This is very similar to the solution Mintmaster suggested:
On cartesian coordinates, sphere center at origin (0,0,0), points and triangle vertices at surface of unit sphere.
a, b, c = triangle vertices (in clockwise order)
x = point on sphere
p1 = dot(x, cross(a, a-c))
p2 = dot(x, cross(b, b-a))
p3 = dot(x, cross(c, c-b))
If all p1, p2 and p3 are positive, the point is inside the triangle. Otherwise it's outside.
On cartesian coordinates, sphere center at origin (0,0,0), points and triangle vertices at surface of unit sphere.
a, b, c = triangle vertices (in clockwise order)
x = point on sphere
p1 = dot(x, cross(a, a-c))
p2 = dot(x, cross(b, b-a))
p3 = dot(x, cross(c, c-b))
If all p1, p2 and p3 are positive, the point is inside the triangle. Otherwise it's outside.