To solve this problem under the assumption that the stated quantities are small, you can insert a perturbation expansion for the unknown phi and solve for its expansion coefficients in the smallness parameter:

First I'm copying the definitions from the question to be self-contained:

Ecp = (s1*s3*Abs[J]/DE^2/2 - 
         1/DE*s2*s3*KroneckerDelta[1, s1])*(t1^2 - t2^2) /. 
      KroneckerDelta[1, s1] -&gt; (1 + s1)/2 /. t1 -&gt; tt1*DE /. 
    t2 -&gt; tt2*DE /. xi -&gt; xii*DE /. J -&gt; Jj*DE
Ec2p = s2/DE^2*KroneckerDelta[1, s1] (t1^2 - t2^2)^2/Abs[J] /. 
      KroneckerDelta[1, s1] -&gt; (1 + s1)/2 /. t1 -&gt; tt1*DE /. 
    t2 -&gt; tt2*DE /. xi -&gt; xii*DE /. J -&gt; Jj*DE
Esp = s2*s3/DE^2*t1*t2*xi*
        Abs[J]/(Abs[xi] KroneckerDelta[-1, s1] + 
           Abs[J] KroneckerDelta[1, s1]) /. 
       KroneckerDelta[1, s1] -&gt; (1 + s1)/2 /. 
      KroneckerDelta[-1, s1] -&gt; (1 - s1)/2 /. t1 -&gt; tt1*DE /. 
    t2 -&gt; tt2*DE /. xi -&gt; xii*DE /. J -&gt; Jj*DE


Now I define the derivative that you're trying to set to zero:

deriv = 
 D[E00 + Cos[phi]*Ecp + Cos[phi]^2*Ec2p + Sin[phi]*Esp, phi]

(*
==&gt; (DE s2 s3 tt1 tt2 xii Abs[DE Jj] Cos[phi])/(
 1/2 (1 + s1) Abs[DE Jj] + 
  1/2 (1 - s1) Abs[DE xii]) - (DE^2 tt1^2 - 
    DE^2 tt2^2) (-(((1 + s1) s2 s3)/(2 DE)) + (s1 s3 Abs[DE Jj])/(
    2 DE^2)) Sin[phi] - ((1 + s1) s2 (DE^2 tt1^2 - DE^2 tt2^2)^2 Cos[
   phi] Sin[phi])/(DE^2 Abs[DE Jj])
*)


Now I make the assumption that tt1, tt2, Jj, xii are all proportional to the small parameter ϵ and that I can then also write phi = ϕ0 + ϵ ϕ1 to first order in that small parameter. The expansion of deriv in powers of ϵ has two nonvanishing terms if we go to third order. Setting these two non-vanishing terms to zero allows us to solve for the two unknowns ϕ0 and ϕ1. This solves the problem of finding phi to first order in the small parameter. 

Solve[
 Thread[CoefficientList[
    Normal@Series[
      Simplify[
       deriv /. 
         Thread[{tt1, tt2, Jj, 
            xii} -&gt; ϵ {tt1, tt2, Jj, xii}] /. 
        phi -&gt; (ϕ0 + ϵ ϕ1), 
       DE Jj ϵ &gt; 0 &amp;&amp; DE xii ϵ &gt; 0], {ϵ, 0,
        3}], ϵ] == 0], {ϕ0, ϕ1}]

(*
==&gt; {{ϕ0 -&gt; 
   0, ϕ1 -&gt; -((
    4 Jj tt1 tt2 xii)/((1 + s1) (tt1^2 - tt2^2) (Jj + Jj s1 + xii - 
       s1 xii)))}, {ϕ0 -&gt; -Pi, ϕ1 -&gt; -((
    4 Jj tt1 tt2 xii)/((1 + s1) (tt1^2 - tt2^2) (Jj + Jj s1 + xii - 
       s1 xii)))}, {ϕ0 -&gt; Pi, ϕ1 -&gt; -((
    4 Jj tt1 tt2 xii)/((1 + s1) (tt1^2 - tt2^2) (Jj + Jj s1 + xii - 
       s1 xii)))}}
*)


In order to get useful results, I had to add some assumptions about certain products of your parameters being positive. You may have to check whether those assumptions are appropriate for your data.

Series expansion of large expression
Im trying to do my chem homework and I'm stumped. 


  Determine the solubility of $\ce{N2}$ in water exposed to air at 25°C, if the
  atmospheric pressure is 1.2 atm, assume that the mole fraction of
  nitrogen is 0.78 in air and the Henry's Law constant for nitrogen in
  water at this temperature is $6.1 × 10^{-4} M/atm$.



I have the answer, but i want to know how to solve for it. I've been reading through my notes and the book trying to find a reference through which to approach this problem and i have found none. If anyone can guide me in the right direction that would suffice

