Q. Let $A$ and $B$ denote the statements $A:\cos\alpha + \cos\beta + \cos{\gamma = 0}$ $B:\sin{\alpha + \sin{\beta + \sin{\gamma = 0}}}$ If $\cos{(\beta - \gamma)} + \cos{(\gamma - \alpha)} + \cos{(\alpha - \beta)} = - \frac{3}{2}$, then
A) $A$ is true and $B$ is false
B) $A$ is false and $B$ is true
C) Both $A$ and $B$ are true
D) Both $A$ and $B$ are false
Explanation
(c)
$\cos{(\beta - \gamma)} + \cos{(\gamma - \alpha)} + \cos{(\alpha - \beta)} = - \frac{3}{2}$
$\Rightarrow \ \ 2\left\lbrack \cos(\beta - \gamma) + \cos{(\gamma - \alpha) + \cos(\alpha - \beta)} \right\rbrack + 3 = 0$
$\Rightarrow \ \ 2\left\lbrack \cos{(\beta - \gamma) + \cos(\gamma - \alpha) + \cos(\alpha - \beta)} \right\rbrack$
$+ \sin^{2}{\alpha + \cos^{2}{\alpha + \sin^{2}{\beta + \cos^{2}{\beta + \sin^{2}{\gamma + \cos^{2}\gamma}}}}}$=0
$\Rightarrow \ \ \left( \sin{\alpha + \sin{\beta + \sin\gamma}} \right)^{2} + \left( \cos{\alpha + \cos{\beta + \cos\gamma}} \right)^{2} = 0$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.