Question

If $\cos \alpha +2\cos \beta +3\cos \gamma =0,\sin \alpha +2\sin \beta +3\sin \gamma =0$ and $\alpha +\beta +\gamma =\pi$, then $\sin 3\alpha +8\sin 3\beta +27\sin 3\gamma =$

• A. $-18$
• B. $0$
• C. $3$
• D. $9$

Answer 1

The correct option is B $0$

Let $z_1=\cos \alpha +i\sin \alpha =e^{i\alpha }$, $z_2=\cos \beta +i\sin \beta =e^{i\beta }$ and $z_3=\cos \gamma +i\sin \gamma =e^{i\gamma }$

Now $z_1+2z_2+3z_2=(\cos \alpha +2\cos \beta +3\cos \gamma )+i(\sin \alpha +2\sin \beta +3\sin \gamma )=0$

Now using the fact: If $a+b+c=0$, then $a^3+b^3+c^3=3abc$

$(z_1{)}^3+(2z_2{)}^3+(3z_3{)}^3=3(z_1\left)\right(2z_2\left)\right(3z_3)$

$\Rightarrow e^{3i\alpha }+8e^{3i\beta }+27e^{3i\gamma }=18e^{i(\alpha +\beta +\gamma )}=18e^{i\pi }=18(\cos \pi +i\sin \pi )=18(-1+0i)$

$\Rightarrow (\cos 3\alpha +8\cos 3\beta +27\cos 3\gamma )+i(\sin 3\alpha +8\sin 3\beta +27\sin \gamma )=-18+i\left(0\right)$

Hence $\sin 3\alpha +8\sin 3\beta +27\sin 3\gamma =0$