Sin 5x+sin x+2 sin^2 x=1

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вопрос

To solve this equation, we will first combine like terms and rewrite it in a more simplified form:

sin 5x + sin x + 2sin^2 x = 1

Write sin 5x in terms of sin x using the multiple angle identity:

sin 5x = sin(3x + 2x) = sin 3x cos 2x + cos 3x sin 2x

Now, substitute this expression back into the equation:

(sin 3x cos 2x + cos 3x sin 2x) + sin x + 2sin^2 x = 1

Now, using the sum-to-product identities for sine and cosine, we can simplify further:

sin 3x cos 2x + cos 3x sin 2x = sin(3x + 2x) = sin 5x

Substitute this back into the equation:

sin 5x + sin x + 2sin^2 x = 1

sin 5x + sin x + 2(1 - cos^2 x) = 1

sin 5x + sin x + 2 - 2cos^2 x = 1

sin 5x + sin x - 2cos^2 x = -1

Now, we can use the Pythagorean identity sin^2 x + cos^2 x = 1:

-2(1 - sin^2 x) = -1

2sin^2 x - 1 = -1

2sin^2 x = 0

sin^2 x = 0

sin x = 0

Therefore, the solution to the equation sin 5x + sin x + 2sin^2 x = 1 is sin x = 0.