How many solutions does the equation sin(4x)=1/2 have on the interval (0 2pi)?

The angle < pi/2  and whose sine is 1/2 is pi/6

so let 4x = pi/6 and we have x = pi/24   which is one solution.

angle in second quadrant = 5pi / 6  so x = 5pi/24 is another solution.

Each of these  will give new angles as we revolve around the circle in steps of pi/2  so for x = pi/24 next one will be 13pi/24  and for x = 5pi/24 next one will be 17pi/24    This will be true for all angles less than  48pi/24  (ie 2pi)

so there will be 8 solutions