  The 'Arcsine' laws of Brownian particles in one dimension describe
distributions of three quantities: the time $t_m$ to reach maximum position,
the time $t_r$ spent on the positive side and the time $t_\ell$ of the last
visit to the origin. Interestingly, the cumulative distribution of all the
three quantities are same and given by Arcsine function. In this paper, we
study distribution of these three times $t_m,~t_r$ and $t_\ell$ in the context
of single run-and-tumble particle in one dimension, which is a simple
non-Markovian process. We compute exact distributions of these three quantities
for arbitrary time and find that all three distributions have delta function
part and a non-delta function part. Interestingly, we find that the
distributions of $t_m$ and $t_r$ are identical (reminiscent of the Brownian
particle case) when the initial velocities of the particle are chosen with
equal probability. On the other hand, for $t_\ell$, only the non-delta function
part is same with the other two. In addition, we find explicit expressions of
the joint distributions of the maximum displacement and the time at which this
maxima occurs. We verify all our analytical results through numerical
simulations.
