Find values of $k$, $l$, and $m$ such that the following equation is satisfied

Question

I am trying to find the (numerical) subset of $(k,l,m)\in[-\pi,\pi)\times[-\pi,\pi)\times[-\pi,\pi)$ where the next expression is valid $$ \tan \Bigl( \frac{k+l+m}{2} \Bigr) = \frac{ \sin k+ \sin l+\sin m }{ \cos k+ \cos l+\cos m }. $$ When I fix $k$ I am able, using ContourPlot, to obtain a set of curves revealing where the previous expression is satisfied. However, I'd like to get not a curve but the values of $l$ and $m$.

Note that using NSolve returns "This system cannot be solved with the methods available to Solve.". NMinimize only gives a point on the curve.

Answer 1

Some mathematical simplifications are needed before plotting in order to deal with divergencies of $\tan x$ for $x=\frac\pi2+n \pi$, where $n\in \mathbb{Z}$. One can use that $$\tan\frac{x+y}{2}=\frac{\sin x+\sin y}{\cos x+\cos y}.$$

The rest is automatic

Clear[eq]
eq[0]=(Sin[k]+Sin[l+m])(Cos[k]+Cos[l]+Cos[m])-(Cos[k]+Cos[l+m])(Sin[k]+Sin[l]+Sin[m])
eq[1]=FullSimplify[eq[0]]
eq[2]=TrigFactor[eq[1]]
Out[1]= -(Cos[k]+Cos[l+m]) (Sin[k]+Sin[l]+Sin[m])+(Cos[k]+Cos[l]+Cos[m]) (Sin[k]+Sin[l+m])
Out[2]= Sin[k-l]+Sin[l]+Sin[k-m]-Sin[k-l-m]+Sin[m]
Out[3]= 2 Cos[k/2-l/2-m/2] (-Sin[k/2-l/2-m/2]+Sin[k/2+l/2-m/2]+Sin[k/2-l/2+m/2])

The first factor is describing planes $k+l+m=\pi+2n\pi$, where $\tan\frac{k+l+m}{2}$ is diverging. The second factor is a regular solution. Now we plot the two factors

ContourPlot3D[Evaluate[{eq[2][[2]]==0,eq[2][[3]]==0}],{k,-π,π},{l,-π,π},{m,-π,π},PlotTheme->{"Minimal"},ContourStyle->{Opacity[0.35],Automatic}]