From the diagram we see there is a 90 degree angle, also the radius from a point B to the center of the circle is the same as the radius from the center of the circle to the right angle.
Let the radius be r.
sin a=opposite/adjacent
sin 76.6=r/(r+112)
0.973(r+112)=r
0.973r+108.976=r
r-0.973r=108.976
0.027r=108.976
r=108.976/0.027
r=4036.15 miles
the radius of earth will r=4036.15 miles
