Full plot of the general solution of an equation

Questions
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1

There are two following equations

\[Sigma] = 
 ArcCos [\[Omega]/(
  Sin [\[Theta] - \[Omega] \[Tau]] \[Epsilon] Sqrt[(1 + \[Alpha]^2)])]


\[CapitalDelta] = 
  2 Cos [\[Theta] - \[Omega] \[Tau]] \[Epsilon] Sqrt[(1 + \[Alpha]^2)]
    Sin[\[Sigma]];

The parameter values are

\[Theta] = \[Pi]/4; \[Tau] = 13.33; \[Epsilon] = 0.021; \[Alpha] = 1;

I need to plot

\[Omega] \[Tau]  vs   `\[CapitalDelta] \[Tau]`

However as given in the notebook I am not getting the complete plot, it may be because there is a general solution for ArcCos[x], but I cannot understand how to include the general solution here. Kindly suggest.

Open in Wolfram Cloud ↗
2

What do you mean by “complete plot”? You are explicitly restricting the range with the PlotRange option, so are you saying that the plot is incorrect in that range? Also, I think that for your chosen range for omega, you will have non-real values for n, so are you asking how to plot complex values?

3

The figure is given below, I want to plot the first one

figure.png
figure.png334×409 7.44 KB

Please suggest, how do I include the general solution of ArcCos [x]? I have considered sigma = ArcCos[x], however it may have other solutions at 2n pi+/- ArcCos[x].

4

Here is a way to get your picture:

x == n /. Solve[y == m, \[Omega]][[1]]
eq = ApplySides[#^2 &, %]
ContourPlot[Evaluate@eq, {x, -1, 1}, {y, -1/2, 1/2}]