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