Answer
$\theta = 48.0^{\circ}$
Work Step by Step
$L = 6077-31~cos~2\theta$
$6080.2 = 6077-31~cos~2\theta$
$31~cos~2\theta = 6077-6080.2$
$cos~2\theta = -\frac{3.2}{31}$
$2\theta = arccos(-\frac{3.2}{31})$
$\theta = \frac{1}{2}~arccos(-\frac{3.2}{31})$
$\theta = 48.0^{\circ}$
