For x ( 0 ,) , the equation sin x + 2 sin 2 x - sin 3 x = 3 has [2014]

Question

For $x \in (0, π),$ the equation $\sin x + 2 \sin 2 x - \sin 3 x = 3$ has [2014]

Smart Achievers · Accepted Answer

(4)

$\sin x + 2 \sin 2 x - \sin 3 x = 3$

$\Rightarrow \sin x + 4 \sin x \cos x - 3 \sin x + 4 \sin^{3} x = 3$

$\Rightarrow \sin x (- 2 + 4 \cos x + 4 \sin^{2} x) = 3$

$\Rightarrow \sin x (- 2 + 4 \cos x + 4 - 4 \cos^{2} x) = 3$

$2 + 4 \cos x - 4 \cos^{2} x = \frac{3}{\sin x}$ $[∵ 0 \leq \sin x \leq 1]$

$\Rightarrow 2 - 4 (\cos^{2} x - 2 \cos x \cdot \frac{1}{2} + \frac{1}{4}) + 1 = \frac{3}{\sin x}$

$\Rightarrow 3 - 4 {(\cos x - \frac{1}{2})}^{2} = \frac{3}{\sin x}$ $∴$ $L.H.S \leq 3 and R.H.S \geq 3$

$Hence, the equation has no solution.$

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