Question

If $\omega$ is a complex cube root of unity, then $\sin \left\{({\omega }^{10}+{\omega }^{23})\pi -\frac{\pi }{4}\right\}$ is equal to

• A. $\frac{1}{\sqrt{2}}$
• B. $\frac{1}{2}$
• C. $1$
• D. $\frac{\sqrt{3}}{2}$

Answer 1

Answer: (A)

Since, $\omega$ is a cube root of unity
$\therefore sin\left\{\left({\omega }^{10}+{\omega }^{23}\right)\pi -\frac{\pi }{4}\right\}$
$=sin\left\{\left(\omega +{\omega }^2\right)\pi -\frac{\pi }{4}\right\}$
$=sin\left(-\pi -\frac{\pi }{4}\right)\left(\because 1+\omega +{\omega }^2=0\right)$
$=-sin\left(\pi +\frac{\pi }{4}\right)=\frac{1}{\sqrt{2}}$