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 \(\beta = 45^\circ \)?

Give your answer to the first decimal place.

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