Why are these plots not the same?

Question

I have the following code:

f = 6;
t0 = 0.0;
t1 = 50.0;
m[t_] := {Cos[2 Pi*f*t], 0.0, 0.0};
Plot[Part[m[t], 1], {t, t0, t1}, Frame -> True,
FrameLabel -> {Style["Time", FontSize -> 16], 
Style["Amplitude", FontSize -> 16]},
PlotRange -> All, PlotStyle -> {Red}]

The result is:

I want to rewrite the code using OptionsPattern as follows:

gt0 = 0.0;
gt1 = 50;
gf = 6;

ClearAll[af];
Options[af] = {
T0 -> gt0,
T1 -> gt1,
Frequency -> gf,
};

af[
OptionsPattern[]
] := Module[
{
t0 = OptionValue[T0],
t1 = OptionValue[T1],
f = OptionValue[Frequency],
m, t
},
m = {Cos[2 Pi*f*t], 0.0, 0.0}
]

rmt = af[];

Plot[Part[rmt[t], 1], {t, gt0, gt1}, Frame -> True,
FrameLabel -> {Style["Time", FontSize -> 16], 
Style["Amplitude", FontSize -> 16]}, PlotRange -> All,
PlotStyle -> {Red}]

The result is:

Why are these two plots not the same?

Answer 1

Your 2nd code example is wrong. It should be

gt0 = 0.0;
gt1 = 1.0; (* I am changing this to get a more reasonable plot *)
gf = 6; 

ClearAll[af];
Options[af] = {Frequency -> gf}; (* you don't use any other options *)
af[OptionsPattern[]] := {Cos[2 Pi*OptionValue[Frequency]*t], 0.0, 0.0}
rmt = af[]

Plot[Part[rmt, 1], {t, gt0, gt1}, 
  Frame -> True, 
  FrameLabel -> {Style["Time", FontSize -> 16], Style["Amplitude", FontSize -> 16]},
  PlotRange -> All, 
  PlotStyle -> {Red}]

Answer 2

Let's see what rmt looks like:

rmt

{Cos[12 π t$34324], 0., 0.}

Notice that the t variable is actually a module variable! Next, let's consider rmt[t]:

rmt[t]

{Cos[12 π t$34324], 0., 0.}[t]

Finally, what does Part[rmt[t], 1] look like:

Part[rmt[t], 1]

t

So, your second example ends up plotting just t, and not Cos[12 Pi t]. This is why they are different.