 Solution

At the north pole, for $v=(v_1,v_2,0)$ and $r=|v|$, ${\exp }_p(v)=\cos rp+(\sin r/r)v$. Polar coordinates give $(\sin r\cos \theta ,\sin r\sin \theta ,\cos r)$. They are one-to-one for $0<r<\pi$, covering the sphere minus the two poles as a coordinate chart, with the north pole added as the polar origin; the south pole is the cut locus.