Question

If $sin\alpha +sin\beta +sin\gamma =0=cos\alpha +cos\beta +cos\gamma$ then $si{n}^2\alpha +si{n}^2\beta +si{n}^2\gamma =$

• A. $-\frac{3}{2}$
• B. $\frac{3}{2}$
• C. $\frac{2}{3}$
• D. none of these

Answer 1

The correct option is B $\frac{3}{2}$

$\cos \alpha +\cos \beta +\cos \gamma =0=\sin \alpha +\sin \beta +\sin \gamma$
Let $a=\cos \alpha +i\sin \alpha ,b=\cos \beta +i\sin \beta$ and $c=\cos \gamma +i\sin \gamma$
$a+b+c=0$ and $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=0$
$\Rightarrow ab+bc+ca=0$
$\Rightarrow a^2+b^2+c^2=0$
$\therefore \cos 2\alpha +\cos 2\beta +\cos 2\gamma =0$
$\Rightarrow 1-2{\sin }^2\alpha +1-2{\sin }^2\beta +1-2{\sin }^2\gamma =0$
$\therefore {\sin }^2\alpha +{\sin }^2\beta +{\sin }^2\gamma =3/2$
Hence, option B.