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

(D) both $A$ and $B$ are false

JEE Main · 2009 · 12 Apr 2009 · Offline · Trigonometric Ratios and Identities

Let

A

and

B

denote the statements
A:

cos α + cos β + cos γ = 0

sin α + sin β + sin γ = 0

cos (β - γ) + cos (γ - α) + cos (α - β) = - \frac{3}{2}

, then

Answer: (C)

Multiply the given relation by 2 and add 3 to it, writing the 3 as

sin^{2} α + cos^{2} α + sin^{2} β + cos^{2} β + sin^{2} γ + cos^{2} γ

. Expanding each cosine of a difference, the terms regroup exactly into

(sin α + sin β + sin γ)^{2} + (cos α + cos β + cos γ)^{2} = 0

. A sum of two squares can vanish only if each is zero, so both

cos α + cos β + cos γ = 0

and

sin α + sin β + sin γ = 0

, making both statements true.

JEE Main · 2009 · 12 Apr 2009 · Offline · Single correct option