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)

both $A$ and $B$ are true

Answer 2

$cos\left(\beta-\gamma\right)+cos\left(\gamma-\alpha\right)+cos\left(\alpha-\beta\right)=-\frac{3}{2}$ \Rightarrow2\left[cos\left(??- ?\right) + cos\left(? - a\right) + cos\left(a - ??right)\right]+ 3 = 0 \Rightarrow 2\left[cos\left(??- ?\right) + cos\left(? - a\right) + cos\left(a - ??right)\right]+sin^{2}\, a + cos^{2}\, a + sin^{2}\,??+ cos^{2}\, ??+ sin^{2}\, ? + cos^{2}\, ? = 0 $\Rightarrow \left(sin\,a + sin\,??+ sin\,?\right)^{2}+\left(cos\,\alpha+cos\,\beta+cos\,\gamma\right)^{2}=0$