  We use the Fokker Planck equation as a starting point for studying the
orientational probability distribution of an Active Brownian Particle (ABP) in
$(d+1)$ dimensions. This Fokker Planck equation admits an exact solution in
series form which is, however, unwieldly to use because of poor convergence for
short and intermediate times. A truncated version of this series is a
reasonable approximation for long times. In this paper, we present an
analytical closed form expression, which gives a good approximate orientational
probability distribution, which is derived using saddle point methods for short
times. However, it works well even for intermediate times. Thus, we have simple
analytical forms for the ${\it entire}$ range of time scales for the
orientational probability distribution of an ABP. Our predictions can be tested
against future experiments and simulations probing orientational probability
distribution of an ABP.
