2*sin^3(x) - cos(2x) - sin(x) = 0

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вопрос

To solve the equation 2sin^3(x) - cos(2x) - sin(x) = 0, we can first simplify the equation by using trigonometric identities.

Recall the double angle identity for cosine: cos(2x) = 1 - 2sin^2(x).

Now, substitute this into the equation:
2sin^3(x) - (1 - 2sin^2(x)) - sin(x) = 0.

Simplify further:
2sin^3(x) - 1 + 2sin^2(x) - sin(x) = 0.

Rearrange the terms:
2sin^3(x) + 2sin^2(x) - sin(x) - 1 = 0.

Now, factor out a sin(x):
sin(x)(2sin^2(x) + 2sin(x) - 1) = 0.

Now, we have two possible solutions:

For sin(x) = 0, x can be any multiple of π.

To solve the second equation, we can use the quadratic formula:
sin(x) = [-2 ± sqrt((2)^2 - 4(2)(-1))]/(2*2)
sin(x) = [-2 ± sqrt(4 + 8)]/4
sin(x) = [-2 ± sqrt(12)]/4
sin(x) = [-2 ± 2sqrt(3)]/4
sin(x) = -1/2 ± sqrt(3)/2.

Therefore, x = π/3, 5π/3.

So, the solutions to the equation 2sin^3(x) - cos(2x) - sin(x) = 0 are x = π(n), π/3 and 5π/3.