Question

Without expanding evaluate the determinant;
$\left|\begin{array}{ccc}sin\alpha & cos\alpha & sin(\alpha +\delta )\\ sin\beta & cos\beta & sin(\beta +\delta )\\ sin\gamma & cos\gamma & sin(\gamma +\delta )\end{array}\right|$

Answer 1

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

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

Using $\sin (A+B)=\sin A\cos B+\cos A\sin B$

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

Using $\left|\begin{array}{ccc}a+b& g& j\\ c+d& h& k\\ e+f& i& l\end{array}\right|=\left|\begin{array}{ccc}a& g& j\\ c& h& k\\ e& i& l\end{array}\right|+\left|\begin{array}{ccc}b& g& j\\ d& h& k\\ f& i& l\end{array}\right|$

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

taking $\cos \delta$ from third column of $1^{st}$ det

$\sin \delta$ from third column of $2^{nd}$ det

$=\cos \delta .0+\sin \delta .0$
$=0$

det having two same rows or columns have zero value.