Graphs of sin x and sin 2x intersect at x = _______ in the interval (0, π)

The graphs of sin x and sin 2x intersect when their y-values are equal, i.e., when sin x = sin 2x.

Step 1: Set the two functions equal to each other:

sin x = sin 2x

Step 2: The sin 2x can be expanded using the double angle formula for sine, which is sin 2x = 2sin x cos x. So the equation becomes:

sin x = 2sin x cos x

Step 3: If sin x = 0, then x = 0, π. But 0 is not in the interval (0, π), so we ignore it.

Step 4: If 2sin x cos x = sin x, then either sin x = 0 (which we've already covered) or 2cos x = 1.

Step 5: Solving 2cos x = 1 gives cos x = 1/2. The solutions to this in the interval (0, π) are x = π/3 and x = 5π/3. But 5π/3 is not in the interval (0, π), so we ignore it.

So, the graphs of sin x and sin 2x intersect at x = π/3 in the interval (0, π).