Covering a reduced spherical body by a disk

In this paper, the following two theorems are proved: $(1)$ every spherical convex body $W$ of constant width $\Delta (W) \geq \frac{\pi}{2}$ may be covered by a disk of radius $\Delta(W) + \arcsin \left( \frac{2\sqrt{3}}{3} \cdot \cos \frac{\Delta(W)}{2}\right) - \frac{\pi}{2}$; $(2)$ every reduced spherical convex body $R$ of thickness $\Delta(R)<\frac{\pi}{2}$ may be covered by a disk of radius $\arctan \left( \sqrt{2} \cdot \tan \frac{\Delta(R)}{2}\right)$.