Rotating curves described by parametric equations

We will rotate the curve C: x=Sin[7t], y=5Cos[t] around the origin thru an angle of aa radians.

First define the functions and plot the curve C itself...

x[t_]=Sin[7t]
y[t_]=5Cos[t]

ParametricPlot[{x[t],y[t]},{t,0,2Pi},Axes->True,Frame->True]

When we rotate we will want to have a square viewing centered at the origin, and the output suggests the viewing window [-6,6]x[-6,6]
will be appropriate to show all of the rotated graphs. Use the PlotRange command to set this window...

ParametricPlot[{x[t],y[t]},{t,0,2Pi},PlotRange->{{-6,6},{-6,6}},Axes->True,Frame->True,Background->Purple]

Now we’ll show the curve C and the curve obtained by from by rotating by aa=.5 radians ...

aa=.5;
ParametricPlot[{{x[t],y[t]},{x[t]*Cos[aa]-y[t]*Sin[aa],x[t]*Sin[aa]+y[t]*Cos[aa]}},{t,0,2Pi},Axes->True,Frame->True,PlotRange->{{-6,6},{-6,6}},Background->Purple]

From this we can easily allow the angle aa to vary from 0 to 2Pi
and use Manipulate to animate the rotation of C all the way around...

Manipulate[ParametricPlot[{{x[t],y[t]},{x[t]*Cos[aa]-y[t]*Sin[aa],x[t]*Sin[aa]+y[t]*Cos[aa]}},{t,0,2Pi},Axes->True,Frame->True,PlotRange->{{-6,6},{-6,6}},Background->Purple],{aa,0,2Pi}]