Question

$Ifcos\alpha +cos\beta +cos\gamma =sin\alpha +sin\beta +sin\gamma =0thencos3\alpha +cos3\beta +cos3\gamma equalsto$

• A.
• B.
• C.
• D.

Answer 1

The correct option is C

$cos\alpha +cos\beta +cos\gamma =0$ $andsin\alpha isin\beta +sin\gamma =0$ $Leta=cos\alpha isin\alpha ;b=cos\beta iisin\beta and$ $c=cos\gamma +isin\gamma .Therefore$ $a+b+c=(cos\alpha +cos\beta +cos\gamma )+i(sin\alpha +sin\beta +sin\gamma )$ $=0+i0=0$ $ifa+b+c=0,then{a}^3+b^3+c^3=3abcor$ $(cos\alpha +isin\alpha {)}^3+(cos\beta +isin\beta {)}^3+(cos\gamma +isin\gamma {)}^3$ $=3(cos\alpha +isin\alpha )(cos\beta +isin\beta )(cos\gamma +isin\gamma )$ $\Rightarrow (cos3\gamma +isin3\gamma )+(cos3\beta +isin3\beta )+(cos3\gamma +isin3\gamma )$ $=3\left[cos\right(\alpha +\beta +\gamma )+isin(\alpha +\beta +\gamma \left)\right]$ $orcos3\alpha +cos3\beta +cos3)$ $\gamma =3cos(\alpha +\beta +\gamma ).$