I Have the following problem (see image below). I would like to calculate the distance between points $A$ and $X$. The radius of the circle is known ($r$); furthermore it is given that the length of $AB$ is $0.5$ greater than the circular arc distance of $BX$.
I tried solving the problem by expressing the lengths of $AO$ and the circular arc distance $BX$ in $\alpha$:
\begin{align} (\text{arc distance }BX) & = \frac{0.5 \pi - \alpha}{2 \pi} \times 2 \pi r \\
AO & = \frac{r}{\sin \alpha} \\
(\text{arc distance }BX) + 0.5 & = AO \\
\frac{0.5 \pi - \alpha}{ 2 \pi} \times 2 \pi r + 0.5 - \frac{r}{ \sin \alpha} & = 0 \end{align}
However, I'm not sure how to solve this for $\alpha$.
Is there perhaps a better way to approach this problem?