Question

If $\cos \alpha +\cos \beta +\cos \gamma =0=\sin \alpha +\sin \beta +\sin \gamma$, then $\sin 3\alpha +\sin 3\beta +\sin 3\gamma$ is equal to

• A. $\sin 3(\alpha +\beta +\gamma )$
• B. $3\sin (\alpha +\beta +\gamma )$
• C. $3\sin 3(\alpha +\beta +\gamma )$
• D. $3\cos 3(\alpha +\beta +\gamma )$

Answer 1

Answer: (B)

$3\sin (\alpha +\beta +\gamma )$

If $a=cos\alpha +isin\alpha ,b=cos\beta +isin\beta ,c=cos\gamma +isin\gamma ,$ then $a+b+c=(cos\alpha +cos\beta +cos\gamma )+i(sin\alpha +sin\beta +sin\gamma )=0+i0=0$
$\Rightarrow a^3+b^3+c^3=3abc$
$\Rightarrow \sum (cos\alpha +isin\alpha {)}^3=3(cos\alpha +isin\alpha \left)\right(cos\beta +isin\beta \left)\right(cos\gamma +isin\gamma )$
$\Rightarrow \sum cos3\alpha +i\sum sin3\alpha =3cos(\alpha +\beta +\gamma )+i3sin(\alpha +\beta +\gamma )$
$\Rightarrow sin3\alpha +sin3\beta +sin3\gamma =3sin(\alpha +\beta +\gamma )$