Final answer:
The equation 0 = sin(3x)cos(x) - sin(x)cos(3x) can be solved using the sum-to-product trigonometric identity to yield solutions x = 0, π/2, π, 3π/2, and 2π within the interval [0, 2π].
Step-by-step explanation:
To find the solutions to the equation 0 = sin(3x)cos(x) - sin(x)cos(3x) on the interval [0, 2π], we can recognize this as an application of the sum-to-product trigonometric identities.
Specifically, the given equation is an example of the identity sin(A)cos(B) - sin(B)cos(A) = sin(A - B).
Using this identity, we can rewrite the equation as:
sin(3x - x) = sin(2x) = 0
To find the x-values where sin(2x) equals zero, we look for all angles where the sine function is zero within the given interval.
The sine function is zero at multiples of π.
Thus, the solutions are when:
2x = nπ, where n is an integer.
Dividing both sides by 2 gives us:
x = n(π/2)
Since x is in the interval [0, 2π], n can be 0, 1, 2, 3, or 4, giving us the following solutions:
- x = 0
- x = π/2
- x = π
- x = 3π/2
- x = 2π
These are the solutions to the equation within the specified interval.