For rudder deflection angles of 5, 10, 20, 30, and 45 degrees, at what (negative) angle-of-attack does a fin-rudder combination (including dorsal fin if present) have a lift coefficient of exactly zero?
For the purpose of this question, define angle-of-attack in relation to the chord line of the fin alone, not including the deflected rudder. If we ignore things like spiral slipstream, deflection of airflow around cabin, etc, it would be the same as the sideslip angle.
Loosely speaking, at this angle-of-attack we could say the force from the deflected rudder is being exactly cancelled by the force from the fin, though in reality each influences the airflow around the other and the fin-rudder combination acts as a single unit.
To allow for a definitive answer, I'll specify to use the fin (including dorsal fin) and rudder from current -production Cessna 172 S, but feel to post an answer addressing the fin and rudder from any other aircraft.
Note that an aircraft could never sustain a sideslip angle that caused the fin plus deflected rudder to have a lift coefficient of zero, unless some other yaw torque due to something like asymmetric thrust or extreme adverse yaw from deflected ailerons was acting to maintain the slip angle.
Approximations are fine-- obviously an exact answer would involve a complicated computer modelliing project, or actual wind tunnel experiments. My hope is that existing information relating to the lift curves of symmetrical airfoils with deployed flaps may be adapted to give an approximation of an answer to this question.