NDSolveValue[{-Derivative[0, 2][Φ][x, t] + Derivative[2, 0][Φ][x, t] == Sin[Φ[x, t]],
  Φ[x, 0] == 0, -Derivative[0, 1][Φ][0, t] + Derivative[1, 0][Φ][0, t] == 2*Sin[2*t]},
  {Φ[x, t], Derivative[1, 0][Φ][x, t]}, {x, 0, 1}, {t, 0, 1}]