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If $\cos x + 2 \cos y + 3 \cos z = \sin x + 2 \sin y + 3 \sin z = 0$ then the value of $\sin 3 x + 8 \sin 3 y + 27 \sin 3 z$ is

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$\sin (x + 2 y + z)$

$3 \sin (x + y + z)$

$\sin (x + y + z)$

$18 \sin (x + y + z)$

answer is C.

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Detailed Solution

$α = c i s x, β = c i s y, γ = c i s z$ given
$α + 2 β + 3 γ = 0 \Rightarrow α^{3} + 8 β^{3} + 27 γ^{3} = 18 α β γ$
$\Rightarrow \sin 3 x + 8 \sin 3 y + 27 \sin 3 z = 18 \sin (x + y + z)$

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