Parametricplot for Series expansion fails to plot a Circle

Question

I'm currently trying to plot a circle in mathematica using ParametricPlot and it fails to produce a circle when i produce the code below

Clear[t]; 
approxcos[t_] = Normal[Series[Cos[t], {t, 0, 8}]]
approxsin[t_] = Normal[Series[Sin[t], {t, 0, 9}]]
ParametricPlot[{approxcos[t], approxsin[t]}, {t, 0, 2 Pi}, PlotStyle -> {{Blue, Thickness[0.01]}}, 
PlotRange -> All, AxesLabel -> {"x", "y"}, AspectRatio -> Automatic]

Any reason as to why it fails to produce a circle using the code above?

Answer 1

It depends on the MacLaurin series which works fine for $t$-values close to $0$.

If you substitute values from $0$ to $2\pi$ the approximation fails to be a fine approximation

For instance for $t=2\pi$ the approximation of $(\sin(t),\cos(t))$ gives $(11.9,21)$.

If you use $t\in[-\pi,\pi]$ the plot is decent

Answer 2

I would like to add a graphical illustration to the proof by @Raffaele where he points that one needs to take more terms in order to have a convenient representation of the unit circle.

Indeed, on this picture are plotted some of the curves starting at point $S(1,0)$, corresponding to partial sums using monomials till degrees $2, 4, 6, \cdots 18$. In particular, your curve would be with $k=9$ (not represented here). It is only with $k\ge18$ that we have an almost exact representation of the unit circle.

Please note the case $k=2$ : it is a parabolic arc.