Question

Dados os complexos a seguir, determine o módulo e o argumento dos respectivos, além de representa-los no plano complexo de Argand-Gauss.
a) Z = -1 + √3i
b) Z = 1 + i

Answer 1

Resposta:

$\textsf{Leia abaixo}$

Explicação passo a passo:

$\mathsf{z = -1 + \sqrt{3}\:i}$

$\mathsf{\rho = \sqrt{a^2 + b^2}}$

$\mathsf{\rho = \sqrt{(-1)^2 + (\sqrt{3})^2}}$

$\mathsf{\rho = \sqrt{1 + 3}}$

$\mathsf{\rho = \sqrt{4}}$

$\boxed{\boxed{\mathsf{\rho = 2}}}\leftarrow\textsf{m{\'o}dulo}$

$\mathsf{sen\:\Theta = \dfrac{b}{\rho} = \dfrac{\sqrt{3}}{2}}$

$\mathsf{cos\:\Theta = \dfrac{a}{\rho} = -\dfrac{1}{2}}$

$\boxed{\boxed{\mathsf{\Theta = 120\textdegree = \dfrac{2\pi }{3}}}}\leftarrow\textsf{argumento}$

$\mathsf{z = 1 + i}$

$\mathsf{\rho = \sqrt{a^2 + b^2}}$

$\mathsf{\rho = \sqrt{(1)^2 + (1)^2}}$

$\mathsf{\rho = \sqrt{1 + 1}}$

$\boxed{\boxed{\mathsf{\rho = \sqrt{2}}}}\leftarrow\textsf{m{\'o}dulo}$

$\mathsf{sen\:\Theta = \dfrac{b}{\rho} = \dfrac{1}{\sqrt{2}} = \dfrac{\sqrt{2}}{2}}$

$\mathsf{cos\:\Theta = \dfrac{a}{\rho} = \dfrac{1}{\sqrt{2}} = \dfrac{\sqrt{2}}{2}}$

$\boxed{\boxed{\mathsf{\Theta = 45\textdegree = \dfrac{\pi }{4}}}}\leftarrow\textsf{argumento}$