As you rightly deduced
T=52πConsider just the basic version of the sine curve.
y=sin(x)The maximum "amplitude" is 1. The locations where the amplitude is 0.5 are
6π,65π,67πand611π. The total "time" between each the first/second and third/fourth points is
64π. Multiply this by 2 because there are two sets of these points -->
68π=34πNow, we divide this by
2π to give the proportion of the period which is greater than half of the total amplitude. In this example
Ttotal has a period of 1.
Tover0.5a=32⋅TtotalSo, in the case of the question...
Tover0.5a=32⋅T=32⋅52π=154π