Answer
$F = \frac{75.6~N}{cos~\theta+0.42~sin~\theta}$
Work Step by Step
For the toy chest to be on the verge of moving, we can assume that the horizontal component of the child's applied force is equal in magnitude to the maximum possible force of static friction.
We can find $F$ in terms of $\theta$:
$F~cos~\theta = (mg-F~sin~\theta)~\mu_s$
$F~(cos~\theta+\mu_s~sin~\theta) = mg~\mu_s$
$F = \frac{mg~\mu_s}{cos~\theta+\mu_s~sin~\theta}$
$F = \frac{(180~N)~(0.42)}{cos~\theta+0.42~sin~\theta}$
$F = \frac{75.6~N}{cos~\theta+0.42~sin~\theta}$
