MathematicsIf cos⁡α+2cos⁡β+3cos⁡γ=sin⁡α+2sin⁡β+3sin⁡γ=0, then the value of sin⁡3α+8sin⁡3β+27sin⁡3γ is

If $\cos α + 2 \cos β + 3 \cos γ = \sin α + 2 \sin β + 3 \sin γ = 0,$ then the value of $\sin 3 α + 8 \sin 3 β + 27 \sin 3 γ$ is

A
$\sin (α + β + γ)$
B
$3 \sin (α + β + γ)$
C
$18 \sin (α + β + γ)$
D
$\sin (α + 2 β + 3)$

Solution:

$a = \cos α + isin α, b = \cos β + isin β, c = \cos γ + isin γ$
Then, $a + 2 b + 3 c = (\cos α + 2 \cos β + 3 \cos γ) + i (\sin α + 2 \sin β + 3 \sin γ) = 0$
$\begin{array}{l} \Rightarrow a^{3} + 8 b^{3} + 27 c^{3} = 18 abc \\ \Rightarrow \cos 3 α + 8 \cos 3 β + 27 \cos 3 γ = 18 \cos (α + β + γ) \end{array}$
And $\sin 3 α + 8 \sin 3 β + 27 \sin 3 γ = 18 \sin (α + β + γ)$

Solution:

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