For x \in (0, π) the equation sin x + 2 sin 2x

Engineering

Mathematics

General Solution of Trigonometric Equation

Question

For x $\in$ (0, π) the equation sin x + 2 sin 2x – sin 3x = 3 has

one solution

three solutions

no solution

infinitely many solutions

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sin x + 2 sin 2x – sin 3x = 3

⇒ 2 sin 2x – 2 cos 2x sin x = 3

⇒ 2 cos 2x sin x – 2 sin 2x + 1 + 1 + 1 = 0

⇒ 2 cos 2x sin x – 2 sin 2x + (sin²x + cos²x) + (sin²2x + cos²2x) + 1 = 0

⇒ (1 – sin 2x)² + (sin x + cos 2x)² + cos²x = 0, simultaneously all 3 are zero is not possible

$∴$ No solution