Take the radius of the great circle and the angle you have to rotate (so 60 deg) and use that to find the distance between the two points
Then just set that as the base of an isosceles trangle with the planar radius of the small circle as the legs.
Oh wait, I'm messing up what you're given.
If you're doing this as a function of the angle between the axes, then you could:
Draw the sphere as a circle, draw the great/small circles as lines (because they're being viewed directly from the side)
Find the point where the side-on circle's lines intersect.
Take the distance from the axis of the small circle to the point you just found.
You now have the height of an isosceles triangle formed by radii in the small circle.
Some basic trig later, and you've got the angle
Actually, your function should take 3 arguments, Radius of the sphere, radius of the small circle, and angle between the axes.
hSmall = (rSphere**2 - rSmall**2)**.5
x = hSmall * tan(90 - theta)
alpha =acos( x / rSmall)
What each line does:
First find the length of the axis of the small circle
Then find the height of the isosceles triangle
Find half the peak angle using trig
double it and return