Question

If $\tan A$ and $\tan B$ are the roots of $x^2-3x-7=0$, then the value of $\sin 2(A+B)$ is

• A. $\frac{48}{55}$
• B. $\frac{16}{73}$
• C. $\frac{48}{73}$
• D. $\frac{16}{55}$

Answer 1

The correct option is C $\frac{48}{73}$

$\tan A,\tan B$ are roots of $x^2-3x-7=0$

$\Rightarrow \tan A+\tan B=3$
$\Rightarrow \tan A\cdot \tan B=-7$

So, $\tan (A+B)=\frac{3}{1-(-7)}=\frac{3}{8}$

$\Rightarrow \sin 2(A+B)=\frac{2\tan (A+B)}{1+{\tan }^2(A+B)}=\frac{3/4}{1+9/64}$
$=\frac{48}{73}$