When tracing boundaries by combinatorial methods it is useful to have their initial directions.
For inverse rational functions defined by an equation zg(w)-h(w) = 0 the tangent directions of boundaries at a node are obtained in terms of the direction of a cut from the associated branch-point.
Let W be not a zero of either g or h and let Zg(W)-h(W) = 0  and also let Zg'(W)-h'(W) = 0 where as usual the superscript indicates differentiation. The two equations make W a node and Z its branch-point.  To save notation we shall write G for g(W) and H for h(W). Then
(z-Z)Gg(w)  = [Gh(w) - Hg(w)].
Expanding in terms of w-W gives
(z-Z)GΣ(w-W)ⁿg⁽ⁿ⁾  (W)/n! = Σ(w-W)ⁿ[Gh⁽ⁿ⁾ (W) - Hg⁽ⁿ⁾ (W)]/n!
As w tends to W z-Z tends to some power of w-W. Retaining only the lowest power on either side gives the approximation
(z-Z)G² ≅ (w-W)^M [Gh^(M) (W) - Hg^(M) (W)]/M!
where M is the index of the first non-zero term. Note that from the definitions M is 2 or greater. This is valid for z on a cut from Z and w on a boundary-element from W. Replacing the approximation by an equation puts w on a tangent to the boundary-element which meets it at W. Then taking arguments gives
arg(z-Z)+2argG = Marg(w-W)+arg[Gh^(M) (W) - Hg^(M) (W)].
This can be solved for arg(w-W) in terms of arg(z-Z). Note that there are M solutions representing that M boundary-elements meet at W. The angle between each one and its neighbours is (360/M)°. If the cut is specialized to go from Z towards infinity then arg(z-Z) = argZ = argH-argG and the LHS becomes
argG+argH.
When working an example it is usually better to work from first principles than rely on substitution into a formula which may not be remembered correctly or indeed may not be stated correctly.