Let (\vec{a}, \vec{b}) and (\vec{c}) be three unit vectors such that (\vec{a} \times (\vec{b} \times \vec{c}) = \sqrt{3}2(\vec{b} + \vec{c})). If (\vec{b}) is not parallel to (\vec{c}), then the angle between (\vec{a}) and (\vec{b}) is:

(\vec{a} \times (\vec{b} \times \vec{c}) = \sqrt{3}2(\vec{b} + \vec{c})) (1)
(\vec{a} \times (\vec{b} \times \vec{c}) = (\vec{a} . \vec{c})\vec{b} - (\vec{a} . \vec{b})\vec{c}) (2)
(1) = (2)
((\vec{a} . \vec{c})\vec{b} - (\vec{a} . \vec{b})\vec{c} = \sqrt{3}2(\vec{b} + \vec{c}))
Equating LHS = RHS
((\vec{a} . \vec{c}) = \sqrt{3}2)
((\vec{a} . \vec{b}) = -\sqrt{3}2)
1 \times 1 \cos\theta = -\sqrt{3}2
\theta = 5\pi6