Answer
The angle is $120^{\circ}$
Work Step by Step
The force exerted by each half of the string against the archer must be $F/2$. Since the net force is $F$, the force perpendicular to the archer's applied force must be $\frac{F\sqrt 3}{2}$. Since the string elongates in the direction of the net force and the forces here form a t30-60-90 triangle with the $60^{\circ}$ angle at the archer's hand, the answer desired is $2*60 = 120$.
