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Question:

For x ∈ (0, π), the equation sinx + 2sin2x − sin3x = 3 has:

infinitely many solutions

three solutions

one solution

no solution

Solution:

sinx + 2sin2x − sin3x = 3
sinx + 4sinxcosx − (3sinx − 4sin³x) = 3
sinx + 4sinxcosx + 4sin³x − 3sinx = 3
sinx(4cos²x + 4sin²x − 2) = 3
sinx[4(cos²x + sin²x) − 2] = 3
sinx[4(1) − 2] = 3
2sinx = 3
sinx = 3/2
Since the range of sinx is [-1, 1], there is no solution for sinx = 3/2.
Hence, no solution.