Solve the equation sin(3x) = 0 for x in the interval [0, 2π].
Correct Answer: x = 0, π, 2π, and x = nπ/3 for n = 0, 1, 2, 3, 4, 5.
- Step 1: Understand the equation sin(3x) = 0. This means we need to find values of 3x where the sine function equals zero.
- Step 2: Recall that sine equals zero at integer multiples of π. So, we can write the equation as 3x = nπ, where n is any integer.
- Step 3: Solve for x by dividing both sides of the equation by 3: x = nπ/3.
- Step 4: Determine the values of n that keep x within the interval [0, 2π].
- Step 5: Calculate the values of x for n = 0, 1, 2, 3, 4, 5: This gives us x = 0, π/3, 2π/3, π, 4π/3, 5π/3, and 2π.
- Step 6: List the solutions: x = 0, π, 2π, and x = nπ/3 for n = 0, 1, 2, 3, 4, 5.
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