Find all solutions of the equation 2 sin x cos 2x-cos 2x=0 over the interval 0<x<π

Answer 1

To find all solutions of the equation 2sin(x)cos(2x) - cos(2x) = 0 over the interval 0 < x < π, we can first factor out cos(2x) from the equation:

cos(2x)(2sin(x) - 1) = 0.

Now we have two possibilities:

1) cos(2x) = 0
2) 2sin(x) - 1 = 0

For cos(2x) = 0, we know that cos(2x) = 0 when 2x = π/2 or 2x = 3π/2. Therefore, x = π/4 or x = 3π/4.

For 2sin(x) - 1 = 0, we get sin(x) = 1/2 which occurs at x = π/6 or x = 5π/6.

Therefore, the solutions of the equation 2sin(x)cos(2x) - cos(2x) = 0 over the interval 0 < x < π are x = π/4, x = 3π/4, x = π/6, and x = 5π/6.