To solve the equation sin 2x = cos x, we can use the identity sin 2A = 2 sin A cos A. This allows us to rewrite the equation as 2 sin x cos x - cos x = 0.
Factoring out the common factor of cos x, we have cos x (2 sin x - 1) = 0.
Now, we can set each factor equal to zero and solve for x:
1) cos x = 0:
We know that the cosine function is zero at π/2 (90°) and 3π/2 (270°) in the given domain (0 ≤ x ≤ 2π).
2) 2 sin x - 1 = 0:
Solving this equation, we get sin x = 1/2. In the first and second quadrants, the sine function is positive, and it is equal to 1/2 at π/6 (30°) and 5π/6 (150°) within the given domain.
Thus, the solutions for 0 ≤ x ≤ 2π are:
x = π/2, 3π/2, π/6, 5π/6.
If you want a general solution, you can express it by adding 2πk to each of the four solutions, where k is an integer.