Question

The value of
$\left|\begin{array}{ccc}\sin \alpha & \cos \alpha & \sin (\alpha +\delta )\\ \sin \beta & \cos \beta & \sin (\beta +\delta )\\ \sin \gamma & \cos \gamma & \sin (\gamma +\delta )\end{array}\right|$ is

• A. $0$
• B. $1$
• C. $2$
• D. $-1$

Answer 1

The correct option is A $0$

$\Delta =\left|\begin{array}{ccc}\sin \alpha & \cos \alpha & \sin (\alpha +\delta )\\ \sin \beta & \cos \beta & \sin (\beta +\delta )\\ \sin \gamma & \cos \gamma & \sin (\gamma +\delta )\end{array}\right|$

$\Rightarrow \Delta =\left|\begin{array}{ccc}\sin \alpha & \cos \alpha & \sin \alpha \cos \delta +\sin \delta \cos \alpha \\ \sin \beta & \cos \beta & \sin \beta \cos \delta +\sin \delta \cos \beta \\ \sin \gamma & \cos \gamma & \sin \gamma \cos \delta +\sin \delta \cos \gamma \end{array}\right|$
Using the transformation, $C_3\to C_3-\cos \delta C_1-\sin \delta C_2$
$\Rightarrow \Delta =\left|\begin{array}{ccc}\sin \alpha & \cos \alpha & 0\\ \sin \beta & \cos \beta & 0\\ \sin \gamma & \cos \gamma & 0\end{array}\right|$
$\Rightarrow \Delta =0$