Browse Questions

# If $\cos^{-1}\alpha+\cos^{-1}\beta+\cos^{-1}\gamma=3\pi$,then $\alpha(\beta+\gamma)+\beta(\gamma+\alpha)+\gamma(\alpha+\beta)$ equals$(A)\;0\quad(B)\quad 1\quad(C)\quad 6\quad(D)\quad 12$

Toolbox:
• Principal interval of cos is [$0,\pi$]
• $cos\pi=-1$
Ans - ( C ) 6
Max value of $cos^{-1}x=\pi$ in the principal interval
and given that
$cos^{-1}\alpha+cos^{-1}\beta+cos^{-1}\gamma=3\pi$

$\Rightarrow cos^{-1}\alpha = cos^{-1}\beta = cos^{-1}\gamma=\pi$
$\Rightarrow \alpha = \beta = \gamma =cos\pi= 1$
$\alpha(\beta+\gamma)+\beta(\gamma+\alpha)+\gamma(\alpha+\beta)=$
$-1(-2)-1(-2)-1(-2)=2+2+2=6$

edited Mar 9, 2013