plt=Graphics[{Line[{{0,0},{Cos[#2],Sin[#2]}}],Text[Style[#1,16],1.05{Cos[#2],Sin[#2]}]}&@@@({{"η","ϕn","ϕi","θi","θn"},Range[0,9(π/5),2(π/5)]}\[Transpose])];
Manipulate[Module[{pl},
pl[1]=Last@WebPlot[os,Blue];
pl[2]=Last@WebPlot[f[αn,i,βa]];
os=First@WebPlot[f[αn,i,βa]];
Show[plt,pl[1],pl[2]]
],
{{i,10,"i"},-60,+60,ContinuousAction->False},
{{αn,10,"αn"},-30,+30,ContinuousAction->False},
{{βa,10,"βa"},0,+65,ContinuousAction->False},
{{os,f[10,10,10]},ControlType->None},
Initialization:>(
f[αn_,i_,βa_]:={2 αn,i+αn,βa,βa+i,αn+βa+i};
WebPlot[{η_,ϕn_,ϕi_,θi_,θn_},col_:Red]:=Module[{pts},
pts={(1/40)Abs[{η,ϕn,ϕi,θi,θn}],Range[0,9(π/5),2(π/5)]}\[Transpose];
{{η,ϕn,ϕi,θi,θn},Graphics[{col,Thickness[0.01],Line[#1{Cos[#2],Sin[#2]}&@@@AppendTo[pts,pts[[1]]]]}]}
]),
TrackedSymbols:>{i,αn,βa}
]