Geometry Level 4

A ladder rests against a wall at an angle $$\alpha$$ to the horizontal. Its foot is pulled away from the wall through a distance $$a$$, so that it slides a distance $$b$$ down the wall, making a new angle $$\beta$$ with the horizontal. What is the value of $$\frac{a}{b}$$ when $$\alpha = 60^\circ$$ and $$\beta = 45^\circ$$?