soln=Solve[{-2(1-L)+2 lambda Cos[theta]+mu Sin[theta]==0,
(1/2) theta-2 lambda L Sin[theta]+mu L Cos[theta]==0},{lambda,mu}]

Simplify[%]

theta=Pi/8;

ParametricPlot[{Cos[theta]-L Cos[theta]+(theta Sin[theta])/(4L),
-(theta Cos[theta])/(2 L)-2 (-1+L) Sin[theta]},{L,.25,1.75},
PlotRange->{{.3,.6},{-.05,.05}}]