  A class of thick branes in the background of sine-Gordon kinks with a scalar
potential $V(\phi)=p(1+\cos\frac{2\phi}{q})$ was constructed by R. Koley and S.
Kar [Classical Quantum Gravity \textbf{22}, 753 (2005)]. In this paper, in the
background of the warped geometry, we investigate the issue of localization of
spin half fermions on these branes in the presence of two types of
scalar-fermion couplings: $\eta\bar{\Psi}\phi\Psi$ and $\eta\bar{\Psi}\sin\phi
\Psi$. By presenting the mass-independent potentials in the corresponding
Schr\"{o}dinger equations, we obtain the lowest Kaluza--Klein (KK) modes and a
continuous gapless spectrum of KK states with $m^2>0$ for both types of
couplings. For the Yukawa coupling $\eta\bar{\Psi}\phi\Psi$, the effective
potential of the right chiral fermions for positive $q$ and $\eta$ is always
positive, hence only the effective potential of the left chiral fermions could
trap the corresponding zero mode. This is a well-known conclusion which had
been discussed extensively in the literature. However, for the coupling
$\eta\bar{\Psi}\sin\phi \Psi$, the effective potential of the right chiral
fermions for positive $q$ and $\eta$ is no longer always positive. Although the
value of the potential at the location of the brane is still positive, it has a
series of wells and barriers on each side, which ensures that the right chiral
fermion zero mode could be trapped. Thus we may draw the remarkable conclusion:
for positive $\eta$ and $q$, the potentials of both the left and right chiral
fermions could trap the corresponding zero modes under certain restrictions.
