We study the statistics of the first-passage time of a single
run-and-tumble particle (RTP) in one spatial dimension, with or without
resetting, to a fixed target located at L > 0. First, we compute the
first-passage time distribution of a free RTP, without resetting or in a
confining potential, but averaged over the initial position drawn from
an arbitrary distribution p(x). Recent experiments used a
noninstantaneous resetting protocol that motivated us to study in
particular the case where p(x) corresponds to the stationary
non-Boltzmann distribution of an RTP in the presence of a harmonic trap.
This distribution p(x) is characterized by a parameter v > 0, which
depends on the microscopic parameters of the RTP dynamics. We show that
the first-passage time distribution of the free RTP, drawn from this
initial distribution, develops interesting singular behaviors, depending
on the value of v. We then switch on resetting, mimicked by relaxation
of the RTP in the presence of a harmonic trap. Resetting leads to a
finite mean first-passage time and we study this as a function of the
resetting rate for different values of the parameters v and b = L/c,
where c is the position of the right edge of the initial distribution
p(x). In the diffusive limit of the RTP dynamics, we find a rich phase
diagram in the (b, v) plane, with an interesting reentrance phase
transition. Away from the diffusive limit, qualitatively similar rich
behaviors emerge for the full RTP dynamics.
