Question

Determine o argumento principal de cada um dos seguintes números complexos: A) Z= -2+2i B) Z= -2+2i√3 C) Z= √6-i√2

Answer 1

Explicação passo-a-passo:

a) $z=-2+2i$

$|z|=\sqrt{(-2)^2+2^2}$

$|z|=\sqrt{4+4}$

$|z|=\sqrt{8}$

$|z|=2\sqrt{2}$

Temos que:

$\text{sen}~\theta=\dfrac{2}{2\sqrt{2}}~\longrightarrow~\text{sen}~\theta=\dfrac{\sqrt{2}}{2}$

$\text{cos}~\theta=\dfrac{-2}{2\sqrt{2}}~\longrightarrow~\text{cos}~\theta=\dfrac{-\sqrt{2}}{2}$

Logo, $\text{arg}(z)=\dfrac{3\pi}{4}~\text{rad}$

b) $z=-2+2i\sqrt{3}$

$|z|=\sqrt{(-2)^2+(2\sqrt{3})^2}$

$|z|=\sqrt{4+12}$

$|z|=\sqrt{16}$

$|z|=4$

Temos que:

$\text{sen}~\theta=\dfrac{2\sqrt{3}}{4}~\longrightarrow~\text{sen}~\theta=\dfrac{\sqrt{3}}{2}$

$\text{cos}~\theta=\dfrac{-2}{4}~\longrightarrow~\text{cos}~\theta=\dfrac{-1}{2}$

Logo, $\text{arg}(z)=\dfrac{2\pi}{3}~\text{rad}$

c) $z=\sqrt{6}-i\sqrt{2}$

$|z|=\sqrt{(\sqrt{6})^2+(-\sqrt{2})^2}$

$|z|=\sqrt{6+2}$

$|z|=\sqrt{8}$

$|z|=2\sqrt{2}$

Temos que:

$\text{sen}~\theta=\dfrac{-\sqrt{2}}{2\sqrt{2}}~\longrightarrow~\text{sen}~\theta=\dfrac{-1}{2}$

$\text{cos}~\theta=\dfrac{\sqrt{6}}{2\sqrt{2}}~\longrightarrow~\text{cos}~\theta=\dfrac{\sqrt{3}}{2}$

Logo, $\text{arg}(z)=\dfrac{7\pi}{6}~\text{rad}$