a
(a) It is required to calculate the time from extreme position.
Hence, in this case equation for displacement of particle can be written as \(x = a\sin (\omega \,t + \frac{\pi }{2}) = a\cos \omega \,t\)

==> \(\frac{a}{2} = a\cos \omega \,t\)

==> \(\omega \,t = \frac{\pi }{3}\)

==> \(\frac{{2\pi }}{T}.t = \frac{\pi }{3}\)

==> \(t = \frac{T}{6}\)