Let A and B denote the statements A: cos α + cos β + cos γ = 0 B: sin α + sin β + sin γ = 0 If cos(β-γ)+cos(γ-α)+cos(α-β)=-(3/2), then

Question

Let A and B denote the statements A: cos α + cos β + cos &gamma; = 0 B: sin α + sin β + sin &gamma; = 0 If cos(β-&gamma;)+cos(&gamma;-α)+cos(α-β)=-(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

Tardigrade Admin · Accepted Answer

cos(β-&gamma;)+cos(&gamma;-α)+cos(α-β)=-(3/2) ⇒ 2[cos(Î² â Î³) + cos(Î³ â Î±) + cos(Î± â Î²)]+ 3 = 0 ⇒ 2[cos(Î² â Î³) + cos(Î³ â Î±) + cos(Î± â Î²)]+sin2 Î± + cos2 Î± + sin2 Î² + cos2 Î² + sin2 Î³ + cos2 Î³ = 0 ⇒ (sin Î± + sin Î² + sin Î³)2+(cos α+cos β+cos &gamma;)2=0

Q. Let $A$ and $B$ denote the statements
$A : cos α + cos β + cos γ = 0$
$B : s in α + s in β + s in γ = 0$
If $cos (β - γ) + cos (γ - α) + cos (α - β) = - \frac{3}{2}$ , then

$A$ is true and $B$ is false

$A$ is false and $B$ is true

both $A$ and $B$ are true

both $A$ and $B$ are false

Q. Let A and B denote the statements A:cosα+cosβ+cosγ=0 B:sinα+sinβ+sinγ=0 If cos(β−γ)+cos(γ−α)+cos(α−β)=−23​, then

A is true and B is false

A is false and B is true

both A and B are true

both A and B are false

cos(β−γ)+cos(γ−α)+cos(α−β)=−23​ ⇒2[cos(β−γ)+cos(γ−α)+cos(α−β)]+3=0 ⇒2[cos(β−γ)+cos(γ−α)+cos(α−β)]+sin2α+cos2α+sin2β+cos2β+sin2γ+cos2γ=0 ⇒(sinα+sinβ+sinγ)2+(cosα+cosβ+cosγ)2=0

Q. Let $A$ and $B$ denote the statements
$A : cos α + cos β + cos γ = 0$
$B : s in α + s in β + s in γ = 0$
If $cos (β - γ) + cos (γ - α) + cos (α - β) = - \frac{3}{2}$ , then

$A$ is true and $B$ is false

$A$ is false and $B$ is true

both $A$ and $B$ are true

both $A$ and $B$ are false