y = 4sin(3pt - 6px) - 4sin(3pt + 6px)

Now consider the trig. identity sin(x) - sin(y) = 2sin[(x-y)/2]cos[(x+y)/2].

This gives y = 8sin(-6px)cos(3pt)

The negative sign can come out of the sine function to give:

y = -8cos(3pt)sin(6px)

(b) (i) t = 0

y = -8cos(3p0)sin(6px)

y = -8sin(6px)

(ii) t = T/4

T=2p/w =
2p/3p =
2/3 therefore t = 1/6 and the standing wave is

y = -8cos(3p/6)sin(6px) = 0

(iii) t = 3/15 T

at t = 3T/15 = 3x2/15x3 = 2/15

The standing wave is y = -8cos[3p(2/15)]sin(6px)
= -8cos(2p/5)sin(6px)
= -2.47sin(6px)