Final answer:
The trigonometric equation sin 2x - cos x = 0 is solved using double-angle identities, factoring, and simple trigonometric equations. The solutions in the interval [0, 2π) are x = π/2, 3π/2, π/6, and 5π/6.
Step-by-step explanation:
To solve the trigonometric equation sin 2x - cos x = 0 within the interval [0, 2π), we use trigonometric identities to simplify the equation. First, recognize that sin 2x can be rewritten using the double-angle identity as 2 sin x cos x. So, the equation becomes 2 sin x cos x - cos x = 0. We can factor out the cos x, giving us cos x (2 sin x - 1) = 0.
Next, we solve for the values of x that satisfy each part of the equation. Setting cos x to zero gives us the solutions where x = π/2 and 3π/2. Setting 2 sin x - 1 to zero and solving for sin x gives us the solution x = π/6 and 5π/6. Combining these, the solutions to the equation sin 2x - cos x = 0 are x = π/2, 3π/2, π/6, and 5π/6 within the given interval.