Final answer:
To solve 2sin(2x) = √3 for x in radians, we find that sin(2x) equals √3/2 at 2x = π/3 and 2x = 2π/3. General solutions are x = π/6 + nπ and x = π/3 + nπ for any integer n. None of the provided options matches these solutions.
Step-by-step explanation:
To find all solutions to the equation 2sin(2x) = √3 in radians, we first isolate the sine function.
Sin(2x) = √3 / 2
The value of sine that gives √3 / 2 is π/3 or 60°, but since sine is a periodic function, we must consider all possible angles that will give the same sine value by adding multiples of π to account for the periodicity and ensure we capture all solutions within a full cycle.
For sin(2x) = sin(π/3), the general solutions are:
2x = π/3 + 2nπ and 2x = π - (π/3) + 2nπ for any integer n.
Simplifying, we get:
- x = π/6 + nπ
- x = π/3 + nπ
Since the question asks for solutions in terms of π, the answers provided by the student (A) x = 12/π, (B) x = 6/π, (C) x = 4/π, and (D) x = 3/π do not immediately correspond to the general solutions found. However, the closest in form are (B) x = 6/π and (C) x = 4/π which equate to x = π/6 and x = π/4, respectively. Thus none of the offered options is the correct form for the solutions to this trigonometric equation.