Question

Plzzz ans fast..... Complete soln plzzz​

Answer 1

Answer:

your answer is

$2\pi \div 3$

Answer 2

Given :

$\sf{z=\dfrac{-4+2\sqrt3\;i}{5+\sqrt 3 \;i }}$

To find :

value of arg(z) = ?

Solution :

Converting the given complex number into standard form

$\implies \sf{z=\dfrac{-4+2\sqrt3\;i}{5+\sqrt 3 \;i }}$

$\implies \sf{z=\dfrac{-4+2\sqrt3\;i}{5+\sqrt 3 \; i}\times \dfrac{5-\sqrt3\;i}{5-\sqrt3\;i}}$

$\implies \sf{z=\dfrac{(-4)(5)+(-4)(-\sqrt3 \;i)+(2\sqrt3\;i)(5)+(2\sqrt3\;i)(-\sqrt3\;i)}{(5)^2 - (\sqrt3)^2\;i^2}}$

$\implies \sf{z=\dfrac{-20+4\sqrt3\;i+10\sqrt3\;i-6\;i^2}{25 - 3\;i^2}}$

putting i² = - 1

$\implies \sf{z=\dfrac{-20+14\sqrt3\;i-6(-1)}{25 - 3(-1)}}$

$\implies \sf{z=\dfrac{-14+14\sqrt3\;i}{28}}$

$\implies \sf{z=\dfrac{-1+\sqrt3\;i}{2}}$

$\implies \sf{z=\dfrac{-1}{2}+\dfrac{\sqrt3}{2}\;i}$

Now comparing it to z = x + i y and calculating arg (z)

we will get,

$\bullet\;\;\sf{x=\dfrac{-1}{2}\;\;and\;\;y=\dfrac{\sqrt3}{2}}$

we can see that, x < 0 and y > 0

therefore,

point (x,y) belongs to second quadrant of complex plane.

so,

$\implies\sf{arg(z)=\pi - tan\;\alpha}$

(where α is the angle as shown in attachment)

$\sf{since\;\:tan\;\alpha=\dfrac{y}{x}\;\;therefore,}$

$\implies\sf{arg(z)=\pi - tan^{-1}\left(\dfrac{y}{x}\right)}$

$\implies\sf{arg(z)=\pi - tan^{-1}\left(\dfrac{\frac{\sqrt3}{2}}{\frac{1}{2}}\right)}$

$\implies\sf{arg(z)=\pi - tan^{-1}(\sqrt3)}$

$\implies\sf{arg(z)=\pi - \dfrac{\pi}{3}}$

$\implies\boxed{\boxed{\red{\sf{arg(z)=\dfrac{2\pi}{3}}}}}$

Therefore,

Argument of complex number z is $\dfrac{2\pi}{3}$.