ClearAll["Global`*"]
Plot[Sin[[Pi]*x]/([Pi]*x), {x, -2, 2}, PlotRange -> Full, Exclusions -> None, GridLines -> Automatic, ImageSize -> Full]
Normal[Series[Sin[[Pi]*x]/([Pi]*x), {x, 0, 12}]]
Plot[Evaluate[Normal[Series[Sin[[Pi]*x]/([Pi]*x), {x, 0, 12}]]], {x,-2, 2}, PlotRange -> Full, Exclusions -> None, GridLines -> Automatic, ImageSize -> Full]
N[Solve[Normal[Series[Sin[[Pi]*x]/([Pi]*x), {x, 0, 12}]] == 1/2, x]]

ClearAll["Global`*"]
Subscript[f, c] = 2*10^9;
k = 30*10^15;
Subscript[f, 0] = 0.5*10^9;
Subscript[s, out] = !(*SubsuperscriptBox[([Integral]), (0), (t)](Cos[2*[Pi]**SubscriptBox[(f), (c)]*[Tau]]*Cos[2*[Pi]*((k*t + *SubscriptBox[(f), (0)]))*[Tau]] [DifferentialD][Tau]))
Plot[Subscript[s, out], {t, 49*10^-9, 51*10^-9}, PlotRange -> Full,  Exclusions -> None, GridLines -> Automatic, ImageSize -> Full]
soutexpansion = Expand[Normal[Series[Subscript[s, out], {t, 50*10^-9, 2}]]]
Plot[Evaluate[soutexpansion], {t, 49.6*10^-9, 50.4*10^-9}, PlotRange -> Full, Exclusions -> None, GridLines -> Automatic, ImageSize -> Full]
Solve[soutexpansion == (2.5*10^-8)/2, t]

Subscript[s, out2] = !(*SubsuperscriptBox[([Integral]), (0), (t)](*FractionBox[(1), (2)]*((Cos[2*[Pi]*((k*t + *SubscriptBox[(f), (0)] - *SubscriptBox[(f), (c)]))*[Tau]] + Cos[2*[Pi]*((k*t + *SubscriptBox[(f), (0)] + *SubscriptBox[(f), (c)]))*[Tau]])) [DifferentialD][Tau]))
Plot[Subscript[s, out2], {t, 49.6*10^-9, 50.4*10^-9}, PlotRange -> Full, Exclusions -> None, GridLines -> Automatic, ImageSize -> Full]
Subscript[s, out2series] = Normal[Series[Subscript[s, out2], {t, 50*10^-9, 30}]]
Plot[Evaluate[Subscript[s, out2series]], {t, 49.6*10^-9, 50.4*10^-9}, PlotRange -> Full, Exclusions -> None, GridLines -> Automatic, ImageSize -> Full]
Solve[Subscript[s, out2series] == (2.5*10^-8)/2, t]