Ttgjyuk

I have this image:

That is made with this code:

 documentclassarticle
 usepackagetikz
usepackageamssymb,amsthm,amsmath
usetikzlibraryshapes,positioning,intersections,quotes

begindocument

begincenter
 begintikzpicture
 draw [rotate=270] (0,0) arc (180:0:1);
 draw [rotate=270] (-2,0) arc (180:0:1);
 draw (0,0) circle (2cm);
 draw[dashed] (0,2.5) -- (0,-2.5);
 draw[dashed](-2.5,0) -- (2.5,0);
 draw (0,0) -- (-1.1547,1.9);
 draw (0,0) -- (-1.01,2);
 draw [rotate=120](0,0) -- (-1.1547,1.9);
 draw [rotate=120](0,0) -- (-1.01,2);
 filldraw[black] (1.2,0) circle (0pt)
 node[anchor=west] $S_0$;
 filldraw[black] (-1.2,0) circle (0pt)
 node[anchor=west] $S_3$;
 filldraw[black] (0,1.2) circle (0pt)
 node[anchor=west] $S_1$;
 filldraw[black] (0,-1.2) circle (0pt)
 node[anchor=west] $S_2$;

 filldraw[black] (-1,2.1) circle (0pt)
 node[anchor=west] tiny$R_1$;
 endtikzpicture
endcenter




enddocument

And I want to know if there is a way to color in gray the zone that is between the 2 lines (in both zones). Any ideas? Thanks.

Sure. (But why do you not use polar coordinates?)

documentclassarticle
usepackagetikz
begindocument
section*Two methods that do not require libraries
subsection*Method 1: clip
begincenter
 begintikzpicture
 draw [rotate=270] (0,0) arc (180:0:1);
 draw [rotate=270] (-2,0) arc (180:0:1);
 draw (0,0) circle (2cm);
 draw[dashed] (0,2.5) -- (0,-2.5);
 draw[dashed](-2.5,0) -- (2.5,0);
 draw (0,0) -- (-1.1547,1.9);
 draw (0,0) -- (-1.01,2);
 draw [rotate=120](0,0) -- (-1.1547,1.9);
 draw [rotate=120](0,0) -- (-1.01,2);
 path (1.2,0) node[anchor=west] $S_0$;
 path (-1.2,0) node[anchor=west] $S_3$;
 path (0,1.2) node[anchor=west] $S_1$;
 path (0,-1.2) node[anchor=west] $S_2$;
 path (-1,2.1) node[anchor=west] tiny$R_1$;
 clip (0,0) circle[radius=2cm-pgflinewidth/2];
 draw[fill=gray] (0,0) -- (-1.01,2) -- (-1.1547,1.9);
 draw[fill=gray,rotate=120] (0,0) -- (-1.01,2) -- (-1.1547,1.9);
 endtikzpicture
endcenter
subsection*Method 2: compute the polar coordinates of the points and use them
begincenter
 begintikzpicture
 draw [rotate=270] (0,0) arc (180:0:1);
 draw [rotate=270] (-2,0) arc (180:0:1);
 draw (0,0) circle (2cm);
 draw[dashed] (0,2.5) -- (0,-2.5);
 draw[dashed](-2.5,0) -- (2.5,0);
 draw (0,0) -- (-1.1547,1.9);
 draw (0,0) -- (-1.01,2);
 pgfmathsetmacromyangleOneatan2(1.9,-1.1547)
 %pgfmathsetmacromyradiusveclen(1.9,-1.1547)
 pgfmathsetmacromyangleTwoatan2(2,-1.01)
 draw[fill=gray] (0,0) -- (myangleOne:2) arc(myangleOne:myangleTwo:2)
 --cycle;
 %typeoutmyangleOne,myangleTwo
 draw [rotate=120](0,0) -- (-1.1547,1.9);
 draw [rotate=120](0,0) -- (-1.01,2);
 draw[fill=gray,rotate=120] (0,0) -- (myangleOne:2) arc(myangleOne:myangleTwo:2)
 --cycle;
 path (1.2,0) node[anchor=west] $S_0$;
 path (-1.2,0) node[anchor=west] $S_3$;
 path (0,1.2) node[anchor=west] $S_1$;
 path (0,-1.2) node[anchor=west] $S_2$;
 path (-1,2.1) node[anchor=west] tiny$R_1$;
 endtikzpicture
endcenter
enddocument