Let $A$ and $B$ denote the statements A: $\cos \alpha+\cos \beta+\cos \gamma=0$ B: $\sin \alpha+\sin…

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
  1. $A$ is true and $B$ is false
  2. $A$ is false and $B$ is true
  3. both $A$ and $B$ are true
  4. both $A$ and $B$ are false

Solution

$ \begin{aligned} & \cos (\beta-\gamma)+\cos (\gamma-\alpha)+\cos (\alpha-\beta)=-\frac{3}{2} \\ & \Rightarrow 2[\cos (\beta-\gamma)+\cos (\gamma-\alpha)+\cos (\alpha-\beta)]+3=0 \\ & \Rightarrow 2[\cos (\beta-\gamma)+\cos (\gamma-\alpha)+\cos (\alpha-\beta)]+\sin ^2 \alpha+\cos ^2 \alpha+\sin ^2 \beta+\cos ^2 \beta+\sin ^2 \gamma+\cos ^2 \gamma=0 \\ & \Rightarrow(\sin \alpha+\sin \beta+\sin \gamma)^2+(\cos \alpha+\cos \beta+\cos \gamma)^2=0 \end{aligned} $

Asked in: JEE Main 2009

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