Trigonometric EquationHard
Question
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)= -
, then
A : cosα + cosβ + cosγ = 0
B : sinα + sinβ + sinγ = 0
If cos (β - γ) + cos (γ - α) + cos(α - β)= -
, thenOptions
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
Solution
cos(β - γ) + cos(γ - α) + cos(α - β) = -
⇒ 2[cos(β - γ) + cos(γ - α) + cos(α - β)] + 3 = 0
⇒ 2[cos(β - γ) + cos(γ - α) + cos(α - β)] + sin2 α + cos2α + sin2β + cos2β + sin2γ + cos2γ = 0
⇒ (sinα + sinβ + sinγ)2 + (cosα + cosβ + cosγ)2) = 0
⇒ 2[cos(β - γ) + cos(γ - α) + cos(α - β)] + 3 = 0
⇒ 2[cos(β - γ) + cos(γ - α) + cos(α - β)] + sin2 α + cos2α + sin2β + cos2β + sin2γ + cos2γ = 0
⇒ (sinα + sinβ + sinγ)2 + (cosα + cosβ + cosγ)2) = 0
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