Vtyjkyuk

Consider the following MWE:

documentclass[tikz]standalone
begindocument
 foreach n in 0,1,2,...,72
 begintikzpicture
 useasboundingbox (-2.5,-2.5) rectangle (2.5,2.5);
 pgfmathsetmacroq10*n
 draw[densely dotted] (0,-2.5) -- (0,2.5);
 draw (0,0) circle(1);
 draw (q:1) -- (0,2.5*sin(q));
 fill[black,radius=.1] (q:1) circle;
 fill[red,radius=.1] (0,2.5*sin(q)) circle; %use 1*sin(·) …
 endtikzpicture
 
enddocument

My question is: How can I put a trace at the red point (so that a line represents the path of the red point)?

As far as I can see, you could draw the line by dividing it into three cases:

documentclass[tikz]standalone
begindocument
 foreach n in 0,1,2,...,27
 begintikzpicture
 useasboundingbox (-2.5,-2.5) rectangle (2.5,2.5);
 pgfmathsetmacroq10*n
 draw[densely dotted] (0,-2.5) -- (0,2.5);
 draw (0,0) circle(1);
 draw (q:1) -- (0,2.5*sin(q));
 fill[black,radius=.1] (q:1) circle;
 fill[red,radius=.1] (0,2.5*sin(q)) circle; %use 1*sin(·) …
 draw[ultra thick,red] (0,0) -- (0,2.5*sin(q));
 ifnumn>9
 draw[ultra thick,red] (0,0) -- (0,2.5);
 fi
 ifnumn>28
 draw[ultra thick,red] (0,-2.5) -- (0,2.5);
 fi
 endtikzpicture
 
enddocument

THIS IS REALLY JUST FOR FUN. No competitor to @samcarter's nice answer from which most of this is stolen.

documentclass[tikz]standalone
usetikzlibrarydecorations.footprints
begindocument
tikzsetcat prints/.style=decorate,decoration=footprints,foot length=2pt,stride
 length=5pt,foot sep=1pt,foot of=felis silvestris,red
 foreach n in 0,1,2,...,27
 begintikzpicture
 useasboundingbox (-2.5,-2.5) rectangle (2.5,2.5);
 pgfmathsetmacroq10*n
 draw[densely dotted] (0,-2.5) -- (0,2.5);
 draw (0,0) circle(1);
 draw (q:1) -- (0,2.5*sin(q));
 fill[black,radius=.1] (q:1) circle;
 fill[red,radius=.1] (0,2.5*sin(q)) circle;
 ifnumn<10
 draw[cat prints] (0,0) -- (0,2.5*sin(q));
 else
 ifnumn>9
 draw[cat prints] (0,0) -- (0,2.5);
 draw[cat prints] (0,2.5cm-0.5pt) -- (0,2.5*sin(q));
 fi
 ifnumn>28
 draw[cat prints] (0,0) -- (0,2.5);
 draw[cat prints] (0,2.5cm-0.5pt) -- (0,-2.5cm+0.5pt);
 draw[cat prints] (0,2.5) -- (0,2.5*sin(q));
 fi
 fi
 endtikzpicture
 
enddocument

UPDATE: Fading footprints. Plus duck queen. ;-)

documentclass[tikz,border=5mm]standalone
usetikzlibrarydecorations.footprints,ducks
newsaveboxDuck
sboxDucktikzduck[crown,laughing, bill=red,longhair=black]
newcounterMyFootSteps
pgfdeclaremetadecorationfading footprintsinitial stateinitial[width=0pt, 
next state=footprint] 
 pgfmathparse10-3*int(pgfmetadecoratedpathlength/pgfmetadecorationsegmentlength)
 setcounterMyFootStepspgfmathresult
 pgfsetlinewidth0.2pt
 pgfset/pgf/decoration/foot length=pgfmetadecorationsegmentlength/5,/pgf/decoration/stride
 length=pgfmetadecorationsegmentlength/3,
 /pgf/decoration/foot sep=1pt,/pgf/decoration/foot of=bird
 
statefootprint[
switch if less than=pgfmetadecorationsegmentlength to final, 
width=pgfmetadecorationsegmentlength/3, next state=footprint] 
stepcounterMyFootSteps
ifnumtheMyFootSteps>0
pgfsetfillopacity0.1*theMyFootSteps
decorationfootprints
pgfusepathqfill
fi

statefinalpgfusepathqfill

begindocument
tikzsetfading cat prints/.style=decorate,decoration=fading footprints,meta-segment length=15pt
foreach n in 0,1,2,...,35
 begintikzpicture
 useasboundingbox (-2.5,-2.5) rectangle (2.5,2.5);
 pgfmathsetmacroq10*n
 draw[densely dotted] (0,-2.5) -- (0,2.5);
 draw (0,0) circle(1);
 draw (q:1) -- (0,2.5*sin(q));
 fill[black,radius=.1] (q:1) circle;
 ifnumn<10
 draw[fading cat prints] (0,-2.5) -- (0,2.5*sin(q));
 else
 ifnumn>9
 ifnumn<29
 draw[fading cat prints] (0,0) -- (0,2.5) -- (0,2.5*sin(q));
 else 
 draw[fading cat prints] (0,0) -- (0,2.5) -- (0,-2.5) -- (0,2.5*sin(q));
 fi
 fi
 fi
 node[scale=0.5] at (0,2.5*sin(q)) useboxDuck;
 endtikzpicture

enddocument