Question

Determine o argumento principal de cada um dos seguintes números complexos: A)z= - 1/2 - 1/2i B)z=2i C)z= -6 D)z= - i/4​ ​E) Z= -3 -3i√3

Answer 1

Explicação passo-a-passo:

a) $z=\dfrac{-1}{2}-\dfrac{i}{2}$

$|z|=\sqrt{\left(\dfrac{-1}{2}\right)^2+\left(\dfrac{-1}{2}\right)^2}$

$|z|=\sqrt{\dfrac{1}{4}+\dfrac{1}{4}}$

$|z|=\sqrt{\dfrac{2}{4}}$

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

Temos que:

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

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

Logo, $\text{arg}(z)=\dfrac{5\pi}{4}~\text{rad}$

b) $z=2i$

$|z|=\sqrt{0^2+2^2}$

$|z|=\sqrt{0+4}$

$|z|=\sqrt{4}$

$|z|=2$

Temos que:

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

$\text{cos}~\theta=\dfrac{0}{1}~\longrightarrow~\text{cos}~\theta=0$

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

c) $z=-6$

$|z|=\sqrt{(-6)^2+0^2}$

$|z|=\sqrt{36+0}$

$|z|=\sqrt{36}$

$|z|=6$

Temos que:

$\text{sen}~\theta=\dfrac{0}{6}~\longrightarrow~\text{sen}~\theta=0$

$\text{cos}~\theta=\dfrac{-6}{6}~\longrightarrow~\text{cos}~\theta=-1$

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

d) $z=-\dfrac{i}{4}$

$|z|=\sqrt{0^2+\left(\dfrac{-1}{4}\right)^2}$

$|z|=\sqrt{0+\dfrac{1}{16}}$

$|z|=\sqrt{\dfrac{1}{16}}$

$|z|=\dfrac{1}{4}$

Temos que:

$\text{sen}~\theta=\dfrac{-\frac{1}{4}}{\frac{1}{4}}~\longrightarrow~\text{sen}~\theta=-1$

$\text{cos}~\theta=\dfrac{0}{\frac{1}{4}}~\longrightarrow~\text{cos}~\theta=0$

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

e) $z=-3-3i\sqrt{3}$

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

$|z|=\sqrt{9+27}$

$|z|=\sqrt{36}$

$|z|=6$

Temos que:

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

$\text{cos}~\theta=\dfrac{-3}{6}~\longrightarrow~\text{cos}~\theta=\dfrac{-1}{2}$

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