Solve for x (exact solutions):

sin x - sin 3x + sin 5x = 0

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Thankyou. I tried find the solutions on my graphics calculator but however, the question is to find the EXACT solution(s) for the equation so I guess it will be in a fractional form or a square root form. Any help is appreciated of course.

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  1. Use these identities and see what you get
    sin 3x = 3 sinx - 4 sin^3x
    sin 5x = 5 sinx- 20sin^3x + 16sin^5x

    This means that
    3 sinx + 16 sin^3x + 16 sin^5x = 0
    sin x*(16 sin^4x + 16 sin^2x +3 = 0)
    One solution is sinx = 0; x = 0.
    The other solution can be obtained by letting sin^2x = u and solving the quadratic equation
    16 u^2 + 16 + 3 = 0
    This can be factored into
    (4u + 3)(4u + 1) = 0

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