Answer
Relative to himself, the smallest angle that the rugby player can throw the ball behind him is $41.8^{\circ}$
Relative to the +x axis, the smallest angle that the rugby player can throw the ball is $131.8^{\circ}$
Work Step by Step
If the rugby player throws the ball exactly sideways relative to himself, the ball will have a component of velocity in the +x direction of $4.0~m/s$ relative to the ground. This pass would be illegal.
To minimize the angle, the rugby player must throw the ball backwards relative to himself so that the ball's component of velocity in the +x direction is 0.
Let $\theta$ be the angle relative to himself that the rugby player throws the ball behind him:
$sin ~\theta = \frac{4.0~m/s}{6.0~m/s}$
$\theta = sin^{-1}~\frac{4.0~m/s}{6.0~m/s}$
$\theta = 41.8^{\circ}$
Relative to himself, the smallest angle that the rugby player can throw the ball behind him is $41.8^{\circ}$
Relative to the +x axis, the smallest angle that the rugby player can throw the ball is $90^{\circ}+41.8^{\circ}$ which is $131.8^{\circ}$
