Question

If $\cos \alpha +\cos \beta +\cos \gamma =\sin \alpha +\sin \beta +\sin \gamma =0$, then the value of $\cos 3\alpha +\cos 3\beta +\cos 3\gamma$ is

• A. $0$
• B. $\cos (\alpha +\beta +\gamma )$
• C. $3\cos (\alpha +\beta +\gamma )$
• D. $3\sin (\alpha +\beta +\gamma )$

Answer 1

Answer: (C)

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

Let $a=\cos \alpha +i\sin \alpha$,
$b=\cos \beta +i\sin \beta$
and $c=\cos \gamma +i\sin \gamma$
Then, $a+b+c=(\cos \alpha +\cos \beta +\cos \gamma )+i(\sin \alpha +\sin \beta +\sin \gamma )$ $=0+i0=0$
$\Rightarrow a^3+b^3+c^3=3abc$
$\Rightarrow (\cos 3\alpha +i\sin 3\alpha )+(\cos 3\beta +i\sin 3\beta )+(\cos 3\gamma +i\sin 3\gamma )$
$=3[\cos (\alpha +\beta +\gamma )+i\sin (\alpha +\beta +\gamma )]$
$\Rightarrow \cos 3\alpha +\cos 3\beta +\cos 3\gamma =3\cos (\alpha +\beta +\gamma )$