
### Instruction: help me answer this question If sin x + sin 3x + sin 5x = 0, find the value of x., think step by step
### Assistant: We can write sin x + sin 3x + sin 5x = 0 as (sin x + sin 5x) + sin 3x = 0. Using the formula 2 sin A cos B = sin (A + B) + sin (A - B), we can simplify it to sin 3x (2 cos 2x + 1) = 0. Therefore, sin 3x = 0 or 2 cos 2x + 1 = 0. If sin 3x = 0, then x = nπ/3, where n ∈ Z. If 2 cos 2x + 1 = 0, then cos 2x = – 1/2 = – cos π/3 = cos (π – π/3), so 2x = 2π/3 or 4π/3, and x = π/3 or 2π/3. However, x = 2π/3 is not a solution because sin 3x is negative when x = 2π/3. , The answer is x = nπ/3, where n ∈ Z
