Question

Let $A$ and $B$ denote the statements
$A: cos\,\alpha + cos\,\beta + cos \,\gamma = 0$
$B: sin \alpha + sin\, \beta + sin\,\gamma = 0$
If $cos\left(\beta-\gamma\right)+cos\left(\gamma-\alpha\right)+cos\left(\alpha-\beta\right)=-\frac{3}{2}$, 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

Answer 1

Answer: (C)

$ cos\left(\beta-\gamma\right)+cos\left(\gamma-\alpha\right)+cos\left(\alpha-\beta\right)=-\frac{3}{2}$
$\Rightarrow 2\left[cos\left(β − γ\right) + cos\left(γ − α\right) + cos\left(α − β\right)\right]+ 3 = 0$
$\Rightarrow 2\left[cos\left(β − γ\right) + cos\left(γ − α\right) + cos\left(α − β\right)\right]+sin^{2}\, α + cos^{2}\, α + sin^{2}\,β + cos^{2}\, β + sin^{2}\, γ + cos^{2}\, γ = 0$
$\Rightarrow \left(sin\,α + sin\,β + sin\,γ\right)^{2}+\left(cos\,\alpha+cos\,\beta+cos\,\gamma\right)^{2}=0$