If k = 2π/45, what is the period?

Answer 1

If you are talking about something like sin (2 pi t/45)
a period is from sin (0) to sin (2 pi)
[ also like from pi to 3 pi but keep it simple]
in other words once around the circle
so from (2 pi t/45) = 0 to (2 pi t/45) = 2 pi
or
from t = 0 to t = 45
which is 45 seconds or centuries or whatever.

Answer 2

in general, the period of sin(kx) is 2π/k
so, if k = 2π/45, then the period is
(2π)/(2π/45) = 45