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

D=asin (wt)

w=2 pi/T

T=2 pi/w

T=2 pi/4 pi

T=1/2

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.