Final answer:
With an equation sin(2x) = 12, there are no real solutions as the sine function's range is between -1 and 1. If it's a typo and should be sin(2x) = 1/2, then the correct solutions within the interval [0,2π) are π/12, 5π/12, 13π/12, and 17π/12.
Step-by-step explanation:
The equation given is sin(2x) = 12, which immediately suggests there's an error because the sine function can never be greater than 1 or less than -1. Therefore, there are no solutions as the equation is not possible within the realm of real numbers. If this is a typo and the equation intended was sin(2x) = 1/2, we can proceed to find real solutions over the interval [0,2π).
To solve sin(2x) = 1/2, we need to find angles for which the sine value is equal to 1/2. This occurs at π/6 and 5π/6 for sin(x). However, since we have sin(2x), we need to divide these angles by 2 to find the solutions for x.
Solutions are: x = π/12, 5π/12, 13π/12, and 17π/12. These angles fall within the given interval. We also have to consider the periodicity of the sine function, which is 2π for sin(x), meaning for sin(2x), the period is π. But, since the interval is from 0 to 2π, additional solutions beyond one period would exceed the interval, thus are not included.
Conclusion:
The potential correct solutions, if the equation were sin(2x) = 1/2, would be option (c) x = π/12, 5π/12, 13π/12, 17π/12.