$\Sigma M_O = 0$
$100 R \cos \alpha = 200 R \cos \beta$
$\cos \alpha = 2 \cos \beta$
From the figure
$\alpha + \beta = 90^\circ$
$\alpha = 90^\circ - \beta$
Thus,
$\cos (90^\circ - \beta) = 2 \cos \beta$
$\sin \beta = 2 \cos \beta$
$\dfrac{\sin \beta}{\cos \beta} = 2$
$\tan \beta = 2$
$\beta = 63.43^\circ$ answer
$\alpha = 90^\circ - 63.43^\circ$
$\alpha = 26.57^\circ$ answer