c = 
        Plot[Sin[t], {t, 99*Pi, 101*Pi}, 
          PlotRange -> {{99*Pi, 101*Pi}, {-1, 1}}, 
          PlotStyle -> {RGBColor[1, 0, 1], Thickness[0.006]}, 
          DisplayFunction -> Identity];
d = 
        Plot[0.99*Sin[0.99*t], {t, 99*Pi, 101*Pi}, 
          PlotRange -> {{99*Pi, 101*Pi}, {-1, 1}}, 
          PlotStyle -> {RGBColor[0, 0, 1], Thickness[0.006]}, 
          DisplayFunction -> Identity];
Show[c, d, DisplayFunction -> $DisplayFunction]

e = 
        Plot[Sin[t], {t, 99*Pi, 101*Pi}, 
          PlotRange -> {{314, 315}, {-1, 1}}, 
          PlotStyle -> {RGBColor[1, 0, 1], Thickness[0.006]}, 
          DisplayFunction -> Identity];
f = 
        Plot[0.99*Sin[0.99*t], {t, 99*Pi, 101*Pi}, 
          PlotRange -> {{314, 315}, {-1, 1}}, 
          PlotStyle -> {RGBColor[0, 0, 1], Thickness[0.006]}, 
          DisplayFunction -> Identity];
Show[e, f, DisplayFunction -> $DisplayFunction]

FindRoot[(10000/199) Sin[t] - (10000/199)*0.99*Sin[(99/100)*t], {t, 
        314, 315}]

{t -> 314.1592653589793`}
tm = 314.1592653589793`

y[t_] = 
        N[(20000/199)*Sin[(199/200) t]*Sin[(1/200)*t], 
          40];
y[tm]
Sin[(199/2)*Pi]

-100.50251256281408`

-100.50251256281408`
N[(20000/199), 40]

N[Abs[Abs[314.1592653589793`] - 100*Pi]/Abs[314.1592653589793`]*100, 
      10000000]

N[Abs[Abs[-100.50251256281408`] - (20000/199)]/Abs[-100.50251256281408`]*100, 10000000]

0.`

N[tm, 10]
t -> 314.1592653589793`

DSolve[{y''[t] + y[t] == Sin[0.99  t], y'[0] == 0, 
        y[0] == 0}, y[t], t]
a = 
      Plot[%[[1, 1, 2]], {t, 0, 1300}, PlotRange -> All, 
        PlotStyle -> {Thickness[0.0001], RGBColor[1, 0, 0]}, 
        AspectRatio -> 1/10]