Clear["Global`*"]
EDIT: Corrected intervals in Solve
sol = Solve[{3 y Cos[y] Cosh[4 x] + (5 + 2 x) Cosh[x] Sin[y] == 0,
3 x Cos[y] Cosh[x] - (5 + 2 x) Cos[y] Sinh[x] == 0, -10 < x < 0,
0 < y < 15}, {x, y}] /. r_Root :> N[r]
(* {{x -> -(5/2), y -> π/2}, {x -> -(5/2), y -> (3 π)/2}, {x -> -(5/2),
y -> (5 π)/2}, {x -> -(5/2), y -> (7 π)/2}, {x -> -(5/2),
y -> (9 π)/2}, {x -> -0.737455, y -> 1.66515}, {x -> -0.737455,
y -> 4.74558}, {x -> -0.737455, y -> 7.87399}, {x -> -0.737455,
y -> 11.0099}, {x -> -0.737455, y -> 14.1483}} *)
If Solve
or NSolve
doesn't work, use FindRoot
sol = Flatten[
Outer[FindRoot[{3 y Cos[y] Cosh[4 x] + (5 + 2 x) Cosh[x] Sin[y] ==
0, 3 x Cos[y] Cosh[x] - (5 + 2 x) Cos[y] Sinh[x] ==
0}, {{x, #1}, {y, #2}}] &, {-3, -1}, {2, 4, 8, 11, 14}], 1]
(* {{x -> -2.5, y -> 1.5708}, {x -> -2.5, y -> 4.71239}, {x -> -2.5,
y -> 7.85398}, {x -> -2.5, y -> 10.9956}, {x -> -2.5,
y -> 14.1372}, {x -> -0.737455, y -> 1.66515}, {x -> -0.737455,
y -> 4.74558}, {x -> -0.737455, y -> 7.87399}, {x -> -0.737455,
y -> 11.0099}, {x -> -0.737455, y -> 14.1483}} *)
Plotting,
ContourPlot[{3 y Cos[y] Cosh[4 x] + (5 + 2 x) Cosh[x] Sin[y] == 0,
3 x Cos[y] Cosh[x] - (5 + 2 x) Cos[y] Sinh[x] == 0}, {x, -10, 0}, {y, 0,
15}, Epilog -> {Red, AbsolutePointSize[4], Point[{x, y} /. sol]},
PlotPoints -> 75, MaxRecursion -> 5,
PlotLegends -> Placed["Expressions", Top]]
