I think you can write also (a shorter way):
In general if you've an oscillation given by:
Then the pulsation is where T is the period. In this case therefore
An oscillating particle has displacement y metres, where y is given by y=asin(kt + A)°, where a measured in metres, t is measured in seconds and k and A are constants. The time for a complete oscillation is T seconds.
Find k in terms of T.
so T must be 4x(length of time taken for the particle to move from y=0 to y=a its maximum displacement)
This length of time therefore can be found using sin(kt + A)°=1
so t=(90 - A)/k
so T=4t=(360-4A)/k
now to eliminate A
y=0 at t=0, meaning 0=asin(0 + A)
so sinA = 0, A=0
T=360/k
Is this reasoning correct?