Question

For the simple harmonic motion equation d=8 sin(4 pi t) , what is the period? If necessary, use the slash (/) to represent a fraction

Answer 1

d=asin(wt)
w=2 pi/T
T=2 pi/w
T=2 pi/4 pi
T=1/2

Answer 2

I assume your question amounts to finding the period of f(t)=8sin(4 pi t). A period L is a positive number such that f(t+L)=f(t) for all t. Since sin(x) is 2 pi perodic we can find the least number L such that 4pi(t+L)=4*pi*t+2*pi. Multiplying out we get
4pi(t+L)=4*pi*t+4*pi*L and we want to find L such that we get 4*pi*t+2*pi, notice if we take L=1/2 we get the desiered result 4*pi*t+4*pi*1/2=4*pi*t+2*pi. This proves that L=1/2 and your function has a period of 1/2.