Find the number of solutions for the equation $sin 5 x + sin 3 x + sin x = 0$ for $0 \leq x \leq π$

x = \frac{n π}{3}

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x = n π \pm \frac{π}{3}

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x = \frac{n π}{6}

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x = n π \pm \frac{π}{6}

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Solution

The correct options are
A $x = \frac{n π}{3}$
B $x = n π \pm \frac{π}{3}$
Given, $sin 5 x + sin 3 x + sin x = 0$

$\Rightarrow sin 5 x + sin x + sin 3 x = 0$
$\Rightarrow 2 sin (\frac{5 x + x}{2}) cos (\frac{5 x - x}{2}) + sin 3 x = 0$
$\Rightarrow 2 sin 3 x cos 2 x + sin 3 x = 0$

$\Rightarrow sin 3 x (2 cos 2 x + 1) = 0$
$\Rightarrow sin 3 x = 0$ or $cos 2 x = - \frac{1}{2}$
$\Rightarrow 3 x = n π$ or $2 x = 2 n π \pm \frac{2 π}{3}$
$\Rightarrow x = \frac{n π}{3}$ or $x = n π \pm \frac{π}{3}$

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Find the number of solutions for the equation sin5x+sin3x+sinx=0 for 0≤x≤π

The correct options are A x=nπ3 B x=nπ±π3Given, sin5x+sin3x+sinx=0⇒sin5x+sinx+sin3x=0⇒2sin(5x+x2)cos(5x−x2)+sin3x=0⇒2sin3xcos2x+sin3x=0⇒sin3x(2cos2x+1)=0⇒sin3x=0 or cos2x=−12⇒3x=nπ or 2x=2nπ±2π3⇒x=nπ3 or x=nπ±π3

Find the number of solutions for the equation $sin 5 x + sin 3 x + sin x = 0$ for $0 \leq x \leq π$