Complex Numbers And Quadratic Equations question 155
Question: If $ \cos \alpha +\cos \beta +\cos \gamma =\sin \alpha +\sin \beta +\sin \gamma =0 $ then $ \cos 3\alpha +\cos 3\beta +\cos 3\gamma $ equals to [Karnataka CET 2000]
Options:
A) 0
B) $ \cos (\alpha +\beta +\gamma ) $
C) $ 3\cos (\alpha +\beta +\gamma ) $
D) $ 3\sin (\alpha +\beta +\gamma ) $
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Answer:
Correct Answer: C
Solution:
$ \cos \alpha +\cos \beta +\cos \gamma =0 $ and $ \sin \alpha +\sin \beta +\sin \gamma =0 $ Let $ a=\cos \alpha +i\sin \alpha ;b=\cos \beta +i\sin \beta $ and $ c=\cos \gamma +i\sin \gamma . $ Therefore $ a+b+c=(\cos \alpha +\cos \beta +\cos \gamma ) $ $ +i(\sin \alpha +\sin \beta +\sin \gamma ) $ $ =0+i0=0 $ If $ a+b+c=0, $ then $ a^{3}+b^{3}+c^{3}=3abc $ or $ {{(\cos \alpha +i\sin a)}^{3}}+{{(\cos \beta +i\sin \beta )}^{3}}+{{(\cos \gamma +i\sin \gamma )}^{3}} $ $ =3(\cos \alpha +i\sin \alpha )(\cos \beta +i\sin \beta )(\cos \gamma +i\sin \gamma ) $
$ \Rightarrow (\cos 3\alpha +i\sin 3\alpha )+(\cos 3\beta +i\sin 3\beta )+(\cos 3\gamma +i\sin 3\gamma ) $ $ =3[\cos (\alpha +\beta +\gamma )+i\sin (\alpha +\beta +\gamma )] $ or $ \cos 3\alpha +\cos 3\beta +\cos 3\gamma =3\cos (\alpha +\beta +\gamma ). $
BETA
