I'm trying to compare the unit normal definition in calculus texts (i.e., $\vec N=\vec T'/||\vec T'||$) where $\vec T$ is the unit tangent vector, with the unit normal vector returned by the FrenetSerretSystem. Using the calculus definition, this works:

    Manipulate[
     
     Module[{r, unitT, unitN},
      
      r[x_] = {x, Sin[x]};
      unitT[x_] = r'[x]/Sqrt[r'[x].r'[x]];
      unitN[x_] = unitT'[x]/Sqrt[unitT'[x].unitT'[x]];
      
      ParametricPlot[r[x], {x, 0, 2 Pi},
       Epilog -> { 
         Orange, Thick, Arrow[{r[tt], r[tt] + unitN[tt]}],
         Red, PointSize[Large], Point[r[tt]]
         }, PlotRange -> {{0, 2 Pi}, {-4, 4}}]],
     
     {tt, 0.01, 2 Pi - 0.01}]

But trying the FrenetSerretSystem, this doesn't.

    Manipulate[
     
     Module[{r, unitT, unitN, sys},
      
      r[x_] = {x, Sin[x]};
      sys = FrenetSerretSystem[r[x], x];
      unitT[x_] = sys[[2, 1]];
      unitN[x_] = sys[[2, 2]];
      
      ParametricPlot[r[x], {x, 0, 2 Pi},
       Epilog -> { 
         Orange, Thick, Arrow[{r[tt], r[tt] + unitN[tt]}],
         Red, PointSize[Large], Point[r[tt]]
         }, PlotRange -> {{0, 2 Pi}, {-4, 4}}]
      
      ],
     
     {tt, 0.01, 2 Pi - 0.01}]

I get an error such as this:

Coordinate {0.01 - Cos[$CellContext`x] (1 + Cos[$CellContext`x]^2)^Rational[-1, 2], 0.009999833334166664 + (1 + Cos[$CellContext`x]^2)^Rational[-1, 2]} should be a pair of numbers, or a Scaled or Offset form.

And this gives a similar error.

    DynamicModule[{r, unitT, unitN, sys},
     
     r[x_] = {x, Sin[x]};
     sys = FrenetSerretSystem[r[x], x];
     unitT[x_] = sys[[2, 1]];
     unitN[x_] = sys[[2, 2]];
     
     Manipulate[
      
      ParametricPlot[r[x], {x, 0, 2 Pi},
       Epilog -> { 
         Orange, Thick, Arrow[{r[tt], r[tt] + unitN[tt]}],
         Red, PointSize[Large], Point[r[tt]]
         }, PlotRange -> {{0, 2 Pi}, {-4, 4}}],
      
      {tt, 0.01, 2 Pi - 0.01}]]

Once I get this repaired, I am going to see the unit normal vector (calculus definition) pointing in the opposite direction of the unit normal vector (FrenetSerretSystem) from 0 to $\pi$ and I am wondering why. I think it might have something to do with the fact that the calculus books tend to used the curvature, whereas the FrenetSerretSystem uses the signed curvature, but I am still trying to figure out the different directions of the unit normal.

Thanks for the help.