Final answer:
The correct answer to the trigonometric equation is not provided among the choices. Solving the equation by simplification suggests that y is directly proportional to x. Thus, y is ±3x, which is not in the form of any arcsine function provided.
Step-by-step explanation:
Let's solve the trigonometric equation for y in terms of x. The given equation is:
sin(3x) × 2cos(2x)sin(3x) = sin²(y) × sin²(x)
First, we can simplify the left side using the trigonometric identity sin(2α) = 2sin(α)cos(α), applied to 3x:
sin(3x) × 2cos(2x)sin(3x) = sin(3x) × sin(6x)
Now, let's assume that y and x are in the range where the sine function is bijective and has an inverse (for example, [-π/2, π/2]), so that we can apply the inverse sine function. Note that the equation now simplifies to:
sin²(3x) = sin²(y) × sin²(x)
Since we are looking for the solution where sin(y) is equal to sin(3x) or -sin(3x), we get:
y = sin^(-1)(sin(3x))
But since sin(y) could be negative, we also consider:
y = -sin^(-1)(sin(3x))
However, given the principal value range for sin^(-1), the correct representation for y would be:
y = ±3x
Now, since we're solving for y in terms of x and not sin(3x), the provided options are comparing the value of y to known angles. Thus, the correct answer is none of the provided multiple-choice answers, because y is not dependent on the arcsine of a value, but rather directly related to x itself.