If omega is a complex cube root of unity,then | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(A) Given that $\omega$ is a complex cube root of unity,we have $\omega^3 = 1$ and $1+\omega+\omega^2 = 0$. First,simplify the powers of $\omega$: $\o

Vedclass · Accepted Answer

$\frac{1}{\sqrt{2}}$

Vedclass · Answer

(A) Given that $\omega$ is a complex cube root of unity,we have $\omega^3 = 1$ and $1+\omega+\omega^2 = 0$. First,simplify the powers of $\omega$: $\omega^{10} = (\omega^3)^3 \cdot \omega = 1^3 \cdot \omega = \omega$ $\omega^{23} = (\omega^3)^7 \cdot \omega^2 = 1^7 \cdot \omega^2 = \omega^2$ Substituting these into the expression: $\sin \left\{(\omega + \omega^2) \pi - \frac{\pi}{4}ight\}$ Since $1 + \omega + \omega^2 = 0$,we have $\omega + \omega^2 = -1$. Thus,the expression becomes: $\sin \left\{(-1) \pi - \frac{\pi}{4}ight\} = \sin \left(-\pi - \frac{\pi}{4}ight)$ $= \sin \left(-\left(\pi + \frac{\pi}{4}ight)ight) = -\sin \left(\pi + \frac{\pi}{4}ight)$ Using the identity $\sin(\pi + 	heta) = -\sin 	heta$: $-(-\sin \frac{\pi}{4}) = \sin \frac{\pi}{4} = \frac{1}{\sqrt{2}}$

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

Similar Questions

If the circle $x^2+y^2=a^2$ intersects the hyperbola $xy=c^2$ in four points $(x_i, y_i)$,for $i=1, 2, 3, 4$,then $y_1+y_2+y_3+y_4$ equals

Which of the following is paramagnetic?

The length of the common chord of the two circles $x^2+y^2-4y=0$ and $x^2+y^2-8x-4y+11=0$ is

Identify $X$ in the following sequence of reaction:
$C_6H_5-Cl$ $\xrightarrow[CuCN]{KCN} P$ $\xrightarrow{LiAlH_4} Q$ $\xrightarrow{HNO_2} X$

An electromagnetic wave with frequency $\omega$ and wavelength $\lambda$ travels in the $+y$ direction. Its magnetic field is along the $+x$ axis. The vector equation for the associated electric field (of amplitude $E_0$) is:

Vedclass Test Series

Exam Paper Generator

Online Exam Module