Question

If $a=\sin \theta$, $b=\sin (\theta +2\pi /3)$, $c=\sin (\theta +4\pi /3)$, $x=\cos \theta$, $y=\cos (\theta +2\pi /3)$, $z=\cos (\theta +4\pi /3)$, then value of
$\Delta =\left|\begin{array}{ccc}a& b& c\\ x& y& z\\ bc& ca& ab\end{array}\right|$
is

• A. $\frac{1}{8}$
• B. $\frac{3\sqrt{3}}{4}$
• C. $\frac{3\sqrt{3}}{8}$
• D. $\frac{\sqrt{3}}{4}$

Answer 1

Answer: (C)

$\frac{3\sqrt{3}}{8}$

$\Delta =\left|\begin{array}{ccc}a& b& c\\ x& y& z\\ bc& ca& ab\end{array}\right|=\left|\begin{array}{ccc}\sin \theta & \sin (\theta +2\pi /3)& \sin (\theta +4\pi /3)\\ \cos \theta & \cos (\theta +2\pi /3)& \cos (\theta +4\pi /3)\\ \sin (\theta +2\pi /3)\sin (\theta +4\pi /3)& \sin (\theta +4\pi /3)\sin \theta & \sin \theta \sin (\theta +2\pi /3)\end{array}\right|=\frac{1}{2}\left|\begin{array}{ccc}\sin \theta & \sin (\theta +2\pi /3)& \sin (\theta +4\pi /3)\\ \cos \theta & \cos (\theta +2\pi /3)& \cos (\theta +4\pi /3)\\ \cos (-2\pi /3)-\cos (2\theta +2\pi )& \cos \left(4\pi /3\right)-\cos (2\theta +4\pi /3)& \cos (-2\pi /3)-\cos (2\theta +2\pi /3)\end{array}\right|=\frac{1}{2}\left|\begin{array}{ccc}\sin \theta & \sin (\theta +2\pi /3)& \sin (\theta +4\pi /3)\\ \cos \theta & \cos (\theta +2\pi /3)& \cos (\theta +4\pi /3)\\ \cos (-2\pi /3)& \cos \left(4\pi /3\right)& \cos (-2\pi /3)\end{array}\right|-\frac{1}{2}\left|\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ \cos (2\theta +2\pi )& \cos (2\theta +4\pi /3)& \cos (2\theta +2\pi /3)\end{array}\right|=\frac{1}{2}\left|\begin{array}{ccc}\sin \theta & \sin (\theta +2\pi /3)& \sin (\theta +4\pi /3)\\ \cos \theta & \cos (\theta +2\pi /3)& \cos (\theta +4\pi /3)\\ -\frac{1}{2}& -\frac{1}{2}& -\frac{1}{2}\end{array}\right|=-\frac{1}{4}\left|\begin{array}{ccc}\sin \theta & \sin (\theta +2\pi /3)& \sin (\theta +4\pi /3)\\ \cos \theta & \cos (\theta +2\pi /3)& \cos (\theta +4\pi /3)\\ 1& 1& 1\end{array}\right|$
Applying $C_2\to C_2-C_1,C_3\to C_3-C_1$
$\Delta =-\frac{1}{4}\left|\begin{array}{ccc}\sin \theta & \sin (\theta +2\pi /3)-\sin \theta & \sin (\theta +4\pi /3)-\sin \theta \\ \cos \theta & \cos (\theta +2\pi /3)-\cos \theta & \cos (\theta +4\pi /3)-\cos \theta \\ 1& 0& 0\end{array}\right|=-\frac{1}{4}(-\frac{3\sqrt{3}}{2})=\frac{3\sqrt{3}}{8}$