The value of left(frac{1+sin frac{2 pi}{9}+i | vedclass

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

Question

(B) Let $z = \frac{1+\sin \frac{2 \pi}{9}+i \cos \frac{2 \pi}{9}}{1+\sin \frac{2 \pi}{9}-i \cos \frac{2 \pi}{9}}$.Using $\sin 	heta = \cos(\frac{\pi}

Vedclass · Accepted Answer

$-\frac{1}{2}(\sqrt{3}-i)$

Vedclass · Answer

(B) Let $z = \frac{1+\sin \frac{2 \pi}{9}+i \cos \frac{2 \pi}{9}}{1+\sin \frac{2 \pi}{9}-i \cos \frac{2 \pi}{9}}$.Using $\sin 	heta = \cos(\frac{\pi}{2}-	heta)$ and $\cos 	heta = \sin(\frac{\pi}{2}-	heta)$,we have $\sin \frac{2 \pi}{9} = \cos \frac{5 \pi}{18}$ and $\cos \frac{2 \pi}{9} = \sin \frac{5 \pi}{18}$.So,$z = \frac{1+\cos \frac{5 \pi}{18}+i \sin \frac{5 \pi}{18}}{1+\cos \frac{5 \pi}{18}-i \sin \frac{5 \pi}{18}}$.Using $1+\cos 	heta = 2 \cos^2 \frac{	heta}{2}$ and $\sin 	heta = 2 \sin \frac{	heta}{2} \cos \frac{	heta}{2}$,we get:$z = \frac{2 \cos^2 \frac{5 \pi}{36} + 2i \sin \frac{5 \pi}{36} \cos \frac{5 \pi}{36}}{2 \cos^2 \frac{5 \pi}{36} - 2i \sin \frac{5 \pi}{36} \cos \frac{5 \pi}{36}} = \frac{\cos \frac{5 \pi}{36} + i \sin \frac{5 \pi}{36}}{\cos \frac{5 \pi}{36} - i \sin \frac{5 \pi}{36}}$.$z = \frac{e^{i 5 \pi / 36}}{e^{-i 5 \pi / 36}} = e^{i 10 \pi / 36} = e^{i 5 \pi / 18}$.Therefore,$z^3 = (e^{i 5 \pi / 18})^3 = e^{i 15 \pi / 18} = e^{i 5 \pi / 6}$.$z^3 = \cos \frac{5 \pi}{6} + i \sin \frac{5 \pi}{6} = -\frac{\sqrt{3}}{2} + i \frac{1}{2} = -\frac{1}{2}(\sqrt{3}-i)$.

The value of $\left(\frac{1+\sin \frac{2 \pi}{9}+i \cos \frac{2 \pi}{9}}{1+\sin \frac{2 \pi}{9}-i \cos \frac{2 \pi}{9}}\right)^{3}$ is

Similar Questions

$A$ force acts on a $3\, g$ particle in such a way that the position of the particle as a function of time is given by $x = 3t - 4t^2 + t^3$,where $x$ is in $meters$ and $t$ is in $seconds$. The work done during the first $4\, seconds$ is .............. $mJ$.

$\frac{{\sqrt{2} - \sin \alpha - \cos \alpha }}{{\sin \alpha - \cos \alpha }} = $

The $[Fe(CN)_6]^{3-}$ complex ion

When a spring is stretched by $2 \ cm$,it stores $100 \ J$ of energy. If it is further stretched by $2 \ cm$,the stored energy will be increased by $.... \ J$.

Which of the following is a low spin complex?

Vedclass Test Series

Exam Paper Generator

Online Exam Module