It is expected number of digits that is true but is it right? Forget about Sin. When I evaluate fg and fg2 with 40 digits fg2 has the right digits not fg . Because it is already evaluated before setting the accuracy after machine zero the digits are not right. My goal is to avoid that and evaluate fg when I need to evaluate with right digits.

In[190]:= fg = Pi/2.1;
Print[fg];
fg2 = Pi/(21/10);
Print[fg2];
SetAccuracy[fg, 40]
SetAccuracy[fg2, 40]


During evaluation of In[190]:= 1.4959965017094252

During evaluation of In[190]:= (10 \[Pi])/21

Out[194]= 1.4959965017094252193174952481058426201344

Out[195]= 1.495996501709425351648877801561668040094

As an alternative to the answer you already received on stackexchange:

Panel[
  DynamicModule[{fun = HoldForm[2.1 Sin[Pi z/2.1]]},
    Column[{
      Row[{
        Style["Function ", 12, Blue, Editable -> False],
        InputField[Dynamic[fun], FieldSize -> {55, 3}, BaseStyle -> {12}]
      },
      Spacer[1]],
      Button[Style["Calculate", 14, Green, Editable -> False], Res = upAC[ReleaseHold@fun]],
      Row[{
        Style["Result ", 12, Red, Editable -> False],
        InputField[Dynamic[Res], FieldSize -> {52, 2}, BaseStyle -> {12}]
      },
      Spacer[5]]
    }],
    Initialization :>
      (
        upAC[in_] :=
        Module[
          {out},
          out = If[Accuracy[in] == \[Infinity], in, Rationalize[in, 10^-Accuracy[in]]];
          Return[out]
        ];
      )
  ]
]