Question

Seja o número complexo z= -v3/2 - 1/2i. O argumento principal do conjugado de z é

π/4
π/6
π/3
5π/6
2π/3

Answer 1

Explicação passo-a-passo:

$\sf z=-\dfrac{\sqrt{3}}{2}-\dfrac{i}{2}$

=> Conjugado

$\sf \overline{z}=-\dfrac{\sqrt{3}}{2}+\dfrac{i}{2}$

=> Argumento

$\sf \rho=\sqrt{\Big(-\dfrac{\sqrt{3}}{2}\Big)^2+\Big(\dfrac{1}{2}\Big)^2}$

$\sf \rho=\sqrt{\dfrac{3}{4}+\dfrac{1}{4}}$

$\sf \rho=\sqrt{\dfrac{4}{4}}$

$\sf \rho=\sqrt{1}$

$\sf \rho=1$

Temos que:

$\sf sen~\theta=\dfrac{\frac{1}{2}}{1}~\Rightarrow~sen~\theta=\dfrac{1}{2}$

$\sf cos~\theta=\dfrac{-\frac{\sqrt{3}}{2}}{1}~\Rightarrow~cos~\theta=-\dfrac{\sqrt{3}}{2}$

Logo, $\sf \red{arg(\overline{z})=\dfrac{5\pi}{6}~rad}$

=> 5π/6