In[]=  Phi[a_,b_,s_,opts:OptionsPattern[]] := Block[ {t}, t+ArcTan[(b-a)Tan[t]/(a+b Tan[t]^2)] /. FindRoot[b EllipticE[t,(b^2-a^2)/b^2] == s, {t, 0}, opts]]

In[]=  Phi[4, 2.9`32, 3.31`32, WorkingPrecision->30]

Out[]= 0.87052028743193111752524449959