The sum of all values of x in -0- 2- for which sin x - sin 2x - sin 3x - sin 4x - 0- is equal to

Sin x + sin 4x + sin 2x + sin 3x = 0 ⇒ 2sin5x2(cos3x2+cosx2)=0 ⇒ 4sin5x2cos x cosx2=0 sin5x2=0 ⇒ x=0, 2π5,4π5,6π5,8π5,10π5 or cos x=0⇒ x=π2,3π2 or cosx2=0⇒ x=π ...

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Byju's Answer

Standard XII

Mathematics

nth Term of A.P

The sum of al...

Question

The sum of all values of $x$ in $[0, 2 π]$ , for which $sin x + sin 2 x + sin 3 x + sin 4 x = 0$ , is equal to

11 π

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9 π

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8 π

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12 π

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Solution

The correct option is B $9 π$
$sin x + sin 4 x + sin 2 x + sin 3 x = 0$
$\Rightarrow 2 sin \frac{5 x}{2} (cos \frac{3 x}{2} + cos \frac{x}{2}) = 0$
$\Rightarrow 4 sin \frac{5 x}{2} cos x cos \frac{x}{2} = 0$
$sin \frac{5 x}{2} = 0 \Rightarrow x = 0, \frac{2 π}{5}, \frac{4 π}{5}, \frac{6 π}{5}, \frac{8 π}{5}, \frac{10 π}{5}$
or $cos x = 0 \Rightarrow x = \frac{π}{2}, \frac{3 π}{2}$
or $cos \frac{x}{2} = 0 \Rightarrow x = π$
$∴$ Sum $= 9 π$

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The sum of all values of x in [0,2π], for which sinx+sin2x+sin3x+sin4x=0, is equal to

The correct option is B 9πsinx+sin4x+sin2x+sin3x=0 ⇒2sin5x2(cos3x2+cosx2)=0 ⇒4sin5x2cosxcosx2=0 sin5x2=0⇒x=0,2π5,4π5,6π5,8π5,10π5 or cosx=0⇒x=π2,3π2 or cosx2=0⇒x=π ∴Sum =9π

The sum of all values of $x$ in $[0, 2 π]$ , for which $sin x + sin 2 x + sin 3 x + sin 4 x = 0$ , is equal to