Use a bulid-in function to this:
A = 1;
p = Pi/4;
o = 2 Pi;
FindMinimum[{(A Sin[o t] - (A Sin[o t + p] - 1)), 0 <= t <= 3/2}, {t, 1}]
(* {0.234633, {t -> 0.9375} *)
sol = FindMinimum[{(A Sin[o t] - (A Sin[o t + p] - 1))}, {t, 1}]
(* {0.234633, {t -> 0.9375} *)
Plot[(A Sin[o t] - (A Sin[o t + p] - 1)), {t, 0, Pi},
Epilog -> {Red, PointSize[0.02], Point[{t /. sol[[2]], sol[[1]]}]}]
Symbolic solution:
Minimize[{(A Sin[o t] - (A Sin[o t + p] - 1)), 0 <= t <= 3/2}, t]
(* {1 - Sin[\[Pi]/4 -
2 ArcTan[2/Sqrt[2 - Sqrt[2]] + Sqrt[2]/(-2 + Sqrt[2])]] -
Sin[2 ArcTan[
2/Sqrt[2 - Sqrt[2]] + Sqrt[2]/(-2 + Sqrt[2])]], {t -> (\[Pi] -
ArcTan[2/Sqrt[2 - Sqrt[2]] + Sqrt[2]/(-2 + Sqrt[2])])/\[Pi]}} *)
%//N
(* {0.234633, {t -> 0.9375} *)