I have the following hyperbolic lines and I'm asked to find the angle between them. The lines are $ε_0 = \{ z \in \mathbb{C}: Im(z) >0, Re(z) = 0\}$ and $ε_{1,2} = \{z \in \mathbb{C}: |z-1| = 2\}$.
With some algebra I found that they intersect in $w = 0 + \sqrt3i$. So, drawing a line from the center of $ε_{1,2}$ to $w$ I create triangle with its vertices being the origin(O), the center of $ε_{1,2}$ and $w$(shape below). I'm asked to find the angle($θ$) created by the line I drew from the center of $ε_{1,2}$ and $ε_0$. Using the law of cosines, i find that $1^2 = 2^2 + \sqrt3^2 - 2*2*\sqrt3*cos(θ) \rightarrow cos(θ) = \sqrt3/2 \rightarrow θ = π/6$, but the book's answer is $-π/3$. What am I doing wrong?