Here is the code:
Manipulate[
Module[{\[CurlyEpsilon] = 10^-6, c1 = Tan[a1], c2 = Tan[a2],
c3 = Tan[a3], c4 = Tan[a4], c5 = Tan[a5], c6 = Tan[a6]},
ContourPlot3D[
Evaluate[
c6 Sin[3 x] Sin[2 y] Sin[z] + c4 Sin[2 x] Sin[3 y] Sin[z] +
c5 Sin[3 x] Sin[y] Sin[2 z] + c2 Sin[x] Sin[3 y] Sin[2 z] +
c3 Sin[2 x] Sin[y] Sin[3 z] + c1 Sin[x] Sin[2 y] Sin[3 z] ==
0], {x, \[CurlyEpsilon],
Pi - \[CurlyEpsilon]}, {y, \[CurlyEpsilon],
Pi - \[CurlyEpsilon]}, {z, \[CurlyEpsilon], Pi - \[CurlyEpsilon]},
Mesh -> False, ImageSize -> {400, 400}, Boxed -> False,
Axes -> False, NormalsFunction -> "Average",
PlotPoints -> ControlActive[10, 30],
PerformanceGoal -> "Speed"]], {{a1, 1,
"\!\(\*SubscriptBox[\(\[Alpha]\), \(1\)]\)"}, -Pi/2 - 0.01,
Pi/2 + 0.01,
ImageSize -> Tiny}, {{a2, 1,
"\!\(\*SubscriptBox[\(\[Alpha]\), \(2\)]\)"}, -Pi/2 - 0.01,
Pi/2 + 0.01,
ImageSize -> Tiny}, {{a3, 1,
"\!\(\*SubscriptBox[\(\[Alpha]\), \(3\)]\)"}, -Pi/2 - 0.01,
Pi/2 + 0.01,
ImageSize -> Tiny}, {{a4, 1,
"\!\(\*SubscriptBox[\(\[Alpha]\), \(4\)]\)"}, -Pi/2 - 0.01,
Pi/2 + 0.01,
ImageSize -> Tiny}, {{a5, 1,
"\!\(\*SubscriptBox[\(\[Alpha]\), \(5\)]\)"}, -Pi/2 - 0.01,
Pi/2 + 0.01,
ImageSize -> Tiny}, {{a6, 1,
"\!\(\*SubscriptBox[\(\[Alpha]\), \(6\)]\)"}, -Pi/2 - 0.01,
Pi/2 + 0.01, ImageSize -> Tiny}, AutorunSequencing -> {1, 3, 5},
ControlPlacement -> Left]
