"A", "\[Lambda]"}] Feb 1 '21 at 13:15

Help Solve with TrigExpand.

Function is periodic, lets find period.

A[\[Lambda]_] = 1 + Pi*Csc[(1 + Pi)*\[Lambda]]^2*
(Pi + Sin[2*(1 + Pi)*\[Lambda]])

Plot[A[\[Lambda]], {\[Lambda], 0, 3}]

\[Lambda]period =
lap /. First@
Solve[0 < lap < 1 && A[1/2] - A[1/2 + lap] == 0 &&
A[1/2] - A[1/2 + 2 lap] == 0 // TrigExpand, lap]

(*   (-1 + 3*Pi -
4*ArcTan[Sec[1/2]*Sqrt[Cos[1/2]^2 + Sin[1/2]^2] -
Tan[1/2]])/(2 + 2*Pi)   *)

\[Lambda]sol[AA_] = \[Lambda] /. Solve[TrigExpand[
0 < \[Lambda] < 2*\[Lambda]period &&
A[\[Lambda]] == AA], \[Lambda], Reals]

(*   {ConditionalExpression[
(2*Pi + 2*ArcTan[Root[Pi^2 + 4*Pi*#1 +
(4 - 4*AA + 2*Pi^2)*#1^2 - 4*Pi*#1^3 +
Pi^2*#1^4 & , 1]])/(1 + Pi),
Element[ArcTan[Root[Pi^2 + 4*Pi*#1 +
(4 - 4*AA + 2*Pi^2)*#1^2 - 4*Pi*#1^3 +
Pi^2*#1^4 & , 1]], Reals] && AA > Pi^2],
ConditionalExpression[
(2*Pi + 2*ArcTan[Root[Pi^2 + 4*Pi*#1 +
(4 - 4*AA + 2*Pi^2)*#1^2 - 4*Pi*#1^3 +
Pi^2*#1^4 & , 2]])/(1 + Pi),
Element[ArcTan[Root[Pi^2 + 4*Pi*#1 +
(4 - 4*AA + 2*Pi^2)*#1^2 - 4*Pi*#1^3 +
Pi^2*#1^4 & , 2]], Reals] && AA > Pi^2],
ConditionalExpression[
(2*ArcTan[Root[Pi^2 + 4*Pi*#1 + (4 - 4*AA + 2*Pi^2)*
#1^2 - 4*Pi*#1^3 + Pi^2*#1^4 & , 3]])/(1 + Pi),
Element[ArcTan[Root[Pi^2 + 4*Pi*#1 +
(4 - 4*AA + 2*Pi^2)*#1^2 - 4*Pi*#1^3 +
Pi^2*#1^4 & , 3]], Reals] && AA > Pi^2],
ConditionalExpression[
(2*ArcTan[Root[Pi^2 + 4*Pi*#1 + (4 - 4*AA + 2*Pi^2)*
#1^2 - 4*Pi*#1^3 + Pi^2*#1^4 & , 4]])/(1 + Pi),
Element[ArcTan[Root[Pi^2 + 4*Pi*#1 +
(4 - 4*AA + 2*Pi^2)*#1^2 - 4*Pi*#1^3 +
Pi^2*#1^4 & , 4]], Reals] && AA > Pi^2]}   *)

pl1 = Plot[Evaluate[\[Lambda]sol[AA]], {AA, 0, 200},
PlotRange -> All, PlotStyle -> {Green, Red, Orange,
Magenta}]

pl2 = ParametricPlot[Evaluate[{A[\[Lambda]], \[Lambda]}],
{\[Lambda], 0, 3}, AspectRatio -> 1]

Show[pl2, pl1]