Angular structure of equiangular cones in Euclidean spaces

Let K be an n-dimensional equiangular cone in a Euclidean vector space. Let 0 < phi <pi be the common angle between any pair of vectors taken among the generators of the cone. We obtain an explicit formula for the maximum angle of K in terms of n and phi. We explain also how to identify a pair {u,v}of unit vectors in K achieving the maximum angle. Finally, we analyze a counterintuitive phenomenon that occurs when n is large: the equiangular cone K can be constructed so that u and v are almost opposite, whereas phi is near pi/2.