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Question
If α=cos α+i sin α,b=cos β+i sin β,c=cos γ+isin γ and bc+ca+ab=1 then cos(β−γ)+cos(γ−α)+cos(α−β)isequalto
If α=cos α+i sin α,b=cos β+i sin β,c=cos γ+isin γ and bc+ca+ab=1 then cos(β−γ)+cos(γ−α)+cos(α−β)isequalto
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Solution
The correct option is C 1
bc=cos β+isin βcos γ+isin γ×cos γ−isin γcos γ−isin γ
⇒bc=cos(β−γ)+isin(β−γ) ......(i)
Similarly, ca=cos(γ−β)+sin(α−β) ......(ii)
and ab=cos(α−β)+isin(γ−β) .....(iii)
from (i) + (ii) + (iii)
cos(β−γ)cos(γ−α)+cos(α−β)+i[sin(β−γ)+som(γ−α)+sin(α−β)]=1
Equating real and imaginary parts,
cos(β−γ)+cos(γ−α)+cos(α−β)=1
bc=cos β+isin βcos γ+isin γ×cos γ−isin γcos γ−isin γ
⇒bc=cos(β−γ)+isin(β−γ) ......(i)
Similarly, ca=cos(γ−β)+sin(α−β) ......(ii)
and ab=cos(α−β)+isin(γ−β) .....(iii)
from (i) + (ii) + (iii)
cos(β−γ)cos(γ−α)+cos(α−β)+i[sin(β−γ)+som(γ−α)+sin(α−β)]=1
Equating real and imaginary parts,
cos(β−γ)+cos(γ−α)+cos(α−β)=1
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