#1_2015

Algebra Level 3

If $tan\alpha +tan\beta +tan\gamma =tan\alpha .tan\beta .tan\gamma$ and $x=cos\alpha +isin\alpha \quad ,\quad y=cos\beta +isin\beta \quad ,\quad z=cos\gamma +isin\gamma$ , then $xyz$ equals

cos\beta +isin\gamma

1

but not

-1

i

-1

but not

1

+1

-1

cos\alpha +isin\gamma

cos\alpha +isin\beta

1 solution

Anandhu Raj
Jan 10, 2015

$xyz$ = ( $cos\alpha +isin\alpha )(cos\beta +isin\beta )(cos\gamma +isin\gamma )$

$\Rightarrow xyz\quad =\quad cos(\alpha +\beta +\gamma )+isin(\alpha +\beta +\gamma )$

$\Rightarrow xyz\quad =\quad cos(n\pi )+isin(n\pi )$

$\Rightarrow xyz\quad =\quad +1\quad +\quad 0$ (when $n\pi =\quad 0,\quad 2\pi ,\quad 4\pi \quad ....)$

$xyz\quad =\quad -1\quad +\quad 0$ (when $n\pi =\quad 1\pi,\quad 3\pi ,\quad 5\pi \quad ....)$

[Note : If a + b + c = $n\pi$ , then tan a + tan b + tan c = tan a . tan b .tan c ]

#1_2015

1 solution

0 pending reports