Question

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

• A. sin (a + b + )
• B. 3sin (a + + )
• C. 18 sin (+ + )
• D. sin (+ + 3 )

Answer 1

The correct option is C 18 sin (+ + )

Let a = cos $\alpha +i\sin \alpha$
b = cos $\beta +i\sin \beta ,c=\cos \gamma +i\sin \gamma$
Then,
a + 2b + 3c = $\left(\cos \alpha +2\cos \beta +3\cos \gamma \right)+i\left(\sin \alpha +2\sin \beta +3\sin \gamma \right)=0$
$⟹a^3+{8b}^3+{27c}^3=18abc$
$⟹\cos 3\alpha +8\cos 3\beta +27\cos 3\gamma =18\cos (\alpha +\beta +\gamma )$ and
$sin3\alpha +8\sin 3\beta +27\sin 3\gamma =18\sin (\alpha +\beta +\gamma )$