If 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 $\mathrm{cos}\mathrm{\alpha }+2\mathrm{cos}\mathrm{\beta }+3\mathrm{cos}\mathrm{\gamma }=\mathrm{sin}\mathrm{\alpha }+2\mathrm{sin}\mathrm{\beta }+3\mathrm{sin}\mathrm{\gamma }=0$ then the value of sin $3\mathrm{\alpha }+8\mathrm{sin}3\mathrm{\beta }+27\mathrm{sin}3\mathrm{\gamma }$ is

1. A

$\mathrm{sin}\left(\mathrm{\alpha }+\mathrm{\beta }+\mathrm{\gamma }\right)$

2. B

$3\mathrm{sin}\left(\mathrm{\alpha }+\mathrm{\beta }+\mathrm{\gamma }\right)$

3. C

$18\mathrm{sin}\left(\mathrm{\alpha }+\mathrm{\beta }+\mathrm{\gamma }\right)$

4. D

$\mathrm{sin}\left(\mathrm{\alpha }+2\mathrm{\beta }+\mathrm{\gamma }\right)$

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### Solution:

Let $\mathrm{a}=\mathrm{cos}\mathrm{\alpha }+\mathrm{isin}\mathrm{\alpha }$
$\begin{array}{r}\mathrm{b}=\mathrm{cos}\mathrm{\beta }+\mathrm{isin}\mathrm{\beta }\\ \mathrm{c}=\mathrm{cos}\mathrm{\gamma }+\mathrm{isin}\mathrm{\gamma }\end{array}$
Then, $\mathrm{a}+2\mathrm{b}+3\mathrm{c}=\left(\mathrm{cos}\mathrm{\alpha }+2\mathrm{cos}\mathrm{\beta }+3\mathrm{cos}\mathrm{\gamma }\right)$
$+\mathrm{i}\left(\mathrm{sin}\mathrm{\alpha }+2\mathrm{sin}\mathrm{\beta }+3\mathrm{sin}\mathrm{\gamma }\right)=0$
$\begin{array}{l}⇒{\mathrm{a}}^{3}+8{\mathrm{b}}^{3}+27{\mathrm{c}}^{3}=18\mathrm{abc}\\ ⇒\mathrm{cos}3\mathrm{\alpha }+8\mathrm{cos}3\mathrm{\beta }+27\mathrm{cos}3\mathrm{\gamma }=18\mathrm{cos}\left(\mathrm{\alpha }+\mathrm{\beta }+\mathrm{\gamma }\right)\end{array}$
and $\mathrm{sin}3\mathrm{\alpha }+8\mathrm{sin}3\mathrm{\beta }+27\mathrm{sin}3\mathrm{\gamma }=18\mathrm{sin}\left(\mathrm{\alpha }+\mathrm{\beta }+\mathrm{\gamma }\right)$

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