1
Question
If the system of equations
xsinα+ysinβ+zsinγ=0
xcosα+ycosβ+zcosγ=0
xcos3α+ycos3β+zcos3γ=0
has solution other than trivial solution , then which of the following is(are) possible?
xsinα+ysinβ+zsinγ=0
xcosα+ycosβ+zcosγ=0
xcos3α+ycos3β+zcos3γ=0
has solution other than trivial solution , then which of the following is(are) possible?
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Solution
The correct options are
B α=β=γ
D α+β+γ=π2
Δ=∣∣
∣
∣∣sinαsinβsinγcosαcosβcosγcos3αcos3βcos3γ∣∣
∣
∣∣=0
=cosαcosβcosγ∣∣
∣
∣∣tanαtanβ−tanαtanγ−tanβ111cos2αcos2βcos2γ∣∣
∣
∣∣
=−cosαcosβcosγ∣∣
∣
∣∣tanβtanβ−tanαtanγ−tanβ100cos2αcos2β−cos2αcos2γ−cos2β∣∣
∣
∣∣
=−cosαcosβcosγ∣∣∣tanβ−tanαtanγ−tanβcos2β−cos2αcos2γ−cos2β∣∣∣
=−cosαcosβcosγ∣∣
∣∣sin(β−α)cosαcosβsin(γ−β)cosβcosγsin2α−sin2βsin2β−sin2γ∣∣
∣∣
=sin(α−β)sin(β−γ)[sin(β+γ)cosγ−sin(α+β)cosα]
=12sin(α−β)sin(β−γ)[sin(β+2γ)+sinβ−sin(β+2α)−sinβ]
=12sin(α−β)sin(β−γ).2cos(α+β+γ)sin(γ−α)
Hence, Δ is zero if either α=β=γ or α+β+γ=π2
B α=β=γ
D α+β+γ=π2
Δ=∣∣ ∣ ∣∣sinαsinβsinγcosαcosβcosγcos3αcos3βcos3γ∣∣ ∣ ∣∣=0
=cosαcosβcosγ∣∣ ∣ ∣∣tanαtanβ−tanαtanγ−tanβ111cos2αcos2βcos2γ∣∣ ∣ ∣∣
=−cosαcosβcosγ∣∣ ∣ ∣∣tanβtanβ−tanαtanγ−tanβ100cos2αcos2β−cos2αcos2γ−cos2β∣∣ ∣ ∣∣
=−cosαcosβcosγ∣∣∣tanβ−tanαtanγ−tanβcos2β−cos2αcos2γ−cos2β∣∣∣
=−cosαcosβcosγ∣∣ ∣∣sin(β−α)cosαcosβsin(γ−β)cosβcosγsin2α−sin2βsin2β−sin2γ∣∣ ∣∣
=sin(α−β)sin(β−γ)[sin(β+γ)cosγ−sin(α+β)cosα]
=12sin(α−β)sin(β−γ)[sin(β+2γ)+sinβ−sin(β+2α)−sinβ]
=12sin(α−β)sin(β−γ).2cos(α+β+γ)sin(γ−α)
Hence, Δ is zero if either α=β=γ or α+β+γ=π2
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