Final answer:
The values for X on the interval 0 ≤ x ≤ 360 degrees where 4sin²(x) - 4sin(x) + 1 = 0 can be found by treating the equation like a quadratic in terms of sin(x), yielding x=30°, 150°, 210°, 330° as solutions.
Step-by-step explanation:
To find all values for X on the interval 0 ≤ x ≤ 360 degrees where 4sin²(x) - 4sin(x) + 1 = 0, we notice that this quadratic-like equation can be solved by treating sin(x) as a variable, let's call it 'u' for simplicity. With u = sin(x), our equation becomes 4u² - 4u + 1 = 0, which resembles a quadratic equation and can be factored or solved using the quadratic formula.
Factoring gives us (2u - 1)² = 0, which implies that 2u - 1 = 0, so u = 1/2. Going back to our substitution, we get sin(x) = 1/2. We know from trigonometry that sin(x) = 1/2 at x = 30° and x = 150° in the first revolution. However, since we are looking in the range from 0 to 360 degrees, we also include their second revolution counterparts, which gives us solutions at x = 210° and x = 330°.
Therefore, the correct answer that includes all possible solutions is option A: x=30°, 150°, 210°, 330°.