Question

If $\omega$ is cube root of unity then $\{({\omega }^{200}+\frac{1}{{\omega }^{200}})\pi +\frac{\pi }{4}\}$ equals

• A. $-\frac{3\pi }{4}$
• B. $\frac{\pi }{2}$
• C. $0$
• D. $\pi$

Answer 1

The correct option is A $-\frac{3\pi }{4}$

We have,
$({\omega }^{200}+\frac{1}{{\omega }^{200}})\pi +\frac{\pi }{4}=\left[{\omega }^2\right({\omega }^{198})+\frac{1}{{\omega }^2\left({\omega }^{198}\right)}]\pi +\frac{\pi }{4}$
$[\therefore {\omega }^{198}=({\omega }^3{)}^{66}=1]$
$=[{\omega }^2+\frac{\omega }{{\omega }^2.\omega }]\pi +\frac{\pi }{4}$
$=-\pi +\frac{\pi }{4}$
$=\frac{-3\pi }{4}$