In the diagram, triangle ABC is right-angled at C and M is the mid-point of BC. It is given that angle ABC = \(\frac{1}{3} \pi\) radians and angle BAM = \(\theta\) radians. Denoting the lengths of BM and MC by x,

1. find AM in terms of x,
2. show that \(\theta = \frac{1}{6} \pi - \tan^{-1} \left( \frac{1}{2 \sqrt{3}} \right)\).